diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzjiio" "b/data_all_eng_slimpj/shuffled/split2/finalzzjiio" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzjiio" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nThe minimum dominating set (MDS) problem is notoriously difficult and yet extremely important because of its numerous applications. Recall that for a graph $G=(V,E)$, set $D\\subseteq V$ is called a {\\it dominating set} if every vertex in $V\\setminus D$ has a neighbor in $D$. For general graphs $G$ even finding a $C\\log{n}$-approximation for some constant $C$ where $n$ is the order of $G$ is NP-hard \\cite{RS}. At the same time, the problem becomes much more tractable when restricted to certain classes of graphs. In particular, assumptions about the sparsity of graphs, measured in various ways, can make the MDS problem easier to approximate. The situation is not much different in the distributed setting, where on one hand, only a $O(\\log{\\Delta})$-approximation for general graphs is known \\cite{KMW}, and on the other, the problem becomes significantly easier for special cases of graphs, like for example planar graphs \\cite{LOW}.\n\nIn this paper, we shall consider an important generalization of the MDS problem, the distance-$k$ minimum dominating set problem, and we shall give fast deterministic distributed approximations in the {\\it Local} model in the case the underlying network satisfies certain sparsity conditions. \n\nThe term distance-$k$ dominating set was given by Henning et al. \\cite{henning}. For a graph $G=(V,E)$ and $k\\in \\mathbb{Z}^+$, set $D\\subseteq V$ is called a {\\it distance-$k$ dominating set} if every vertex $v\\in V$ is within distance $k$ of a vertex from $D$. In particular, $1$-distance dominating set is a dominating set. The problem has many applications in networking and other areas of computer science. Maybe the most natural applications of distance-$k$ dominating sets arise when considering the problem of allocating centers in a network that can share resources with the remaining vertices of the graph when needed \\cite{alloc}.\n\\subsection{Related Work}\nIn the distributed setting, the MDS problem has been extensively studied for many different classes of sparse networks. Lenzen et al. \\cite{LOW} gave a constant-factor distributed approximation of a minimum dominating set that runs in a constant number of rounds in planar graphs in the {\\it Local} model of computations. Using more careful analysis, Wawrzyniak \\cite{ww-ipl} improved the approximation ratio and showed that this algorithm gives in fact a 52-approximation. Amiri et al. \\cite{amiri} showed that a small modification of the algorithm from \\cite{LOW} also gives a constant-factor approximation of a minimum dominating set in graphs of the bounded genus, and even more generally in graphs with no $K_{3,t}$-minor for some constant $t$. In fact, a further generalization is given in \\cite{CHWW} where the authors give a constant-time distributed algorithm for $K_t$-minor-free graphs. In addition, using the methods from \\cite{CHW}, it is possible to improve the approximation factor in these classes of graphs at the expense of the time complexity. Specifically, it can be proved that there is a distributed algorithm which given $\\epsilon>0$ finds a $(1+\\epsilon)$-approximation of a MDS in a graph $G=(V,E)$ that is $K_t$-minor-free in $O(\\log^*{|V|})$ rounds.\nFor graphs of a constant arboricity, a much more general class of graphs, there is a randomized algorithm of Lenzen and Wattenhofer \\cite{LW-arb} that finds a constant approximation in time which is $O(\\log{|V|})$ rounds with high probability.\nIn addition, tight results are known for outerplanar graphs. Recently, using an analysis of a maximal counterexample, Bonamy, Cook, Groenland, and Wesolek \\cite{B} manged to prove very tight bounds for the approximation ratio for MDS in the case of outerplanar graphs. Specifically they showed the following two facts.\n\\begin{itemize}\n \\item There is a deterministic $5$-approximation of the MDS in outerplanar graphs.\n \\item There is no $(5-\\epsilon)$-approximation for outerplanar graphs for any $\\epsilon>0$.\n\\end{itemize}\nVery little is known about distributed algorithms for distance-$k$ dominating sets when $k>1$ as the problem becomes significantly different when $k$ increases making it impossible to adapt solutions for $k=1$. Amiri et al. \\cite{amiri-ossona} gave a constant-factor approximation for the minimum distance-$k$ dominating set problem in graphs $G=(V,E)$ of bounded expansion in $O(\\log{|V|})$ rounds (for a fixed $k$) in the more restrictive {\\it Congest$_{BC}$} model. \n\nThe main motivation for our work comes from the recent paper by Amiri and Wiederhake \\cite{AmiriWieder} who managed to provide a first constant approximation algorithm in a constant number of rounds for distance-$k$ domination in graphs of bounded expansion of high girth (i.e. graphs that are sparse and are trees locally). In fact, we will use the very same procedure from \\cite{AmiriWieder}, but give a different argument in the first part of the paper as we will examine a different class of graphs. Note that the girth assumption in \\cite{AmiriWieder} was related to a previous work on lower bounds that were established for graphs of high girths and \nis absolutely critical to their analysis. It is this assumption that we will get rid of in the current paper (at the expense of dealing with graphs with no $K_{2,t}$-minor rather than a much more general class of graphs of bounded expansion). Therefore, our paper is a step towards constant time, constant approximation algorithms for minimum distance-$k$ domination in graphs of arbitrarily small girth for which constant time approximation are known.\n\nIt is worth mentioning that the problem for $k>1$ seems to be genuinely different than the classical MDS problem, especially in the realm of sparse graphs. For example, the probabilistic algorithm from \\cite{LW-arb} is specific to the case $k=1$, and the methods from \\cite{CHW} used to ameliorate the approximation ratio when a constant approximation is furnished are again applicable only to the regular distance-$1$ domination. \n\n\\subsection{Summary of results}\nWe will work in the {\\it Local} model of computations and assume throught the paper that $k\\geq 2$. Although the first algorithm is identical to the algorithm from \\cite{AmiriWieder} which works in the {\\it Congest} model, the algorithm of Amiri et al. exploits the fact that graphs are locally trees to allow for a {\\it Congest} model implementation. Since the graphs considered in this paper can have many short cycles, the algorithm works only in the {\\it Local} model. In addition, our algorithm for the $(1+\\epsilon)$-factor approximation heavily relies on the assumptions of the {\\it Local} model. In this model, vertices correspond to computational units, and computations are synchronized. In each round, a vertex can send, receive messages from its neighbors, and can perform individual computations. In addition, we assume that vertices have unique identifiers and denote the identifier of $v$ by $ID(v)$.\n\n\nAlthough our results are stated for graphs with no $K_{2,t}$-minor, an important subclass of this class is outerplanar graphs that have no $K_{2,3}$-minor and no $K_4$-minor, that is graphs that admit a planar embedding in $\\mathbb{R}^2$ such that all vertices lie on the boundary of the outer face. It would be possible to phrase the main result of the first part of the paper in a more general language of graphs of bounded expansion that are locally $K_{2,t}$-minor-free, but this would require additional terminology and the benefit seems quite minuscule. \n\nWe will prove the following results. First, we will show that there is a distributed algorithm which finds a constant-approximation of a minimum distance-$k$ dominating set in graphs with no $K_{2,t}$-minor in a constant time which depends on $t$ and $k$ (Theorem \\ref{const-approx}). Second, we will show that a suitable modification of methods from \\cite{CHW} gives a $(1+\\epsilon)$-approximation of the $k$-MDS problem in $O(\\log^*{|V|})$ rounds in graphs $G=(V,E)$ that are $K_{2,t}$-minor-free (Theorem \\ref{main-approx-thm}).\n\nFinally we show that it is possible to find a $(1+\\epsilon)$-factor approximation that runs in $O(\\log^*{|V|})$ rounds in $K_t$-minor-free graphs $G=(V,E)$ of a constant maximum degree (Theorem \\ref{const-deg-thm}).\n\nThe rest of the paper is structured as follows. In the next section, we shall fix some terminology and prove a fact about $K_{2,t}$-minor-free graphs that will be useful in the main part of the paper. Section \\ref{sec-const} contains the analysis of the constant approximation algorithm and Section \\ref{sec-eps} discusses the $(1+\\epsilon)$-approximation. \n\n\\section{Preliminaries}\nLet $G=(V,E)$ and $H=(W,F)$ be graphs. We say that $G$ contains an $H$-minor if $H$ can be obtained from a subgraph of $G$ by a sequence of edge contractions. More formally, $H$ is a minor of $G$ if for some subgraph $G'=(V',E')$ of $G$ we can partition $V'$ into sets $V_1, \\dots, V_l$ so that each $G'[V_i]$ is connected and the graph obtained from $G'$ by contracting every $V_i$ to a vertex is isomorphic to $H$. (Note that we discard all parallel edges or loops if they appear when contracting connected subgraphs.) We will be mainly interested in graphs $G$ that are $H$-minor-free (i.e. have no $H$-minor) for $H=K_{2,t}$ where $t\\in \\mathbb{Z}^+$ is a constant. Clearly, if a graph has no $K_{2,t}$-minor then it has no $K_{2,t+1}$-minor and so assuming there is no $K_{2,t+1}$-minor is weaker than supposing no $K_{2,t}$-minor. Recall that if $G$ is planar, then $G$ has not $K_{3,3}$-minor and if it is outerplanar, then it has no $K_{2,3}$-minor. Consequently, our results apply to outerplanar graphs as $t$ can be a large but fixed positive integer.\n\nA subdivision of a graph $H$, denoted $TH$, is obtained from $H$ by replacing its edges with internally disjoint paths of length at least one.\n\nWe will follow terminology from \\cite{diestel} but will recall main concepts used throughout the paper. In particular, a path between two vertices does not contain a vertex more than once, a walk can contain repeated vertices or edges.\n\nFor two distinct vertices $u,v\\in V$, a $u,v$-path is a path which ends in $u$ and $v$. \nWe use $d_G(u,v)$ to denote the distance between $u$ and $v$ in $G$, that is, the length of a shortest $u,v$-path (allowing for $u=v$). \nFor a subset $Q\\subseteq V$, a $Q$-path is a path $P$ such that $V(P)\\cap Q$ contains only the endpoints of $P$. In particular, every vertex of $Q$ is a trivial $Q$-path. For two disjoint sets $Q_1,Q_2$, a $Q_1, Q_2$-path is a path that has one endpoint in each of the $Q_i$'s and no other vertices in $Q_1\\cup Q_2$. In the case $Q_1=\\{u\\}$, we will use $u,Q_2$-paths for $\\{u\\}, Q_2$-paths. We denote by $uPv$ a the subpath of $P$ between vertices $u$ and $v$.\n\nFor a vertex $v\\in V$, $N(v), N[v]$ denote the neighborhood of $v$ and the closed neighborhood of $v$ respectively, that is $N[v]=\\{v\\} \\cup N(v)$. In addition, for $l\\in \\mathbb{Z}^+$, let $N^l[v]$ denote the set of vertices within distance $l$ of $v$ and we set $N^l(v)=N^l[v]\\setminus \\{v\\}$.\nSimilarly, for a set of vertices $X\\subseteq V$, we let $N^l[X]=\\bigcup_{v\\in X}N^l[v]$.\n\nFinally, we will use $\\gamma_k(G)$ to denote the size of a smallest distance-$k$ dominating set in $G$.\n\nGraphs with no $K_{2,t}$-minor are sparse. To be precise, we have the following fact \\cite{CRS}.\n\\begin{lemma}\\label{crs-lem}\nLet $t\\geq 2$ and let $H$ be a graph of order at least one with no $K_{2,t}$-minor, then $|E(H)|\\leq \\frac{1}{2}(t+1)(|V(H)|-1).$\n\\end{lemma}\nAlthough we are not going to attempt to optimize the constants and do not really need the full power of the previous lemma, we will use it to obtain a bound for the number of vertices on $Q$-paths of length at most $h$.\n\n\\begin{lemma}\\label{main-lem-estimate}\nLet $t\\geq 2$, $h\\in \\mathbb{N}$ and let $H=(V,E)$ be a graph with no $K_{2,t}$-minor. Let $Q\\subseteq V$ and let $Q_h$ denote the set of vertices which are on $Q$-paths in $H$ of length at most $h$. Then $|Q_h|\\leq \\alpha_{h,t} |Q|$ for some $\\alpha_{h,t}$ that depends on $h$ and $t$ only. \n\\end{lemma}\n\\begin{proof} We will induct on $h$. If $h=0$ then $Q_0=Q$, and $\\alpha_{0,t} = 1$. For the inductive step, let $\\mathcal{P}$ be a maximal set of $Q$-paths of length at most $h$ which are internally disjoint. For $u,v\\in Q$ let $P_{u,v}$ denote $u,v$-paths in $\\mathcal{P}$ and note that $|P_{u,v}|\\leq t+1$ because otherwise, graph $H$ has a subdivision of $K_{2,t}$ (as there can be only one edge $uv$, and every other path has length at least two), and so a $K_{2,t}$-minor. Contract paths from $P_{u,v}$ to edge $uv$ and apply Lemma \\ref{crs-lem} to conclude that the number of edges in the contracted graph is less than $(t+1)|Q|\/2$. Consequently, since each path has length at most $h$, the number of vertices on paths from $\\mathcal{P}$ is less than $(t+1)^2(h+1)|Q|\/2$. Let $Q'$ denote the set of vertices on paths from $\\mathcal{P}$ (including paths of length $0$, or $1$, so notice that it does contain every vertex from set $Q$). \n\nIf $S$ is a $Q$-path of length at most $h$ which does not belong to $\\mathcal{P}$, then $S$ contains an internal vertex of a path from $\\mathcal{P}$ and so, all vertices on $S$ that have not been already counted belong to $Q'$-paths of length at most $h-1$ (see Figure~\\ref{fig1}). Thus, by induction, the number of vertices on such paths is at most $\\alpha_{h-1, t}|Q'|$. Consequently, by (\\ref{eqQ}), the number of vertices on all paths is at most \n\\begin{equation}\\label{eqQ2}\n\\begin{split}\n|Q_h| & \\leq (1+\\alpha_{h-1,t})|Q'| \\\\\n& <(1+\\alpha_{h-1,t})(t+1)^2(h+1)|Q|\/2 \\\\\n& \\leq (t+1)^2(h+1)\\alpha_{h-1,t}|Q|,\n\\end{split}\n\\end{equation}\nwhich gives us rough estimate on $\\alpha_{h,t} \\leq (t+1)^{2h}(h+1)!$, and so\n\\end{proof}\n\\begin{equation}\\label{eqQ}\n |Q'|\\leq (t+1)^2(h+1)|Q|\/2.\n\\end{equation}\n\\begin{figure}\n\\begin{center}\n \\scalebox{0.7}{\\input{picture1.pdf_t}}\n\\caption{\\normalsize Example for Lemma 2. The set $Q$ consists of the five black vertices, $t = 3$, $h = 5$. Every $Q$-path not in $P$ has its vertices covered by $P$ and $Q'$-path.}\n\\label{fig1}\n\\end{center}\n\\end{figure}\n\\section{Constant factor approximation}\\label{sec-const}\nIn this section, we will show that the simple algorithm (Algorithm 1 from \\cite{AmiriWieder}) finds a distance-$k$ dominating set of size $O(\\gamma_k(G))$ in graphs $G$ with no $K_{2,t}$-minor. \n\nWe work in the {\\it Local} model, and as a result we may assume that each connected component of $G$ has diameter at least $4k$. Indeed, if component $C$ has diameter less than $4k$, then a simple $O(k)$-round algorithm can test that $C$ is a component and will compute an optimal distance-$k$ dominating set in $C$. In particular, $t\\geq 2$.\nIn the procedure from \\cite{AmiriWieder}, every vertex $v\\in V$ selects vertex $w\\in N^k[v]$ with $|N^k[w]|$ maximum and resolves ties using $ID(w)$. \nMore formally, the algorithm can be described as follows:\\\\\n\\begin{algorithm}[H]\n \\KwData{Graph $G=(V,E)$}\n \\KwResult{Set $D$ }\n \\label{alg1}\n \\caption{{\\sc DomSet}}\n\\begin{enumerate}\n \\item For every $v\\in V$, in parallel, find $q_v=|N^k[v]|.$\n \\item For every $v\\in V$ let $w:=w_v$ be the vertex in $N^k[v]$ such that \n \\begin{itemize}\\item $q_w$ is maximum,\n \\item and subject to this, $ID(w)$ is maximum.\n \\end{itemize}\n \\item Return $D:=\\bigcup \\{w_v\\}$.\n\\end{enumerate}\n\\end{algorithm}\nThe algorithm clearly runs in $O(k)$ rounds and outputs a distance-$k$ dominating set. The only difficulty is to show that it indeed finds a distance-$k$ dominating set $D$ such that $|D|=O(\\gamma_k(G))$, which we will do in the remainder of this section.\n\nIn our analysis we may assume that $G$ is connected because the same argument can be applied to each connected component.\n\nLet $M$ be an optimal distance-$k$ dominating set in $G=(V,E)$. Create Voronoi cells (also called clusters) centered at $M = \\{u_1,...,u_m\\}$ with a vertex $v$ joining cell $C_{i}$ if $d_G(u_i,v)$ is the smallest and with ties resolved by selecting $u_i$ with the maximum ID. This gives a set of cells $\\mathcal{C}=\\{C_1, \\dots, C_m\\}$ such that each $G[C_i]$ is connected. For a cell $C\\in \\mathcal{C}$ let $v_C$ be the vertex in $C$ such that $d_G(v_C,w)\\leq k$ for every $w\\in C$, and subject to this, $|N^k_G(v_C)|$ is maximum, and subject to that, $ID(v_C)$ is maximum. If $C=C_i$, then $v_C$ might be the same as $u_i$ or it can be a different vertex but $u_i$ is always an option.\n\n\\begin{definition}\nLet $C\\in \\mathcal{C}$. A vertex $v\\in C$ is called a border vertex if $v$ has a neighbor in $V\\setminus C$.\n\\end{definition}\nLet $C^*$ denote the set of border vertices in $C$.\nWe have the following simple observation.\n\\begin{lemma}\\label{simple-lem}\nLet $G=(V,E)$ be a connected graph of diameter at least $4k-1$ and let $C$ be a Voronoi cell. If $y,y^*\\in C$ are such that $N^k[y]\\subseteq N^k[y^*]$ and $d_G(y^*,w)0$ let $U_i$ be the set of vertices on $U_{i-1}$-paths of length at most $3k$.\n\\end{definition}\nNote that we have $U_{i-1}\\subseteq U_{i}$ because of the trivial paths.\n\\begin{lemma}\n\\label{lemmaU_k}\nOutput of the {\\sc DomSet} contains only vertices of $U_k$.\n\\end{lemma}\n\\begin{proof}\nAssume towards contradiction that for some $C$ there is a vertex $y\\in C$ selected by {\\sc DomSet} such that $y \\notin U_k$. \nWe have $d_{G}(y,v_C)\\leq k$. Fix one $y,v_C$-path of the shortest length in $G$ and call it $P$. Let $l \\leq k$ denote the length of $P$. We will now select a vertex which is closest to $y$ on $P$ and belongs to the sets $U_{l-i}$ for some $i\\geq 0$. \nLet $i$ be the smallest non-negative integer such that there is a vertex $y^*$ in $U_{l-i}\\cap V(P)$ which satisfies $d_P(y,y^*)\\leq i$. We choose a vertex $y^*$ for which $d_P(y,y^*)$ is the smallest. Since $v_C\\in U_0$ and $d_P(v_C,y)=l$, we can deduce that such $y^*$ always exists and $i\\leq l \\leq k$. \n\n\\begin{figure}\n\\label{fig2}\n\\begin{center}\n \\scalebox{0.8}{\\input{picture2.pdf_t}}\n\\caption{\\normalsize Example of Lemma \\ref{lemmaU_k}. a) Path $S$ connecting $y$ and $y^*$. b) Path $R_1$ and $R_2$ having no intersection with $S$. c) Path $R_1$ having an intersection with $S$ in vertex $u$.\n\\end{center}\n\\end{figure}\n\nNote that for the case $i=0$, we have $y=y^*\\in U_l\\subseteq U_k$. Thus, consider cases when $y^*\\in U_{l-i}$ where $i\\geq 1$, and let $S=yPy^*$.\nWe will now analyze possibilities of the placement of $y^*$ in relation to $y$ (see Figure \\ref{fig2}) and prove the following three claims.\n\\begin{enumerate}\n\\item \\begin{clm}\\label{cl1}\nIf $w\\in C^*$ and $d_{G}(y,w)\\leq k$ then $d_G(y^*,w) < d_G(y,w)$.\n\\end{clm}\n\\begin{proof}\nLet $Q$ be a shortest $y,w$-path of length at most $k$ and assume towards contradiction that $Q$ does not contain $y^*$ (otherwise the condition is trivially true). If $Q\\cap S=\\{y\\}$, then $y\\in U_{(l-i)+1}\\subseteq U_k$ because the length of $Q\\cup S$ is at most $2k$ and both $w,y^*\\in U_{l-i}$. \n\nIf $Q\\cap S\\neq\\{y\\}$ then there is another vertex $z\\in Q\\cap S$ with $z\\neq y$. If $d_G(z,y)>d_G(z,y^*)$ then $d_G(y^*,w) < d_G(y,w)$. Thus assume $d_G(z,y)\\leq d_G(z,y^*)$ and $zSy^*$ does not contain any vertices of $Q$. By the choice of path $P$ we have that $d_P(z,y)k$. Thus $z\\in N^k[y^*]\\setminus N^k[y]$.\nIf for every $w\\in C^*$ $d_G(y,w)\\leq k$ then by Claim \\ref{cl1} for every $w\\in C^*$ $d_G(y^*,w)< d_G(y,w)\\leq k$ and so, by Lemma \\ref{simple-lem}, $N^k[y^*]$ is a proper subset of $N^k[y]$.\n\\end{proof}\n\\end{enumerate}\nTherefore, by Claim \\ref{cl3}, $N^k[y]$ is a proper subset of $N^k[y^*]$ and so {\\sc DomSet} chooses $y^*\\in U_k$. \\end{proof}\n\n\\begin{lemma}\\label{main-const-approx}\nFor every $t,k\\in \\mathbb{Z}^+$ there is $\\beta_{k,t}$ such that the number of vertices in $C$ selected by {\\sc DomSet} is at most $\\beta_{k,t}|C^*|.$\n\\end{lemma}\n\\begin{proof} Since output of {\\sc DomSet} is a subset of $U_k$ and in view of Lemma \\ref{main-lem-estimate}, $|U_i|\\leq \\alpha_{3k,t}|U_{i-1}|$.\n\\end{proof}\n\n\\begin{definition}\nLet $V^*=\\bigcup_{C\\in \\mathcal{C}} C^*$.\n\\end{definition}\nUsing a few relatively easy lemmas, we can conclude the analysis of {\\sc DomSet}.\n\\begin{lemma}\\label{lem-simple-W}\nLet $C\\subseteq V$ be such that $G[C]$ is connected, and such that for some vertex $v_C$, $d_{G[C]}(v_C, w)\\leq k$ for every $w\\in C$.\nLet $W\\subseteq C$ be a set which satisfies $|W|> ks^{k}$. Then $G[C]$ contains a subdivision of $K_{1,s}$ with all leaf vertices in $W$. \n\\end{lemma}\n\\begin{proof} Let $T$ denote a spanning BFS tree in $G[C]$ rooted at $v_C$ and let $W_i$ denote the set of vertices in $W$ that are at distance $i$ from $v_C$ in $T$. We have $\\sum_{i=0}^k|W_i| =|W|$, and so there is an $i$ such that $|W_i|\\geq s^{k}$. For $w\\in W_i$, let $P_w$ denote the path $wTv_C$ (path from $w$ to $v_C$ using the edges of the spanning tree $T$), and let $T'$ be the union $\\bigcup_{w\\in W_i} P_w$. Then $T'$ is a tree with leaves in $W_i$ and for every $w\\in W_i$, $d_{T'}(w,v_C)\\leq k$. If there is a vertex $z\\in T'$ such that $deg_{T'}(z)\\geq s$, then $T'$, and so $G[C]$, contains a subdivision of $K_{1,s}$ with all leaf vertices in $W$. Otherwise, the number of vertices in $W_i$ is less\nthan $s^{k}$. \\end{proof}\n\\begin{lemma}\\label{lem-two-clusters-edges}\nLet $C,C'$ be two Voronoi cells as in Lemma \\ref{lem-simple-W}. Then the number of edges between $C$ and $C'$ is at most $k^2t^{2kt^k}$.\n\\end{lemma}\n\\begin{proof} If there is a vertex $z\\in C$ which has more than $kt^{k}$ neighbors in $C'$, then, by Lemma \\ref{lem-simple-W} applied to $W=N(z)\\cap C'$, $G$ contains a subdivision of $K_{2,t}$. If there is a matching $Q$ between $C$ and $C'$ of size larger than $kt^{kt^k+1}$, then we can apply Lemma \\ref{lem-simple-W} twice. Apply it first with $s=kt^k+1$ and $W= V(Q)\\cap C'$ to get a subdivision of $K_{1,s}$, $T$, in $C'$. Then apply it again with $s=t$ and $W\\subseteq V(Q)\\cap C$ which contains vertices matched by $Q$ with the leaves of $T$ to get a subdivision of $K_{1,t}$ in $C$. Therefore the number of edges between $C$ and $C'$ is at most $k^2t^{2kt^k}.$ \\end{proof}\n\nFinally, we have with the following observation (we recall that $M$ is an optimal distance-$k$ dominating set in $G$).\n\n\\begin{lemma}\\label{bound-lemma}\n$|V^*|\\leq k^2t^{2kt^k}(t+1)|M|.$\n\\end{lemma}\n\\begin{proof} Contracting each $C\\in \\mathcal{C}$ to a vertex gives a minor of $G$ which by Lemma \\ref{crs-lem} has at most $(t+1)|M|\/2$ edges. By Lemma \\ref{lem-two-clusters-edges}, the number of edges with endpoints in two different Voronoi cells is at most $k^2t^{2kt^k}(t+1)|M|\/2$. Consequently, the number of vertices that belong to these edges is at most $k^2t^{2kt^k}(t+1)|M|$. \\end{proof}\n\nWe will now combine the previous facts to prove the main result of this section.\n\n\\begin{theorem}\\label{const-approx}\nLet $t, k\\in Z^+$. Then there exists $\\delta=\\delta(t,k)$ such that given a connected graph $G$ with no $K_{2,t}$-minor and such that $diam(G)\\geq 4k$, algorithm {\\sc DomSet} finds in $O(k)$ rounds a distance-$k$ dominating set $D$ in $G$ such that $|D|\\leq \\delta \\cdot \\gamma_k(G).$ \n\\end{theorem}\n\\begin{proof}\nLet $M$ be an optimal distance-$k$ dominating set in $G$.\nBy Lemma \\ref{main-const-approx}, the set $D$ obtained by {\\sc DomSet} satisfies $|D|\\leq \\beta_{k,t}|V^*|$ for some $\\beta_{k,t}$. In view of Lemma \\ref{bound-lemma}, $|V^*|\\leq k^2t^{2kt^k}(t+1)|M|$ and so, $|D|\\leq \\delta |M|$ for $\\delta =\\beta_{k,t} k^2t^{2kt^k}(t+1)$. \\end{proof}\n\n\\section{The $(1+\\epsilon)$-factor approximation}\\label{sec-eps}\nIn this section, we will first give a distributed $(1+\\epsilon)$-factor approximation of an optimal $k$-MDS in the case when $G$ is $K_{2,t}$-minor-free. This algorithm runs in $O(\\log^*{|V|})$ rounds in the {\\it Local} model.\nAs noted in the introduction, adapting methods from \\cite{CHW} is not automatic. However, there are some instances when this can be accomplished with relatively little effort. In the second part of this section, we give one example of such a situation when a graph $G$ is $K_t$-minor-free and satisfies $\\gamma_k(G)\\leq C \\gamma_1(G)$ for some constant $C$. For example, $K_t$-minor-free graphs of a bounded maximum degree satisfy this condition.\n\\subsection{Graphs with no $K_{2,t}$-minor}\n\nLet $H=(W,F)$ be a graph and let $P=(W_1, \\dots, W_l)$ be an ordered partition of $W$. We define $\\partial(P)$ to be the set of vertices $v\\in W$ such that $v \\in W_i$ and $N(v)\\cap W_j\\neq \\emptyset$ for some $i\\neq j$.\nWe have the following theorem which can be proved by applying methods from \\cite{CHW} and is a special case of the corresponding theorem in \\cite{CHWW1}.\n\\begin{theorem}\\label{waw-thm} Let $s\\in \\mathbb{Z}^+$ and let $\\epsilon>0$. There exists $L$ such that the following holds. Let $H=(W,E)$ be a graph on $n$ vertices with no $K_{s}$-minor. There is a distributed algorithm which finds a partition $P=(W_1, \\dots, W_l)$ such that: \n\\begin{itemize}\n \\item For every $i$, $H[W_i]$ has diameter $O(L)$ and\n \\item $|\\partial(P)|\\leq \\epsilon |W|$.\n \\end{itemize}\nThe algorithm runs in $L\\log^*{n}$ rounds.\n\\end{theorem}\nWe will use the algorithm from Theorem \\ref{waw-thm} to improve the approximation ratio of the algorithm from the previous section. Although the general idea is the same as in \\cite{CHW}, there are a few changes in the analysis that must be made to account for the fact that we are dealing with a distance-$k$ dominating set with $k\\geq 2$. In particular, the assumption that there is no $K_{2,t}$-minor (rather than a more relaxed assumption that there is no $K_{s}$-minor for $s\\geq t+2$) will play a critical role in the analysis via Lemma \\ref{two-cluster-lem}.\n \nLet $\\alpha \\in (0,1)$ be given and let $D$ be the set obtained by {\\sc DomSet}. Then, by Theorem \\ref{const-approx}, \\begin{equation}\\label{eq-C}|D|\\leq \\delta \\cdot \\gamma_k(G)\\end{equation} for some $\\delta$ that depends on $t,k$ only.\nConsider Voronoi cells with centers in vertices from $D$, that is, Voronoi cells $C_v$ for $v\\in D$ with $w$ joining $C_v$ if $d_G(v,w)$ is minimum over all $v\\in D$ and ties resolved by selecting $v$ with maximum $ID(v).$ Let $H=(W,F)$ be obtained from $G$ by contracting each $C_v$ to a vertex. \nSet $\\epsilon :=\\frac{\\alpha}{2\\delta k^2t^{2kt^k}}$ and let $P = (W_1, \\dots, W_l)$ be the partition of $W$ from Theorem \\ref{waw-thm}. We have \\begin{equation}\\label{partial1}|\\partial(P)|\\leq \\epsilon |W| =\\epsilon |D|.\\end{equation}\nPartition $P$ yields partition $P'=(V_1, \\dots, V_l)$ of $V(G)$ by setting $V_i:=\\bigcup_{u\\in W_i} C_u$.\n\\begin{lemma}\\label{two-cluster-lem}\nLet $u,w\\in V(H)$. Then the number of edges in $G$ between $C_u$ and $C_v$ satisfies $|E_{G}(C_u, C_v)|\\leq k^2 t^{2kt^k}.$\n\\end{lemma}\n\\begin{proof} By construction every vertex $w\\in C_u$ is within distance $k$ of $u$. Consequently, by Lemma \\ref{lem-two-clusters-edges}, $|E_G(C_u,C_v)|\\leq k^2t^{2kt^k}.$ \\end{proof}\n\nNow combining Lemma \\ref{two-cluster-lem} and (\\ref{partial1}) we have that $\\epsilon |D|$ Voronoi cells can have $k^2t^{2kt^k}$ edges between them, each connecting a pair of vertices. Thus\n\\begin{equation}\\label{partial2}\n|\\partial(P')|\\leq 2\\epsilon k^2t^{2kt^k}|D|.\n\\end{equation}\nInformally speaking, we use Algorithm 1 to find a seed dominating set. We define Voronoi cells and construct groups of Voronoi cells using Theorem \\ref{waw-thm}, and solve the subgraphs inside these groups optimally. Specifically, we consider the following procedure.\\\\\n\\begin{algorithm}[H]\n \\KwData{Graph $G$ with no $K_{2,t}$-minor, $k\\in Z^+$, $0<\\alpha<1$}\n \\KwResult{Set $Q$}\n\\caption{{\\sc $k$-DomSet Approximation}}\n\\begin{enumerate}\n \\item Find $D$ using {\\sc DomSet}.\n \\item Construct graph $H$ as above and set $\\epsilon:=\\frac{\\alpha}{2\\delta k^2t^{2kt^k}}$.\n \\item Use the algorithm from Theorem \\ref{waw-thm} to find $P$. Let $Q:= \\partial(P')$.\n \\item For every $i=1, \\dots, l$ find a set $Q_i$ in $G[V_i]$ such that $|Q_i|$ is the smallest and $Q_i \\cup (\\partial(P')\\cap V_i)$ distance-$k$ dominates $V_i$ in $G$.\n \\item Return $Q:=Q\\cup \\bigcup_i Q_i.$\n\\end{enumerate}\n\\end{algorithm}\nUsing the above discussion we can now prove the main theorem.\n\\begin{theorem}\\label{main-approx-thm}\nLet $\\alpha\\in (0,1)$ and let $t,k\\in Z^+$. Given a connected graph $G=(V,E)$ with no $K_{2,t}$-minor of diameter at least $4k$, procedure {\\sc $k$-DomSet Approximation} finds in $O(\\log^*{|V|})$ rounds set $Q$ such that $|Q|\\leq (1+\\alpha)\\gamma_k(G).$ \n\\end{theorem}\n\\begin{proof} The algorithm runs in $O(kL \\log^*{|V|})=O(\\log^*{|V|})$ rounds (where $L$ is the constant from Theorem \\ref{waw-thm}) because $diam(G[V_i])=O(kL)$ by Theorem \\ref{waw-thm} and the construction, and so step 4 requires $O(kL)$ rounds in the {\\it Local} model.\n\nLet $M$ be an optimal distance-$k$ dominating set in $G$, $i\\in \\{1, \\dots, l\\}$, and $M_i:=M\\cap V_i$.\nLet $V_i^O$ denote the set of vertices $w\\in V_i$ such that $d_G(w, \\partial(P')\\cap V_i)\\leq k$, and let $V_i^I:=V_i\\setminus V_i^O$. \nClearly every vertex from $V_i^O$ is distance-$k$ dominated by $\\partial(P')\\cap V_i$. In addition, if $w\\in V_i^I$, then $w$ must be distance-$k$ dominated by a vertex from $M_i$. Thus by step 4 of {\\sc $k$-DomSet Approximation}, we have $|Q_i|\\leq |M_i|$, and $|Q|\\leq |\\partial(P')|+\\sum_{i=1}^l |M_i|$, \nwhich in view of (\\ref{partial2}) gives $|Q|\\leq 2\\epsilon k^2t^{2kt^k} |D|+|M|.$\nFurther by (\\ref{eq-C}) and the definition of $\\epsilon$ we have\n$$\n|Q|\\leq (2\\epsilon k^2t^{2kt^k}\\delta+1)\\gamma_{k}(G)=(1+\\alpha)\\gamma_k(G).\n$$\n\n\\end{proof}\n\\subsection{$K_t$-minor free graphs of a constant maximum degree}\nIn this last short section, we show a simple method to find a $(1+\\epsilon)$-factor approximation of the minimum distance-$k$ dominating set if $G$ is $K_t$-minor-free and the maximum degree of $G$ satisfies $\\Delta(G)\\leq L$ for some $L$ independent of $G$. \n\nIn fact, we will give an algorithm for a somewhat more general class of $K_t$-minor-free graphs that we call $(C, \\gamma_k)$-bounded. \n\nNote that obviously, for every graph $G$ and every $i\\in Z^+$, we have $\\gamma_{i}(G)\\geq \\gamma_{i+1}(G)$.\n\nFix $k\\in Z^+$. We say that a graph $G$ is $(C,\\gamma_k)$-bounded if $\\gamma_1(G)\\leq C \\gamma_k(G)$. For example, if for some $L\\geq 3$, graph $G=(V,E)$ is such that $\\Delta(G)\\leq L$, then $\\gamma_{k}(G)> |V|\/L(L-1)^k\\geq \\gamma_1(G)\/L(L-1)^k$ and so $G$ is $(C,\\gamma_k)$-bounded with $C=L(L-1)^k$.\n\nThe algorithm is very simple and we will only outline the main idea.\nFix $C,k, t$ which are known to the algorithm and let $G$ be a graph that is $K_t$-minor-free and $(C,\\gamma_k)$-bounded. Find a constant approximation $S$ of a minimum dominating set in $G$ (i.e. distance-$k$ dominating set with $k=1$) by using the algorithm from \\cite{CHWW}. Partition $V(G)$ into $\\{S_v|v\\in S\\} $ by setting $S_v=\\{v\\}$ and adding $u$ to $S_v$ if $uv\\in E$ and $v$ has the maximum ID over vertices from $S$. Construct $H$ by contracting each $S_v$ to a vertex. Set $\\epsilon$ appropriately and use Theorem \\ref{waw-thm} to find a partition $(W_1,\\dots, W_l)$ of $V(H)$. If a vertex in $S_v\\in W_i$ has a neighbor in $W_j$ for some $j\\neq i$ in $G$ (border set), then add the center of $S_v$, namely $v$, to $D$. Finally, for each $i=1 ,\\dots, l$ find an optimal set $D_i$ in $G$ such that $D_i\\cup D$ distance-$k$ dominates $V_i:=\\bigcup_{S_u\\in W_i} S_u$ in $G$. \n\\begin{theorem}\\label{const-deg-thm}\nLet $C,k,t\\in Z^+$ and let $0<\\alpha <1$. There is a distributed algorithm which given a $K_t$-minor-free graph $G=(V,E)$ which is $(C,\\gamma_k)$-bounded, finds in $O(\\log^*{|V|})$ rounds a set $D\\subseteq V(G)$ such that $|D|\\leq (1+\\alpha)\\gamma_k(G).$ \n\\end{theorem}\n\\begin{proof} (Sketch) The argument is analogous to the proof of Theorem \\ref{main-approx-thm}. In particular, it is easy to see that the number of vertices added to $D$ from border sets $S_v$ is $O(\\epsilon \\gamma_1(G))$ which can be made smaller than $\\alpha \\gamma_k(G)$ using appropriately defined $\\epsilon$ and the fact that $G$ is $(C,\\gamma_k)$-bounded. Now assume $w \\in V_i$ and $w$ is distance-$k$ dominated by a vertex $u\\in V_j$ for some $j\\neq i$. Then a shortest $u,w$-path $P$ in $G$ contains a vertex $x$ from some border set $S_v\\in W_i$. Then however $vxPu$ has length which is less than or equal to the length of $P$ and so the center of $S_v$, which is added to $D$, distance-$k$ dominates $w$. Finally, vertices $w\\in V_i$ that are not distance-$k$ dominated by vertices from other Voronoi cells than $V_i$ are distance-$k$ dominated by $D_i$ and $|D|=\\sum_{i}|D_i|\\leq \\gamma_k(G)$. \\end{proof} \n\n\\section{Conclusions}\nWe finish with a short summary. In this paper we considered the distance-$k$ dominating set problem in a special class of graphs and showed three facts. \n\\begin{itemize}\n \\item There is a (simple) distributed (LOCAL) constant-factor approximation of an optimal distance-$k$ dominating set in graphs $G$ that have no $K_{2,t}$-minor. The algorithm runs in $O_t(k)$ rounds.\n \\item There is a distributed (LOCAL) algorithm which given $\\epsilon>0$ finds in a $K_{2,t}$-minor free graph $G$ of order $n$ a distance-$k$ dominating set of size at most $(1+\\epsilon)\\gamma_k(G).$ The algorithm runs in $O_{\\epsilon, k, t}(\\log^*{n})$ rounds. \n \\item There is a distributed (LOCAL) algorithm which given $\\epsilon>0$ finds in a $K_{t}$-minor free graph $G$ of a constant maximum degree and order $n$ and a distance-$k$ dominating set of size at most $(1+\\epsilon)\\gamma_k(G).$ The algorithm runs in $O_{\\epsilon, k, t,\\Delta(G)}(\\log^*{n})$ rounds. \n \\end{itemize}\n The proofs of the first two statements critically rely on the fact that graph $G$ is $K_{2,t}$-minor free and the third one applies only to very restrictive class of graphs, the class of $K_t$-minor-free graphs of a constant maximum degree. It would be interesting to see if similar facts can be obtained for graphs with no $K_{3,t}$-minor, or in general for graphs which are $K_t$-minor free.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nWith its first detections of gravitational waves~\\cite{Detection1,Detection2,\nDetection3,Detection4,Detection5,Detection6}, the Advanced Laser\nInterferometer Gravitational Wave Observatory (Advanced LIGO) has provided a\nfundamentally new means of observing the Universe. At the heart of\neach of these detections was a merger of compact binaries.\nIn such binaries, each compact object possesses four intrinsic parameters:\nmass, and the three components of the spin vector. Inferring all eight\nintrinsic parameters\\footnote{\\textit{Intrinsic} parameters are fundamental to\nthe underlying physics of the system. In contrast, \\textit{extrinsic}\nparameters are related to the observer (e.g.~polarization, sky location, and\ndistance) and are not considered in this paper. Some authors refer to seven\nintrinsic parameters in the full-dimensional space, which include each spin\ncomponent and the mass ratio of the system. This is because the total mass of\nthe system is simply a scaling factor; we choose to refer to eight parameters\nsince the total mass sets the time and frequency scales and therefore must be\nconsidered in PE.} from a gravitational wave observation, which analysis is\npart of the more general \\textit{parameter estimation} (PE), remains a\nchallenging and computationally expensive enterprise.\n\nThe LIGO\/Virgo Scientific Collaboration (LVC) performs PE in a Bayesian\nframework, implemented within the \\textsc{lalinference} software package that\nis part of the larger open-source software framework\n\\texttt{LALSuite}~\\cite{lalsuite}. In such a framework, we sample the\nposterior distribution by repeatedly calculating the likelihood that a\nparticular waveform matches the data and applying Bayes' theorem. Evaluating\nthe likelihood requires the rapid, sequential generation of as many as \n$\\mysim10^8$ theoretical gravitational wave predictions~\\cite{Veitch2015}.\nGenerating so many predictions via a full solution of the general relativistic\nfield equations (using the tools of numerical relativity) would be far too\ncomputationally expensive. Thus theoretical models adopted for PE generally\nemploy approximate solutions called \\emph{approximants}. State-of-the-art\napproximants adopt post-Newtonian techniques for evaluating the gravitational\nwaveform throughout most of the inspiral and ringdown, and inject information\nfrom numerical relativity calculations for the late inspiral and merger.\n\nOne such gravitational wave approximant is the Spinning Effective One\nBody--Numerical Relativity (SEOBNR) algorithm. This algorithm marries an\neffective-one body inspiral gravitational waveform approximation---with\nunknown higher-order terms fit to numerical relativity-generated gravitational\nwave predictions---to a black hole ringdown model~\\cite{Pan2014}. In\nparticular, SEOBNR starts with the Effective One Body (EOB) approach to\nnon-spinning binary modeling~\\cite{Buonanno1999} by mapping the dynamics of\nthe two-body system to the dynamics of an effective particle moving in a\ndeformed Schwarzschild metric. This work was then extended to include the\neffects of spinning, precessing binaries~\\cite{Buonanno2006}. Implemented\nnumerically, this Spinning EOB procedure adopts a precessing source frame in\nwhich precession-induced variations in amplitude and phase are minimized\nduring inspiral, and a source frame aligned with the spin of the final body\nfor matching the inspiral to the merger-ringdown~\\cite{Pan2014}. \n\nThe other widely adopted approximant within the LVC for PE is the Phenom\nseries of phenomenological waveform models. These waveform models are based on\nthe combination of accurate post-Newtonian inspiral models with late-inspiral\nand merger phenomenological fits to suites of numerical relativity\nsimulations~\\cite{Santamaria2010}. More recently, Phenom models have been \nbuilt to include the effects of precession~\\cite{Hannam2014}. In particular,\nprecession effects are included by using post-Newtonian methods to compute\nprecession angles and then ``twisting'' the underlying non-precessing\nmodel~\\cite{Hannam2014,Husa2016,Khan2016}. Phenom models are simulated\ncompletely in the frequency domain, and therefore simplify some aspects of\nanalysis. The only Phenom model designed to generate gravitational waveform\npredictions across all eight dimensions of parameter space is\nPhenomP~\\cite{Hannam2014}, which was extensively used in the first six\ndetection papers. We remark that Phenom is limited to a relatively small\nnumber of numerical relativity simulations against which it has been\ncalibrated, and it is difficult to determine the degree of systematic\nuncertainty in the model without appealing to another model for comparison. \n\nEvaluating the systematic uncertainties of the Phenom model requires\nconstruction of an independent gravitational waveform model with independent\nsystematics, and the SEOBNR family of models is a good candidate for this\ntask. The only SEOBNR model capable of generating theoretical gravitational\nwaveform predictions in all 8 intrinsic dimensions of parameter space is the\nthird version of the model, v3; the first and second versions were restricted\nto aligned-spin cases. In particular, v3 was built to accommodate arbitrary\nmass ratios, spin magnitudes, and spin orientations and has been calibrated and\nvalidated against a variety of numerical relativity\nsimulations~\\cite{Babak2017}. Thus v3 is vital for precessing compact binary\nmerger PE.\n\nUnfortunately, v3 is too currently too slow for PE. A single waveform\ngeneration across the LIGO band for, say, a black hole-neutron star system\nusing v3 can take as long as an hour on a modern desktop computer. If LIGO\nobserved a black hole-neutron star system merge, a\nsequential-gravitational-wave-generation PE would take thousands of years.\nAttempts to overcome the computational challenge of generating such\ntime-consuming gravitational waveforms include the construction of Reduced\nOrder Model (ROM) approximants. ROMs make use of multidimensional\ninterpolations between sampled points in another underlying approximant. For\nexample, a ROM based on the aligned-spin SEOBNR version 2 (v2)\napproximant~\\cite{Purrer2016,Field2014} is constructed by first generating an\nextensive collection of waveform predictions using v2 that adequately samples\nthe 4D parameter space reliably covered by v2. Then to obtain the\ngravitational waveform at any desired point in parameter space, the ROM simply\ninterpolates within the four dimensions of sampled parameter space. A ROM\nversion of v2 can generate waveforms up to $\\mysim3000$x faster than v2\ndirectly~\\cite{Purrer2016}, which explains in part why ROMs enjoy such\nwidespread use within the LVC for data analysis applications. \n\nWhile ROMs have been constructed with favorable performance characteristics in\naligned-spin situations, the cost of generating a ROM grows exponentially with\nthe dimension of the ROM (though see~\\cite{Field2012} for ideas on combating\nthis using a reduced basis approach). No strategy yet exists that can perform\nthe 8-dimensional (8D) interpolations faster than the 8D approximant; until\nsuch a strategy is invented, the most promising way to improve the performance\nof theoretical waveform generation in the full 8D parameter space will be to\noptimize the approximant directly. As a proof-of-principle, we demonstrated\nthat such an approach is capable of improving the performance of the\naligned-spin v2 approximant by a typical factor of $\\mysim280$x~\\cite{v2opt}.\nWe call our optimized v2 approximant v2\\_opt. The precessing (8D) v3\napproximant was in development as we independently prepared v2\\_opt, and thus\noriginally contained all the same inefficiencies as v2. This suggests that if\nthe full suite of optimizations we implemented in v2 were incorporated into\nv3, v3-based PE timescales might drop by two orders of magnitude at least.\n\nThis paper documents our incorporation of applicable v2 optimizations into v3,\nas well as our implementation of innovative new optimization ideas, which\ntogether act to speed up v3 by $\\mysim340$x. Optimization strategies are\nsummarized in Sec.~\\ref{OptStrats}. Section~\\ref{Results} presents code\nvalidation tests that demonstrate roundoff-level agreement between v3 and our\nlatest optimized version of v3, designated v3\\_Opt, along with benchmarks\nproviding an overview of performance gains across parameter space in v3\\_Opt.\n{\\bf For convenience, Table~\\ref{approx_conv} defines all SEOBNR approximants\nreferenced in this paper.}\n\n \\begin{table}\n \\begin{adjustbox}{max width=\\textwidth}\n \\centering\n \\begin{tabular}{|c|c|l|}\n \\hline\n \\textbf{Base} & \\textbf{Approx.} & \\textbf{Description} \\\\\n \\textbf{Approximant} & \\textbf{Name} & \\\\\n \\hline\\hline\n SEOBNRv2 & v2 & Initial SEOBNRv2 implementation\\tablefootnote{\\label{2cce415}As of publication, the most recent updates to v2\/v2\\_opt are found on commit ID \\texttt{2cce415} in the \\texttt{LALSuite} \\texttt{master} branch.}; see \\cite{Taracchini2014}. \\\\\\cline{2-3}\n (spin-aligned) & v2\\_opt & Optimized v2\\cref{2cce415}; see \\cite{v2opt}. \\\\\n \\hline\n SEOBNRv3 & v3\\_preopt & Initial SEOBNRv3 implementation\\tablefootnote{To generate a waveform with v3\\_preopt, download \\texttt{LALSuite} from the archived repository page \\url{https:\/\/git.ligo.org\/lscsoft\/lalsuite-archive\/tree\/14414694698a2f18c9135445003cade805ad2096} and use approximant tag SEOBNRv3.}; see \\cite{Pan2014}. \\\\\\cline{2-3}\n (precessing) & v3 & Partially optimized v3\\_preopt with bug fixes\\tablefootnote{\\label{19e95b4}As of publication, the most recent updates to v3 and v3\\_opt are found on commit ID \\texttt{19e95b4} in the \\texttt{LALSuite} \\texttt{master} branch.}. \\\\\\cline{2-3}\n & v3\\_pert & v3 with machine-$\\epsilon$ mass perturbation \\cref{19e95b4}. \\\\\\cline{2-3}\n & v3\\_opt &v3 optimized similarly to v2\\_opt \\cref{19e95b4}. \\\\\\cline{2-3}\n & v3\\_Opt & v3\\_opt with new optimization strategies\\tablefootnote{\\label{Opt}Approximants v3\\_opt and v3\\_opt\\_rk4 were updated to run v3\\_Opt and v3\\_Opt\\_rk4, respectively, on commit ID \\texttt{1391f77} in the \\texttt{LALSuite master} branch.}. \\\\\\cline{2-3} \n & v3\\_Opt\\_rk4 & v3\\_Opt implementing RK4 rather than RK8\\cref{Opt}. \\\\\n \\hline\n \\end{tabular}\n \\caption{Approximant naming conventions. These conventions apply\n throughout this paper.}\\label{approx_conv}\n \\end{adjustbox}\n \\end{table}\n\n\\section{SEOBNRv3\\_opt: Optimizations migrated from v2\\_opt}\n\\label{OptStrats}\nOptimizations to v3 were performed in two phases. In the first phase,\ndescribed in Sec.~\\ref{v2opts}, we migrated to v3 all applicable optimizations\ndeveloped during the preparation of v2\\_opt. Sections~\\ref{v3opts:gad} and~\\ref{v3opts:do} detail the second phase of optimization, outlining new strategies\nincorporated into v3\\_Opt.\n\n\\subsection{Migrated Optimizations}\n\\label{v2opts}\nHere we summarize the optimizations to v2 which were migrated to v3 and thus\nimplemented in v3\\_opt.\n\n\\begin{itemize}\n \\item \\emph{Switching compilers}. Switching from the \\texttt{GNU Compiler\n Collection} (\\texttt{gcc})~\\cite{gcc} \\texttt{C} compiler to the\n \\texttt{Intel Compiler Suite} (\\texttt{icc})~\\cite{icc} \\texttt{C}\n compiler improves performance by roughly a factor of 2x. It is well-known\n that the \\texttt{icc} compiler often produces more efficient executables\n than the \\texttt{gcc} compiler\\footnote{We used the following compiler\n flags when compiling with \\texttt{icc}: \\texttt{-xHost}, \\texttt{-O2}, and\n \\texttt{-fno-strict-aliasing}.}.\n\n \\item \\emph{Minimize transcendental function evaluations}. The EOB\n Hamiltonian equations of motion were hand-optimized by minimizing calls to\n some expensive transcendental functions such as \\texttt{exp()},\n \\texttt{log()}, and \\texttt{pow()}.\n\n \\item \\emph{Replacing finite difference with exact derivatives}. When\n solving the EOB Hamiltonian equations of motion, v3 computes partial\n derivatives of the Hamiltonian using finite difference approximations. We\n replaced these with exact, Mathematica-generated expressions for the\n derivatives, using Mathematica's code generation facilities---which\n includes common subexpression elimination (CSE)---to generate the\n \\texttt{C} code~\\cite{Mathematica}. Although this alone acts to\n significantly speed up v3, in this work we further optimize these\n Mathematica-generated derivatives.\n\n \\item \\emph{Increasing the order of the ODE solver}. v3 solves the EOB\n Hamiltonian equations of motion via a Runge-Kutta fourth order (RK4) ODE\n solver. After implementing exact derivatives, we noticed that the number\n of RK4 steps needed dropped significantly---presumably due to the\n effective removal of high numerical noise intrinsic to finite-difference\n derivatives. We then found that adopting a Runge-Kutta eighth order (RK8)\n ODE solver resulted in 2x larger timesteps, so an even larger speed-up was\n observed.\n\n \\item \\emph{Reducing orbital angular velocity calculations}. The orbital\n angular velocity $\\omega$ was calculated for each ($\\ell,m$) mode (as\n defined in \\cite{Pan2014}) inside the ODE solver. As $\\omega$ exhibits no\n dependence on $\\ell$ or $m$, this expensive recalculation was unnecessary\n and needs only be performed once.\n\\end{itemize}\n\nFor more details about these optimizations see our v2 optimization\npaper~\\cite{v2opt}. \n\n\\subsection{Guided Automatic Differentiation: A more efficient way of\n generating symbolic derivatives of the Hamiltonian}\\label{v3opts:gad}\n\nAfter migrating the v2 optimizations described in Sec.~\\ref{v2opts} to v3,\nprofiling analyses indicated that approximately $75\\%$ of v3\\_opt's total\nruntime was spent computing the v3 Hamiltonian~\\cite{Pan2014} and its partial\nderivatives with respect to the twelve degrees of freedom (consisting of three\nspatial degrees $\\{x,y,z\\}$, three momentum degrees $\\{p_x,p_y,p_z\\}$, and\nthree spin degrees for each of the two binary components $i\\in\\{1,2\\}$:\n$\\{s_{i}^{x},s_{i}^{y},s_{i}^{z}\\}$).\n\nIn v3, the ODE solver computes these partial derivatives by direct evaluations\nof the Hamiltonian itself via finite difference\ntechniques~\\cite{Taracchini2014}. In v3\\_opt, these numerical derivatives were\nreplaced with Mathematica-generated exact derivatives. Although these exact\nderivatives unlock significant performance gains, the Mathematica-generated\n\\texttt{C} code was neither particularly human-readable (comprising thousands\nof lines of code output by Mathematica's CSE routines) nor particularly\nwell-optimized (common patterns were still visible and recomputed in the\n\\texttt{C} code). Attempts to gain performance through consolidation of all\nderivatives---as was possible in our optimizations of v2---proved beyond\nMathematica's capabilities when differentiating the v3 Hamiltonian on our\nhigh-performance workstations. Therefore, \\texttt{C} codes for all twelve\nexact derivatives needed to be output separately, resulting in a significant\nnumber of unnecessary re-computations.\n\nWe present here our new strategy for computing partial derivatives of the\nHamiltonian, called \\textit{guided automatic differentiation} (GAD), which\nresults in a significant reduction in computational cost while ensuring the\nresulting code is highly human-readable. GAD is based on forward accumulation\nautomatic differentiation, with the advantage of the subexpressions being\nchosen by hand to minimize the overall number of floating point operations.\n\nThe following describes the process of computing a partial derivative of the\nv3 Hamiltonian $H$ with respect to an \\textit{arbitrary} independent variable\n$x_{1}$ using GAD. We may write $H$ in the following form, where $I$ is a set\nof input quantities:\n\\begin{align*}\nv_{1} &= f_{1}(I) \\\\\nv_{2} &= f_{2}(v_1,I) \\\\\nv_{3} &= f_{3}(v_1,v_2,I) \\\\\n &\\ \\ \\vdots \\\\\nH &= f_{N}(v_{1},v_{2},v_{3},...,I).\n\\end{align*}\nHere $f_\\ell$ is the $\\ell$th function of the set of input quantities $I$ and\npreviously computed subexpressions\n$\\left\\{v_{0},v_{1},\\hdots,v_{\\ell-1}\\right\\}$. Although $N\\approx 200$ for\nv3, \\textit{for the sake of example} we suppose $N=3$, $I=\\{x_{1},x_{2}\\}$,\nand\n\\begin{align*}\nv_1 &= \\sqrt{x_{1}} + ax_{1} \\\\\nv_2 &= \\sqrt{x_{2}} + ax_{2} \\\\\nv_3 &= (v_1 + v_2)\/(v_1v_2) \\\\\nH &= v_3^2.\n\\end{align*}\nWe demonstrate GAD by taking a partial derivative of $H$ with respect to the\nindependent input variable $x_{1}$. Table \\ref{doGAD} displays the evolution\nof this example code under the GAD scheme, which proceeds as follows:\n\\begin{enumerate}[1.]\n \\item We begin with a list of variables and subexpression computations for\n the Hamiltonian, and translate this \\texttt{C} code into the Mathematica\n language.\n \\item We parameterize the terms of each subexpression according to their\n dependence on $x_{1}$.\n \\item Mathematica computes derivatives of each subexpression.\n \\item We convert the Mathematica output into \\texttt{C} code.\n \\item We replace each occurrence of $x_{1}'$ with 1 and remove terms equal to 0.\n\\end{enumerate}\n\nThe resulting \\texttt{C} code is short, optimized, and human readable.\nFurthermore, any terms that are common to all derivative expressions are\ncomputed and saved before computing the partial derivatives, further reducing\nthe computational cost.\n\n\\begin{table}\n \\centering\n \\begin{adjustbox}{max width=\\textwidth}\n \\begin{tabular}{|l|l|}\n \\hline\n \\textbf{Step 1: Convert \\texttt{C} to Mathematica.} & \\textbf{Step 2: Parameterize subexpressions.} \\\\\n \\texttt{v1 = Sqrt[x1] + a*x1} & \\texttt{v1 = Sqrt[x1[x]] + a*x1[x]} \\\\\n \\texttt{v2 = Sqrt[x2] + a*x2} & \\texttt{v2 = Sqrt[x2] + a*x2} \\\\\n \\texttt{v3 = (v1 + v2)\/(v1*v2)} & \\texttt{v3 = (v1[x] + v2[x])\/(v1[x]*v2[x])} \\\\\n \\texttt{H~ = v3*v3} & \\texttt{H = v3[x]*v3[x]} \\\\\n \\hline\n \\multicolumn{2}{|l|}{\\textbf{Step 3: Utilize Mathematica to compute derivatives.}} \\\\\n \\multicolumn{2}{|l|}{\\texttt{v1' = x1'[x]\/(2*Sqrt[x1[x]]) + a*x1'[x]}} \\\\\n \\multicolumn{2}{|l|}{\\texttt{v2' = 0}} \\\\\n \\multicolumn{2}{|l|}{\\texttt{v3' = (v1[x]*v2[x]*(v1'[x] + v2'[x])-((v1[x] + v2[x])*(v1'[x]*v2[x]}} \\\\\n \\multicolumn{2}{|l|}{\\texttt{~~~~~ + v1[x]*v2'[x]))\/(v1[x]*v1[x]*v2[x]*v2[x])}} \\\\\n \\multicolumn{2}{|l|}{\\texttt{H'~ = 2*v3'[x]*v3[x]}} \\\\\n \\hline\n \\multicolumn{2}{|l|}{\\textbf{Step 4: Convert Mathematica to \\texttt{C}; prime notation becomes a protected \\texttt{prm} suffix.}} \\\\\n \\multicolumn{2}{|l|}{\\texttt{v1prm = x1prm\/(2*sqrt(xi)) + a*x1prm}} \\\\\n \\multicolumn{2}{|l|}{\\texttt{v2prm = 0}} \\\\\n \\multicolumn{2}{|l|}{\\texttt{v3prm = (v1*v2*(v1prm + v2prm)-((v1 + v2)*(v1prm*v2 + v1*v2prm))\/(v1*v1*v2*v2)}} \\\\\n \\multicolumn{2}{|l|}{\\texttt{Hprm~ = 2*v3prm*v3}} \\\\\n \\hline\n \\multicolumn{2}{|l|}{\\textbf{Step 5: Replace \\texttt{x1prm} with 1 and remove terms equaling 0.}} \\\\\n \\multicolumn{2}{|l|}{\\texttt{v1prm = 1.\/(2*sqrt(x1)) + a}} \\\\\n \\multicolumn{2}{|l|}{\\texttt{v3prm = (v1*v2*v1prm-(v1 + v2)*v1prm*v2)\/(v1*v1*v2*v2)}} \\\\\n \\multicolumn{2}{|l|}{\\texttt{Hprm~ = 2*v3prm*v3}} \\\\\n \\hline\n \\end{tabular}\n \\end{adjustbox}\n \\caption{Step-by-step GAD code evolution.}\\label{doGAD}\n\\end{table}\n\nSince each $v_{\\ell}$ is merely an intermediate of $H$, there is significant\nfreedom in our choice of the set of subexpressions\n$\\mathcal{V}\\equiv\\left\\{v_{1},v_{2},\\hdots,v_{N-1}\\right\\}$. Our choices do,\nhowever, have a direct effect on the number of calculations necessary to\ncompute $\\partial_{x_{1}}H$, which we measure in floating point operations\n(FLOPs\\footnote{Not to be confused with ``FLOPs per second'' (FLOPS).}.) Our\ngoal in GAD, therefore, is to choose $\\mathcal{V}$ to minimize the number of\nFLOPs needed to compute $\\partial_{x_{1}}H$.\n\nIn general, the largest contributor to FLOPs is the product rule. If there are\n$M$ different subexpressions multiplied together in a given expression,\ncomputing the derivative will require $\\mathcal{O}(M^2)$ FLOPs. If we\ntherefore choose $\\mathcal{V}$ such that each $v_{\\ell}$ contains no more than\ntwo previous subexpressions multiplied together, we should minimize the\noverall cost. We expect a significant reduction in FLOPs to correspond to a\nsignificant reduction in the time to generate a waveform.\n\nWe estimated the number of FLOPs based on benchmarks provided\nin~\\cite{FLOPsite} for CPUs corresponding to the CPU family in our\nworkstations (Intel Core i7-6700) and generated Table \\ref{countratio}. We\nemphasize that the values listed in Table~\\ref{countratio} are truly rough\nestimates, used only to provide us general direction as we seek an optimal\n$\\mathcal{V}$.\n\n\\begin{table}\n \\centering\n \\begin{adjustbox}{max width=\\textwidth}\n \\begin{tabular}{|c|c|c|c|c|c|c|c|}\n \\hline\n $a+b$ & $a-b$ & $a*b$ & $a=b$ & $a\/b$ & ${\\tt sqrt}(a)$ & ${\\tt log}(a)$ & ${\\tt pow}(a,b)$ \\\\ \\cline{3-5}\n \\hline\n 1 & 1 & 1 & 1 & 3 & 3 & 24 & 24 \\\\ \\cline{3-5}\n \\hline\n \\end{tabular}\n \\end{adjustbox}\n \\caption{Relative FLOPs count of the mathematical\n operations.}\\label{countratio}\n\\end{table}\n\nTable~\\ref{FLOPs} compares the number of Hamiltonian derivative FLOPs under\nGAD to the number in the exact derivatives (EDs) generated by Mathematica's\nCSE code generation algorithm. In principle, the difference in FLOPs between\nED and GAD schemes may be used to predict the waveform generation speedup\nfactor. A direct comparison from Table~\\ref{FLOPs} indicates a 3.6x reduction\nin FLOPs when using GAD. For a double neutron star coalescence, Hamiltonian\nderivative computations constitute about 80\\% of waveform generation time.\nThis suggests a speedup factor of 2.3x. Waveform generation times for three\nscenarios comparing ED and GAD implemented in v3\\_opt are shown in\nTable~\\ref{previewbench}, and demonstrates a speedup factor of about 1.7x. We\nemphasize again that counting FLOPs using the relative values of\nTable~\\ref{countratio} only provides a rough estimate of the reduction in\nFLOPs, and the compiler itself rearranges arithmetic expressions to minimize\nFLOPs as well so the gap between our estimated and observed speed-ups is not\nsurprising.\n\n \\begin{table}\n \\centering\n \\begin{adjustbox}{max width=\\textwidth}\n \\begin{tabular}{|c|ccc|c|}\n \\hline\n Derivative scheme & Space derivative & Momentum derivative & Spin derivative & Total \\\\\n & (FLOPs) & (FLOPs) & (FLOPs) & (FLOPs) \\\\\n \\hline\n ED & 3 x 5073 & 3 x 2319 & 6 x 4333 & 48174 \\\\\n GAD & 3 x 1418 & 3 x 527 & 6 x 1264 & 13419 \\\\\n \\hline\n \\end{tabular}\n \\end{adjustbox}\n \\caption{Number of FLOPs using ED versus GAD methods.}\\label{FLOPs}\n \\end{table}\n\n \\begin{table}\n \\centering\n \\begin{adjustbox}{max width=\\textwidth}\n \\begin{tabular}{|c|c|c|c|}\n \\hline\n & v3\\_opt (s) & v3\\_opt (s) \\\\\n \\textbf{Parameters} & ED & GAD \\\\\n \\hline\n \\textbf{Neutron Star Binary} & 36.75 & 20.49 \\\\\n $1.4M_{\\odot} + 1.4M_{\\odot}$, $s_{1}^{y} = 0.05$ & & x(\\textbf{1.79}) \\\\ \n \\hline\n \\textbf{Black Hole + Neutron Star} & 8.07 & 4.69 \\\\\n $10M_{\\odot} + 1.4 M_{\\odot}$, $s_{1}^{y} = 0.4$ & & x(\\textbf{1.72}) \\\\\n \\hline\n \\textbf{Black Hole Binary} (GW150914-like) & & \\\\\n $36M_{\\odot} + 29M_{\\odot}$ & 0.64 & 0.38 \\\\\n $s_{1}^{y}=0.05,\\ s_{1}^{z}=0.5,\\ s_{2}^{y}=-0.01,\\ s_{2}^{z}=-0.2$ & & x(\\textbf{1.68}) \\\\\n \\hline\n \\end{tabular}\n \\end{adjustbox}\n \\caption{Benchmark comparison of ED to GAD strategies. In each\n scenario, we adopt a 10Hz start frequency.}\\label{previewbench}\n \\end{table}\n\n\\subsection{Dense Output: A more efficient way of interpolating\n sparsely-sampled data}\n\\label{v3opts:do}\n\nAn RK4 ODE solver with adaptive timestep control solves the EOB Hamiltonian\nequations of motion in v3; thus solutions are unevenly sampled in time.\nSubsequent analyses require mapping these data into the frequency domain via\nthe fast Fourier transform (FFT), which expects evenly-sampled data. Rather\nthan restricting the integration timestep, v3 uses cubic splines to\ninterpolate the Hamiltonian solutions after RK4 runs to completion. During\noptimization of v2, the GSL cubic spline interpolation routine was optimized\nand gave significant performance gains. During optimization of v3, it was\ndiscovered that third-order Hermite interpolation made v3\\_Opt more faithful\nto v3 (see Section \\ref{Results}). Hermite interpolation requires only two\nfunction values and the derivatives at those values, which are available at\neach step of RK8. Thus we may interpolate the sparsely-sampled data to the\ndesired evenly-sampled data ``on the fly'' during integration. Such an\nintegration routine is called a \\textit{dense output} method~\\cite{NR}. In\nparticular, suppose the RK8 integrator computes the solution $y(t_{0})$ and\n$y(t_{1})$ at times $t_{0}$ and $t_{1}$ with timestep $h$ and derivative\nvalues $y'_{0}$ $y'_{1}$. Then for any $0\\le\\theta\\le1$, we have\n \\begin{equation*}\n \\begin{adjustbox}{max width=\\textwidth}\n $y(t_{0} + \\theta h) = (1-\\theta)y(t_{0}) + \\theta y(t_{1}) + \\theta(\\theta-1)\\left[(1-2\\theta)(y(t_{1}) - y(t_{0}) + (\\theta-1)hy'(t_{0}) + \\theta hy'(t_{1})\\right]$.\n \\end{adjustbox}\n \\end{equation*}\nAs this cubic Hermite interpolation routine uses both the solution data and\nderivative values at each point, it therefore requires only the output of the\nRK8 integration and no further data storage or function evaluations.\n\n\\section{Results}\\label{Results}\nIn Sec.~\\ref{subsec:faith} we establish that v3\\_Opt produces waveforms which\nagree with v3 at the level of roundoff error. Section \\ref{subsec:bench} then\ndescribes the process of measuring speedup and demonstrates the speedup factor\nachieved.\n\n\\subsection{Determining Faithfulness}\\label{subsec:faith}\nGiven two waveforms $h_1(t)$ and $h_2(t)$ (in the time domain), we determine\nif $h_{1}(t)$ is \\emph{faithful} to $h_{2}(t)$ using the LVC's open-source\n\\textsc{PyCBC} software~\\cite{pycbc-software,Canton2014,Usman2015}. This\ncomputation depends on the following definitions, which we write in the same\nform as~\\cite{v4}. The \\emph{noise-weighted overlap} between $h_{1}$ and\n$h_{2}$ is defined as\n \\[\n (h_1 | h_2) \\equiv 4{\\rm Re} \\int_{f_{l}}^{f_{h}} \\frac{ \\tilde{h}_{1}(f)\\tilde{h}^{*}_{2}(f) }{ S_{n}(f) } {\\rm d}f\n \\]\nwith $\\tilde{h}_{i}(f)$ denoting the Fourier transform of the waveform\n$h_{i}(t)$, $h_{i}^{*}$ denoting the complex conjugate of $h_{i}$, $f_{l}$ and\n$f_{h}$ denoting the endpoints of the range of frequencies of interest, and\n$S_n(f)$ denoting the one-sided power spectral density (PSD) of the LIGO\ndetector noise. We chose $f_{l}=20$ Hz and $S_{n}(f)$ to be Advanced LIGO's \ndesign zero-detuned high-power noise PSD~\\cite{noisecurve}. For each waveform,\n$f_{h}$ is the Nyquist critical frequency~\\cite{NR}. We then define the\n\\emph{faithfulness} between $h_{1}$ and $h_{2}$ to be the overlap between the\nnormalized waveforms maximized over relative time and phase shifts:\n \\[\n \\braket{h_{1} | h_{2}} \\equiv \\max_{\\phi_c, t_c} \\frac{ (h_1(\\phi_c,t_c) | h_2) }{ \\sqrt{(h_1 | h_1)(h_2 | h_2)} }.\n \\]\nHere $t_{c}$ and $\\phi_c$ denote the coalescence time and phase, respectively.\nNote that normalization forces $\\braket{h_{1} | h_{2}} \\in [0,1]$, with\n$\\braket{h_{1} | h_{2}} = 1$ indicating complete overlap (and therefore a\nperfectly faithful waveform) while $\\braket{h_{1} | h_{2}} = 0$ indicates no\noverlap (an \\emph{unfaithful} waveform\\footnote{Another common measure in\nfaithfulness tests is \\textit{mismatch}, defined as\n$1-\\braket{h_{1} | h_{2}}$.}). For each faithfulness test conducted, we\ngenerate a waveform with two different approximants and the same set of input\nparameters.\n\nWe ran 100,000 faithfulness tests for each set of waveform approximants we\nwished to compare. The input parameters for each test are randomly chosen by\n\\textsc{PyCBC} with bounds as outlined in Table \\ref{faithparams}; these\nbounds are chosen to capture the relevant parameter space for v3. Note that\neach of the spin parameters $s_{i}^{x}, s_{i}^{y}, s_{i}^{z}$ are chosen\nrandomly in $(-1,1)$ with the constraint\n\\[\n \\sqrt{ (s_{i}^{x})^{2} + (s_{i}^{y})^{2} + (s_{i}^{z})^{2} } \\le 0.99, \\ \\ i \\in \\{ 1,2\\}.\n\\]\n\n\\begin{table}\n \\begin{tabular}{|c|c|}\n \\hline\n Mass of Object 1 (solar masses) & $m_{1}\\in[1,100]$ \\\\\n \\hline\n Mass of Object 2 (solar masses) & $m_{2}\\in[1,100]$ \\\\\n \\hline\n Spin magnitude of Object 1 (dimensionless) & $\\lvert a_{1}\\rvert\\in[0,0.99]$ \\\\\n \\hline\n Spin magnitude of Object 2 (dimensionless) & $\\lvert a_{2}\\rvert\\in[0,0.99]$ \\\\\n \\hline\n Binary total mass (solar masses) & $m_{\\rm total}\\in[4,100]$ \\\\\n \\hline\n Starting orbital frequency (Hz) & $f=19$ \\\\\n \\hline\n \\end{tabular}\n\\caption{Ranges of values for random input parameters in our faithfulness\n tests.}\\label{faithparams}\n\\end{table}\n\nThe specific faithfulness runs we conducted were organized as follows. The\napproximant v3\\_pert is identical to v3 except $m_{1}$ is replaced with\n$m_{1}\\left( 1 + 10^{-16} \\right)$; such a perturbation should result\nin waveforms that are nearly identical and provides a measure of how sensitive\nv3 is to roundoff error. Thus faithfulness tests comparing v3 and v3\\_pert\nprovide a ``control'' against which we compare the faithfulness of v3\\_Opt to\nv3. As another point of comparison, we also test v3 (which is RK4-based)\nagainst the RK4-based v3\\_Opt\\_rk4. For each approximant comparison we\ncompare the effect of increasingly stricter ODE solver tolerance. By default,\nv3 sets the ODE solver's absolute and relative error tolerances to\n$\\varepsilon\\equiv1\\times10^{-8}$; we compare faithfulness at tolerances of\n $\\varepsilon$, $\\varepsilon\\times10^{-1}$, $\\varepsilon\\times10^{-2}$,\n$\\varepsilon\\times10^{-3}$, and $2\\varepsilon\\times10^{-4}$. Finally, we also\nconsider the effect of compiler choice on faithfulness and so conduct\nfaithfulness runs using both \\texttt{gcc} and \\texttt{icc}. Table\n\\ref{faithresults} summarizes the faithfulness tests conducted and their\nresults; the rightmost column displays the counting error $\\sqrt{n}$ for the\nnumber of waveforms $n$ with $\\braket{\\cdot|\\cdot} < 0.999$.\n\n\\begin{table}\n\\begin{adjustbox}{max width=\\textwidth}\n\\begin{tabularx}{1.3\\textwidth}{c c c *{5}{Y} c}\n\\toprule\n & & \\bf{ODE} & \\multicolumn{5}{c}{\\textbf{Number of waveforms with faithfulness}} & \\textbf{Counting} \\\\\n\\cmidrule(lr){4-8}\n\\bf{Comparison} & \\bf{Compiler} & \\bf{tolerance} & $<0.8$ & $<0.9$ & $<0.95$ & $<0.99$ & $<0.999$ & \\textbf{Error} \\\\\n\\midrule\nv3 vs.~v3\\_pert & \\texttt{gcc} & $\\varepsilon$ & 1 & 5 & 13 & 104 & 399 & $\\pm20$ \\\\\n(per $10^{5}$ for $10^{6}$ tests) & \\texttt{icc} & $\\varepsilon$ & 1.0 & 4.2 & 11.5 & 109.0 & 398.2 & $\\pm6.3$ \\\\\n\\hline\nv3 vs.~v3\\_Opt & \\texttt{gcc} & $\\varepsilon$ & 5 & 28 & 136 & 1184 & 5466 & $\\pm74$ \\\\\n & \\texttt{icc} & $\\varepsilon$ & 5 & 28 & 135 & 1174 & 5509 & $\\pm74$ \\\\\n & \\texttt{icc} & $\\varepsilon\\times10^{-1}$ & 2 & 16 & 44 & 327 & 1510 & $\\pm39$ \\\\\n & \\texttt{icc} & $\\varepsilon\\times10^{-2}$ & 0 & 2 & 12 & 143 & 727 & $\\pm27$ \\\\\n & \\texttt{icc} & $\\varepsilon\\times10^{-3}$ & 1 & 3 & 8 & 80 & 511 & $\\pm23$ \\\\\n & \\texttt{icc} & $2\\varepsilon\\times10^{-4}$& 1 & 1 & 2 & 60 & 457 & $\\pm21$ \\\\\n\\hline\nv3 vs.~v3\\_Opt\\_rk4 & \\texttt{gcc} & $\\varepsilon$ & 1 & 9 & 35 & 427 & 1529 & $\\pm39$ \\\\\n & \\texttt{icc} & $\\varepsilon$ & 0 & 9 & 35 & 420 & 1510 & $\\pm39$ \\\\\n & \\texttt{icc} & $\\varepsilon\\times10^{-1}$ & 1 & 7 & 24 & 223 & 926 & $\\pm30$ \\\\\n & \\texttt{icc} & $\\varepsilon\\times10^{-2}$ & 0 & 0 & 8 & 114 & 585 & $\\pm24$ \\\\\n & \\texttt{icc} & $\\varepsilon\\times10^{-3}$ & 1 & 3 & 8 & 77 & 483 & $\\pm22$ \\\\\n & \\texttt{icc} & $2\\varepsilon\\times10^{-4}$& 1 & 2 & 3 & 52 & 423 & $\\pm21$ \\\\\n\\bottomrule\n\\end{tabularx}\n\\end{adjustbox}\n\\caption{Summary of \\texttt{PyCBC} faithfulness results. Here\n$\\varepsilon=1\\times10^{-8}$ and each row reports the results of a run of\n\\textbf{100,000 faithfulness tests}. \\texttt{icc} refers to Intel compiler\nversion 15.5.223, while \\texttt{gcc} refers to GNU compiler version\n4.9.}\\label{faithresults}\n\\end{table}\n\nWe comment on the values in Table \\ref{faithresults}. For a couple of\nparameters for which $\\braket{\\cdot | \\cdot} < 0.8$ when comparing v3 to\nv3\\_Opt compiled with \\texttt{gcc}, one author back-traced a significant\ndifference between v3 and v3\\_Opt to the ODE stopping condition or the time of\nmaximum amplitude being clearly wrong in v3 but not v3\\_Opt. In particular,\nthere are some algorithms within v3 that are fundamentally non-robust, and\nv3\\_Opt inherits most of these functions. The RK8 integration of v3\\_Opt\nshould be just as accurate as the RK4 integration of v3\\_Opt\\_rk4 when the\ntolerances are equal, but the output from RK8 should be much sparser (by more\nthan a factor of 2) than RK4. Since we observe worse faithfulness with v3\\_Opt\nthan v3\\_Opt\\_rk4, we conclude that most of the truncation error stems from\nthe interpolation of the sparsely-sampled ODE solution to a uniform timestep.\n\nMost importantly, notice that as we make the ODE solver's tolerance $\\varepsilon$\nstricter (resulting in smaller errors and more finely sampled output data from\nthe ODE solver), the faithfulness between v3 and v3\\_Opt improves to the level\nof agreement between v3 and v3\\_pert. Thus we conclude that v3\\_opt generates\nroundoff-level agreement in the limit of $\\varepsilon\\to0$ with errors dominated\nby interpolation otherwise.\n\n\\subsection{Performance Benchmarks}\\label{subsec:bench}\nIn order to capture the full effect of our optimizations to v3, we compared\nwaveform generation times of v3\\_Opt with waveform generation times of\nv3\\_preopt. In particular, v3\\_preopt lacks by-hand optimizations of the EOB\nHamiltonian implemented in the development of v2\\_opt; thus unnecessary\ncomputations of transcendental functions \\texttt{pow()}, \\texttt{log()}, and\n\\texttt{exp()} remain therein. All reported benchmarks were completed on a\nsingle core of a modern desktop computer with an Intel Core i7-7700 CPU and 64\nGB RAM.\n\nTo highlight cases of interest, Table \\ref{benchmarkresults} summarizes\nbenchmarks of v3\\_Opt and v3\\_Opt\\_rk4 in comparison to v3\\_preopt for a\nhandful of scenarios of interest to LIGO. The speedup factors are also\nincluded, with speedup simply defined to be the ratio of time to generate a\nwaveform with v3\\_preopt to the time to generate the same waveform with\nv3\\_Opt or v3\\_Opt\\_rk4.\n\n \\begin{table}\n \\centering\n \\begin{adjustbox}{max width=\\textwidth}\n \\begin{tabular}{|c|c|c|c|c|}\n \\hline\n Physical scenario & v3\\_preopt & v3\\_Opt\\_rk4 & v3\\_Opt & v3\\_Opt \\\\\n & \\texttt{gcc}, (s) & \\texttt{gcc}, (s) & \\texttt{gcc}, (s) & \\texttt{icc}, (s) \\\\\n \\hline\n \\hline\n DNS, $s_{2}^{y}=0.05$ & \\multirow{2}{*}{8618.60} & 98.51 & 42.85 & 21.22 \\\\ \\cline{3-5}\n $1.3M_{\\odot}+1.3M_{\\odot}$ & & x(\\textbf{87.49}) & x(\\textbf{201.1}) & x(\\textbf{406.2}) \\\\\n \\hline\n BHNS, $s_{\\rm NS}^{y}=0.05$ & \\multirow{2}{*}{2760.77} & 20.75 & 8.84 & 4.37 \\\\ \\cline{3-5}\n $10M_{\\odot}+1.3M_{\\odot}$ & & x(\\textbf{133.0}) & x(\\textbf{312}) & x(\\textbf{632}) \\\\\n \\hline\n BHB, $s_{2}^{y}=0.05$ & \\multirow{2}{*}{127.71} & 1.70 & 0.90 & 0.46 \\\\ \\cline{3-5}\n $16M_{\\odot}+16M_{\\odot}$ & & x(\\textbf{75.1}) & x(\\textbf{140}) & x(\\textbf{280}) \\\\\n \\hline\n BHB, $s_{1}^{y}=s_{2}^{y}=0.9$ & \\multirow{2}{*}{168.13} & 1.75 & 0.91 & 0.46 \\\\ \\cline{3-5}\n $16M_{\\odot}+16M_{\\odot}$ & & x(\\textbf{96.1}) & x(\\textbf{180}) & x(\\textbf{370}) \\\\\n \\hline\n BHB, $s_{1}^{y}=s_{2}^{z}=0.9$ & \\multirow{2}{*}{235.53} & 3.48 & 1.55 & 0.76 \\\\ \\cline{3-5}\n $10M_{\\odot}+10M_{\\odot}$ & & x(\\textbf{67.7}) & x(\\textbf{152}) & x(\\textbf{310}) \\\\\n \\hline\n BHB, GW150914-like & \\multirow{4}{*}{31.48} & \\multirow{2}{*}{0.75} & \\multirow{2}{*}{0.51} & \\multirow{2}{*}{0.27} \\\\\n $36M_{\\odot}+29M_{\\odot}$ & & & & \\\\ \\cline{3-5}\n $s_{1}^{y}=0.05$, $s_{1}^{z} = 0.5$ & & \\multirow{2}{*}{x(\\textbf{42})} & \\multirow{2}{*}{x(\\textbf{60})} & \\multirow{2}{*}{x(\\textbf{120})} \\\\\n $s_{2}^{y}=-0.01$, $s_{2}^{z}=-0.2$ & & & & \\\\\n \\hline\n \\end{tabular}\n \\end{adjustbox}\n \\caption{Benchmarks and speedups of v3\\_Opt and v3\\_Opt\\_rk4 compared to\n v3.}\\label{benchmarkresults}\n \\end{table}\n\nTo demonstrate that the advertised speedup factors of Table\n\\ref{benchmarkresults} apply across the parameter space of binaries of\ninterest to the LVC, we completed four benchmark surveys. The first two\nconcern binary black hole systems, one with varying masses and the other with\nvarying spins. The third survey considers mixed binaries (one black hole and\none neutron star), and the fourth binary neutron stars. The parameters tested\nin each run are included in Table \\ref{parambench}. The results of these\nsurveys are plotted in Figure \\ref{speed} and summarized in Table\n\\ref{benchmarkresults}.\n\n \\begin{table}\n \\begin{adjustbox}{max width=\\textwidth}\n \\begin{tabular}{lcccc}\n \\hline\n Ranges & $m_{1}$ ($M_{\\odot}$) & $q$ (dimensionless) & $a_{1}$ (dimensionless) & $a_{2}$ (dimensionless) \\\\\n \\hline\\hline\n BHB$_{\\rm M}$ & $[16.7,100.3]$ & $[1,10]$ & 0.0500001 & 0 \\\\\n BHB$_{\\rm S}$ & 10 & 1 & $[-0.95,0.95]$ & $[-0.95,0.95]$ \\\\\n BHNS & $[7,100]$ & $\\frac{M}{1.4}$ & $[-0.95,0.95]$ & 0 \\\\\n DNS & $[1.2,2.3]$ & $\\frac{M}{m\\in[1.2,2.3]}$ & 0.0500001 & 0 \\\\\n \\hline\n \\end{tabular}\n \\end{adjustbox}\n \\caption{Surveyed parameters: each survey tested 400 parameter\n combinations, with 20 evenly-spaced values taken in each range\n indicated. Here BHB$_{\\rm M}$ indicates the black hole binary mass\n survey, BHB$_{\\rm S}$ the black hole binary spin survey, BHNS the black\n hole neutron star survey, and DNS the double neutron star survey. We\n define $q \\equiv \\frac{m_{1}}{m_{2}}$, the ratio of the mass of object 1\n to the mass of object 2. The dimensionless Kerr spins of each object are\n denoted $a_{1}$ and $a_{2}$, respectively. Each waveform generation\n started with a frequency of 10 Hz used a sample rate of 16,384\n Hz.}\\label{parambench}\n \\end{table}\n\n\\begin{figure}\n \\centering\n \\begin{subfigure}[t]{0.5\\textwidth}\n \\centering\n \\includegraphics[height=5cm]{speedup_plot.eps}\n \\end{subfigure}%\n~\n \\begin{subfigure}[t]{0.5\\textwidth}\n \\centering\n \\includegraphics[height=5cm]{time_plot.eps}\n \\end{subfigure}\n \\caption{\\textbf{Performance benchmarks:} \\textbf{Left panel:} plots speedup\n factor versus number of wavecycles in the binary inspiral. Measuring the\n number of wavecycles allows us to compactly display the results of the\n benchmark tests without explicit reference to mass or spin. \\textbf{Right\n panel:} plots the number of wavecycles versus the time taken to output\n the waveform. Note that the speedup factor in the left panel is simply the\n ratio of the curves in the right panel.}\\label{speed}\n\\end{figure}\n\nWe would like to measure an average speedup based on the four surveys. As\nin~\\cite{v2opt}, we define an overall speedup factor as a waveform\ncycle-weighted average\n \\[\n \\mathcal{S} = \\frac{\\sum_{i}\\mathcal{S}_{i}N_{i}}{\\sum_{i} N_{i}}\n \\]\nwhere $\\mathcal{S}_{i}$ is the speedup factor for generating the\n$i^{\\text{th}}$ waveform and $N_{i}$ is the number of\nwavecycles in the $i^{\\text{th}}$ waveform. We found $\\mathcal{S}\\sim340$.\nThis reduces the time necessary for a black hole binary PE run from\n$\\mathord{\\sim}$100 years (with v3\\_preopt) to $\\mathord{\\sim}$8 months (with v3\\_Opt). We\nexpect lower mass PE runs will be possible on similar timescales with\nadditional optimizations.\n\n\\section{Conclusions and Future Work}\n\\label{Conclusions}\nAnticipating the potential detection by Advanced LIGO of significantly\nprecessing compact binaries, we have optimized v3 to make costly\nprecessing-waveform-approximant-based data analysis applications like PE\npossible in a reasonable amount of time. If an efficient 8D ROM is found, such\noptimizations will make the construction of this ROM faster. After migrating\nv2\/v4 optimizations to v3, we further optimized partial derivatives of the\nHamiltonian using a GAD scheme. This resulted in waveforms that are faithful\nto v3, as evidenced by faithfulness increasing to 1 as ODE tolerance\ndecreases. We achieved an average overall speedup of $\\mysim340$x, ranging\nfrom $\\mysim120$x for GW150914-like black hole binaries to $\\mysim630$x for\nblack hole-neutron star binaries. We expect that further optimizations are\npossible, achieving an additional speedup factor of at least $\\mathord{\\sim}$3x.\nFuture work will focus on transforming Cartesian coordinates to spherical\ncoordinates to lower sampling rates even more during ODE solving and\nintegration.\n\n\\ack\nWe thank O.~Birnholtz, N.~Johnson-McDaniel, R.~Sturani, A.~Taracchini, and\nC.~Haster for helpful comments and discussion during a review of the v3\\_opt\ncode. A.~Taracchini is especially thanked for introducing us to faithfulness\ntesting via \\texttt{PyCBC}, as is S.~Teukolsky for suggesting dense output\nmethods. We also thank R.~Haas for his work optimizing v3 during its\ndevelopment, and I.~Ruchlin for numerous helpful discussions. This work was\nsupported primarily by NSF LIGO Research Support Grant PHY-1607405. Early work\non this project was supported by NSF EPSCoR Grant 1458952 and NASA Grant\n13-ATP13-0077. We are grateful for the computational resources provided by the\nLeonard E.~Parker Center for Gravitation, Cosmology and Astrophysics at\nUniversity of Wisconsin-Milwaukee (NSF Grant 0923409), LIGO Livingston\nObservatory, LIGO Hanford Observatory, and the LIGO-Caltech Computing Cluster.\n\n\\section*{References}\n\n\\sloppy\n\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThe classical theories as well as most of the recent theoretical studies and experiments regarding the efficiency of plasmonic resonances \nreported in the literature are concerned with metal nanoparticles where the exterior domain is lossless, \nsee \\textit{e.g.}\\\/, \\cite{Bohren+Huffman1983,Link+etal2000,Maier2007,Tretyakov2014,Miller+etal2016,Tzarouchis+etal2016}. \nAs \\textit{e.g.}\\\/, in \\cite{Miller+etal2016}, a variational approach is employed in connection with a generalized optical theorem for scattering, absorption and extinction \n\\cite{Lytle+etal2005}, to obtain an upper bound on the absorption that can be achieved inside a scatterer with arbitrary geometry and with a given volume and material property.\nFurthermore, an upper bound on the dipole absorption of an electrically small particle with arbitrary geometry and structural parameters is given in \\cite{Tretyakov2014}.\nThe bound \\cite[Eq.~(16) on p.~937]{Tretyakov2014} is based on the \noptical theorem \\cite{Bohren+Huffman1983} and obtained by optimizing absorption \ndirectly in terms of the complex valued polarizability of the particle. \nBoth results in \\cite{Tretyakov2014} and \\cite{Miller+etal2016} are valid for a lossless surrounding medium only. \n\nHowever, there are many application areas of plasmonics where the exterior losses must be taken into account.\nOne potential application in medicine is the localized electrophoretic heating of a bio-targeted and electrically \ncharged gold nanoparticle (GNP) suspension as a radiotherapeutic hyperthermia based method to treat cancer, \n\\textit{cf.}\\\/, \\cite{Corr+etal2012,Sassaroli+etal2012,Collins+etal2014,Nordebo+etal2017a,Dalarsson+etal2017a}\nor with the related plasmonic photothermal therapy as proposed in \\cite{Huang+etal2008}.\nOther potential application areas include plasmon waveguides, aperture arrays, extraordinary transmission, superlenses, artificial magnetism, negative refractive index,\nand surface-enhanced biological sensing with molecular monolayer spectroscopy, etc., see \\textit{e.g.}\\\/, \\cite{Maier2007}. \n\nThere has been a number of investigations devoted to the scattering, absorption and extinction of small particles embedded \nin a lossy medium, see \\textit{e.g.}\\\/, \\cite{Mundy+etal1974,Chylek1977,Bohren+Gilra1979,Lebedev+etal1999,Sudiarta+Chylek2001,Durant+etal2007a}. \nThis topic has even been subjected to some controversy due to the difficulties to define a general theory\nencompassing the notion of a cross section when the surrounding medium is lossy, see \\textit{e.g.}\\\/, \\cite{Bohren+Gilra1979,Lebedev+etal1999,Sudiarta+Chylek2001,Durant+etal2007a}. \nIn contrast, for a lossless surrounding medium the absorption cross section can be defined\nfrom the power flowing into a conceptual sphere surrounding the particle at an arbitrary radius, and which enables the derivation of an optical theorem valid for arbitrary geometries \\cite[pp.~71 and 140]{Bohren+Huffman1983}. For a lossy medium this theory is no longer valid, which is due to the fact\nthat the absorption in the surrounding medium depends on the geometry of the scatterer.\nHence, the optical theorems for lossy media are typically given only for spheres. As \\textit{e.g.}\\\/, in \\cite{Nordebo+etal2018f} is given \nnew fundamental upper bounds on the multipole absorption and scattering of a rotationally invariant\nsphere embedded in a lossy surrounding medium and which are derived based on the corresponding generalized optical theorem \nas given in \\textit{e.g.}\\\/, \\cite[Eq.~(7) on p.~1276]{Sudiarta+Chylek2001}.\n\nWe are concerned here with the optimal absorption of an electrically small spherical object \nembedded in a lossy surrounding medium when the near-field distribution can be found using the quasistatic approximation. \nFor simplicity, we are considering only the important special case of non-magnetic, dielectric materials which are common in plasmonic applications. \nMagnetic materials can be treated similarly.\nWe are also considering only a single electric dipole resonance, and we are\ndiscarding any possibilities of having an unbounded absorption due to multiple mode super resolution effects, etc.,\nsee \\textit{e.g.}\\\/, \\cite{Valagiannopoulos+etal2015,Maslovski+etal2016,Valagiannopoulos+Tretyakov2016}.\nA quasistatic theory has been developed in \\cite{Nordebo+etal2017a,Dalarsson+etal2017a} giving the optimal plasmonic dipole resonance of small dielectric \nellipsoids in terms of an optimal conjugate match with respect to the background loss. \nHowever, an important limitation of this theory is that it does not give the correct physical answers when the background becomes lossless or has very small losses.\nThis limitation is obvious since the quasistatic optimal absorption becomes unbounded in the case when the external losses vanish\n\\textit{cf.}\\\/, \\textit{e.g.}\\\/, \\cite[Eq.~(33) on p.~5]{Dalarsson+etal2017a}, \nin contrast to the well known fact that the absorption cross section of a small dipole scatterer in a lossless exterior medium is bounded \nby $3\\lambda^2\/8\\pi$, \\textit{cf.}\\\/, \\cite[Eq.~(16) on p.~937]{Tretyakov2014}.\n\nThis limitation, which may even appear as a contradiction, can be understood simply by realizing that the quasistatic model disregards radiation damping. \nMoreover, the plasmonic singularity of the sphere ($\\epsilon=-2$ in vacuum) \nexists only in the sense of a limit as the size of the particle approaches zero \\cite{Tzarouchis+etal2016}.\nIn fact, it turns out that the dipole resonance of a small sphere has a very subtle limiting behavior as the electrical size approaches zero,\nand as \\textit{e.g.}\\\/, in \\cite{Tzarouchis+etal2016} Pad\\'{e} approximants are used to reveal new scattering aspects of small spherical particles.\nIn this paper, an asymptotic analysis based on the Mie theory is employed to study the limiting behavior of the quasistatic optimal resonance \\cite{Nordebo+etal2017a},\nas the electrical size of the sphere as well as the external losses tend to zero. The limitation of the quasistatic theory is then finally assessed by providing explicit\nasymptotic formulas for the validity of the quasistatic model of the optimal resonance. \nWe explicitly find the validity region of the quasistatic model, which is determined by the scattering loss factor.\n\nThe rest of the paper is organized as follows. In Section \\ref{sect:paradox} is given a description of\nthe quasistatic optimal plasmonic resonance of an ellipsoid embedded in a lossy background medium. \nIn Section \\ref{sect:Miesolution} we develop a detailed full electrodynamic analysis with explicit asymptotic results\nconcerning the special case with a sphere.\nNumerical examples are given in Section \\ref{sect:numexamples} and the vector spherical \nwaves are defined in Appendix \\ref{sect:spherical}.\n\n\\section{The quasistatic optimal plasmonic resonance of an ellipsoid in a lossy background medium}\\label{sect:paradox}\n\\subsection{Notation and conventions}\nThe following notations and conventions are used in this paper.\nClassical electrodynamics is considered where the electric and magnetic field intensities $\\bm{E}$ and $\\bm{H}$\nare given in SI-units \\cite{Jackson1999}. \nThe time convention for time harmonic fields (phasors) is given by $\\eu^{-\\iu\\omega t}$\nwhere $\\omega$ is the angular frequency and $t$ the time. \nConsequently, the relative permittivity $\\epsilon$ of a passive isotropic dielectric material has positive imaginary part.\nLet $\\mu_0$, $\\epsilon_0$, $\\eta_0$ and $\\mrm{c}_0$ denote the permeability, the permittivity, the wave impedance and\nthe speed of light in vacuum, respectively, and where $\\eta_0=\\sqrt{\\mu_0\/\\epsilon_0}$ and $\\mrm{c}_0=1\/\\sqrt{\\mu_0\\epsilon_0}$.\nThe wavenumber of vacuum is given by $k_0=\\omega\\sqrt{\\mu_0\\epsilon_0}$.\nThe wavenumber of a homogeneous and isotropic medium with relative permeability $\\mu$ and permittivity $\\epsilon$ is given by $k=k_0\\sqrt{\\mu\\epsilon}$\nand the wavelength $\\lambda$ is defined by $k\\lambda=2\\pi$.\nThe wave impedance of the same medium is given by $\\eta_0\\eta$ where $\\eta=\\sqrt{\\mu\/\\epsilon}$ is the relative wave impedance.\nIn the following, we will consider only non-magnetic, homogeneous and isotropic dielectric or conducting materials, and hence $\\mu=1$ from now on. \nThe spherical coordinates are denoted by $(r,\\theta,\\phi)$, the corresponding unit vectors $(\\hat{\\bm{r}},\\hat{\\bm{\\theta}},\\hat{\\bm{\\phi}})$,\nand the radius vector $\\bm{r}=r\\hat{\\bm{r}}$.\nFinally, the real and the imaginary parts, and the complex conjugate of a complex number $\\zeta$ are denoted \n$\\Re\\left\\{\\zeta\\right\\}$, $\\Im\\left\\{\\zeta\\right\\}$ and $\\zeta^*$, respectively.\n\n\n\\subsection{Optimization under the quasistatic approximation}\nThe maximal absorption of a small dielectric ellipsoid under the quasistatic approximation can readily be calculated as follows, see also \\cite{Nordebo+etal2017a,Dalarsson+etal2017a}.\nConsider a small, homogeneous and isotropic dielectric ellipsoid with relative permittivity $\\epsilon$ which is embedded in \na lossy dielectric background medium with relative permittivity $\\epsilon_\\mrm{b}$. In the quasistatic approximation, \nthe polarizability of the ellipsoid with a uniform excitation $\\bm{E}_\\mrm{i}=E_0\\hat{\\bm{e}}$ along one of its axes is given by the expression\n\\begin{equation}\\label{eq:alphaellipsoiddef}\n\\alpha=V\\frac{\\epsilon-\\epsilon_\\mrm{b}}{\\epsilon_\\mrm{b}+L(\\epsilon-\\epsilon_\\mrm{b})},\n\\end{equation}\nwhere $01\/3$ for a prolate spheroid, the sphere and an oblate spheroid, respectively.\nThe dipole moment of the small ellipsoid is given by $\\bm{p}=\\epsilon_0\\epsilon_\\mrm{b}\\alpha\\bm{E}_\\mrm{i}=\\int_\\Omega\\epsilon_0(\\epsilon-\\epsilon_\\mrm{b})\\bm{E}\\mrm{d}v$\nwhere $\\Omega$ denotes the ellipsoidal domain, and since the resulting internal field $\\bm{E}$ of the ellipsoid is a constant vector parallel to \n$\\bm{E}_\\mrm{i}$ \\cite{Bohren+Huffman1983}, it follows readily that\n\\begin{equation}\n\\bm{E}=\\frac{\\epsilon_\\mrm{b}\\alpha}{V(\\epsilon-\\epsilon_\\mrm{b})}\\bm{E}_\\mrm{i}=\\frac{\\epsilon_\\mrm{b}}{L\\left(\\epsilon+\\epsilon_\\mrm{b}\\frac{(1-L)}{L} \\right)}\\bm{E}_\\mrm{i}.\n\\end{equation}\nThe power absorbed in the ellipsoid can now be calculated from Poynting's theorem as\n\\begin{multline}\\label{eq:Pabsellipsoid1}\nP_\\mrm{abs}=\\frac{\\omega\\epsilon_0}{2}\\Im\\{\\epsilon\\}\\int_\\Omega\\left|\\bm{E} \\right|^2\\mrm{d}v= \\\\\n\\frac{\\omega\\epsilon_0}{2}\\frac{\\left|\\epsilon_\\mrm{b}\\right|^2}{L^2}\\frac{\\Im\\{\\epsilon\\}}{\\left|\\epsilon+\\epsilon_\\mrm{b}\\frac{(1-L)}{L} \\right|^2}\\left| E_0\\right|^2V,\n\\end{multline}\nor \n\\begin{equation}\\label{eq:Pabsellipsoid2}\nP_\\mrm{abs}=\n\\frac{\\omega\\epsilon_0}{2}\\frac{\\left|\\epsilon_\\mrm{b}\\right|^2\\Im\\{\\epsilon\\}}{\\left|\\epsilon_\\mrm{b}+L(\\epsilon-\\epsilon_\\mrm{b})\\right|^2}\\left| E_0\\right|^2V.\n\\end{equation}\nAs \\textit{e.g.}\\\/, , for the sphere ($L=1\/3$), the absorption cross section $C_\\mrm{abs}$ is obtained by normalizing with the power intensity \nof the incoming plane wave, yielding\n\\begin{equation}\\label{eq:Cabssphere1}\nC_\\mrm{abs}=\\frac{P_\\mrm{abs}}{I_\\mrm{i}}\n=12\\pi k_0 a^3\\frac{\\left|\\epsilon_\\mrm{b}\\right|^2}{\\Re\\{\\sqrt{\\epsilon_\\mrm{b}}\\}}\\frac{\\Im\\{\\epsilon\\}}{\\left|\\epsilon+2\\epsilon_\\mrm{b} \\right|^2},\n\\end{equation}\nwhere $I_\\mrm{i}=\\frac{1}{2}\\Re\\{E_0H_0^*\\}=\\left| E_0\\right|^2\\Re\\{\\sqrt{\\epsilon_\\mrm{b}}\\}\/2\\eta_0$ and where $H_0=E_0\/\\eta$.\n\nWhen the background medium as well as the shape of the ellipsoid are fixed, the expression \\eqref{eq:Pabsellipsoid1} can be maximized as follows.\nLet $\\epsilon_\\mrm{opt}$ denote a fixed complex-valued constant with $\\Im\\{\\epsilon_\\mrm{opt}\\}>0$ and consider the real-valued function \n\\begin{equation}\\label{eq:fofepsilondef}\nf(\\epsilon)=\\frac{\\Im\\{\\epsilon\\}}{\\left|\\epsilon-\\epsilon_\\mrm{opt}^* \\right|^2},\n\\end{equation}\nwhere $\\epsilon$ is a complex-valued variable with $\\Im\\{\\epsilon\\}>0$.\nIt can be shown that the function $f(\\epsilon)$ has a local maximum at $\\epsilon_\\mrm{opt}$ \\textit{cf.}\\\/, \\cite[Sect.~2.5, Eqs.~(15) through (17)]{Nordebo+etal2017a}.\nHence, for the ellipsoid the maximizer of $P_\\mrm{abs}$ is given by\n\\begin{equation}\\label{eq:epsilonodef}\n\\epsilon_\\mrm{opt}=-\\epsilon_\\mrm{b}^*\\frac{(1-L)}{L},\n\\end{equation}\nand in particular for the sphere $\\epsilon_\\mrm{opt}=-2\\epsilon_\\mrm{b}^*$, \n\\textit{cf.}\\\/, \\cite{Nordebo+etal2017a,Dalarsson+etal2017a}.\nThe corresponding maximal absorption is given by\n\\begin{equation}\\label{eq:Pabsellipsoid1o}\nP_\\mrm{abs}^\\mrm{qs,opt}(\\epsilon_\\mrm{b},L)=\\frac{\\omega\\epsilon_0}{2}\\frac{1}{4L(1-L)}\\frac{\\left|\\epsilon_\\mrm{b}\\right|^2}{\\Im\\{\\epsilon_\\mrm{b}\\}}\\left| E_0\\right|^2V,\n\\end{equation}\nand for the sphere we obtain the optimal absorption cross section\n\\begin{equation}\\label{eq:Cabssphere1o}\nC_\\mrm{abs}^\\mrm{qs,opt}\n=\\frac{3\\pi}{2}k_0 a^3\\frac{\\left|\\epsilon_\\mrm{b}\\right|^2}{\\Re\\{\\sqrt{\\epsilon_\\mrm{b}}\\}}\\frac{1}{\\Im\\{\\epsilon_\\mrm{b}\\}}.\n\\end{equation}\nThe solution \\eqref{eq:epsilonodef} is referred to as an optimal conjugate match and can be\ninterpreted in terms of an optimal plasmonic resonance for the ellipsoid \\cite{Nordebo+etal2017a,Dalarsson+etal2017a}.\nNote that \\eqref{eq:Pabsellipsoid1o} is unbounded in the cases when the exterior domain becomes lossless and $\\Im\\{\\epsilon_\\mrm{b}\\}\\rightarrow 0$,\nas well as when the ellipsoid collapses and $L\\rightarrow 0$ or $L\\rightarrow 1$.\n\nWhen both media parameters $\\epsilon$ and $\\epsilon_\\mrm{b}$ are fixed, the expression \\eqref{eq:Pabsellipsoid2} can be maximized\nwith respect to the shape parameter $L$. By straightforward differentiation it is found that the maximizing shape parameter $L_\\mrm{opt}$ is given by\n\\begin{equation}\\label{eq:Lodef}\nL_\\mrm{opt}=\\Re\\left\\{\\frac{\\epsilon_\\mrm{b}}{\\epsilon_\\mrm{b}-\\epsilon} \\right\\},\n\\end{equation}\nand the corresponding maximal absorption is given by\n\\begin{equation}\\label{eq:Pabsellipsoid2o}\nP_\\mrm{abs}^\\mrm{qs,opt}(\\epsilon_\\mrm{b},\\epsilon)\n=\\frac{\\omega\\epsilon_0}{2}\\frac{\\left|\\epsilon_\\mrm{b}\\right|^2\\Im\\{\\epsilon\\}\\left|\\epsilon-\\epsilon_\\mrm{b}\\right|^2}{\\left(\\Im\\{\\epsilon_\\mrm{b}^*\\epsilon\\} \\right)^2}\n\\left| E_0\\right|^2V.\n\\end{equation}\nIn the limiting case when the exterior region becomes lossless and $\\Im\\{\\epsilon_\\mrm{b}\\}\\rightarrow 0$, the expression \\eqref{eq:Pabsellipsoid2o} simplifies to\n\\begin{equation}\\label{eq:Pabsellipsoid3o}\nP_\\mrm{abs}^\\mrm{qs,opt}(\\epsilon_\\mrm{b},\\epsilon)\n=\\frac{\\omega\\epsilon_0}{2}\\frac{\\left|\\epsilon-\\epsilon_\\mrm{b}\\right|^2}{\\Im\\{\\epsilon\\}}\n\\left| E_0\\right|^2V,\n\\end{equation}\nand which agrees with the upper bound given in \\cite[Eq.~(32b) on p.~3345 and Eq.~(41) on p.~3349]{Miller+etal2016} under the same quasistatic assumption (incident field is uniform).\n\nWhat is important to note at this point is the unboundedness of the quasistatic maximal absorption \\eqref{eq:Pabsellipsoid1o} as the external loss factor\n$\\Im\\{\\epsilon_\\mrm{b}\\}$ tends to zero. Obviously, this is in contradiction to the well known bound on the \nabsorption of an arbitrary electric dipole scatterer in a lossless background medium given by\n\\begin{equation}\\label{eq:Cabsdip}\nC_\\mrm{abs}^\\mrm{dip}=\\frac{3\\pi}{2}\\frac{1}{k_\\mrm{b}^2},\n\\end{equation}\nwhere $k_\\mrm{b}=k_0\\sqrt{\\epsilon_\\mrm{b}}$, \\textit{cf.}\\\/, \\cite[Eq.~(16) on p.~937]{Tretyakov2014}. As we will see later, this apparent contradiction is due simply\nto the fact that the quasistatic approximation does not take the scattering loss into account. In particular, in the next section we will see\nthat the fully dynamical model for the normalized absorption cross section area of a small dielectric spherical dipole has a very subtle limiting behavior, \nand is in fact unbounded when $\\epsilon=-2\\epsilon_\\mrm{b}^*$ and both the electrical size as well as the exterior losses tend to zero at the same time. \nTo this end, it is also interesting to observe that the limit of \\eqref{eq:Pabsellipsoid2o} as $(\\Im\\{\\epsilon\\},\\Im\\{\\epsilon_\\mrm{b}\\})\\rightarrow (0,0)$ will depend on how\nthis limit is taken. In particular, \\eqref{eq:Pabsellipsoid3o} is obtained for fixed $\\epsilon$ ($\\Im\\{\\epsilon\\}>0$) as $\\Im\\{\\epsilon_\\mrm{b}\\}\\rightarrow 0$ and \\eqref{eq:Pabsellipsoid3o} is\nthen unbounded as $\\Im\\{\\epsilon\\}$ approaches zero. On the other hand, \\eqref{eq:Pabsellipsoid2o} approaches zero for fixed $\\Im\\{\\epsilon_\\mrm{b}\\}>0$ as $\\Im\\{\\epsilon\\}\\rightarrow 0$ \n(and $\\Re\\{\\epsilon\\}\\neq 0$).\n\nThe conclusion of this discussion is that one can not optimize (with respect to $\\epsilon$) the absorption of an ellipsoid under the quasistatic assumption\nwhen the exterior domain is lossless. The natural question that arises is then under which circumstances this optimization model is valid for a lossy exterior domain.\nThis is the topic of the next section, where we restrict the analysis to the absorption of a dielectric sphere in a lossy medium.\n\n\n\\section{The absorption of a small dielectric sphere in a lossy background medium}\\label{sect:Miesolution}\n\\subsection{Electrodynamic solution}\nThe complete electrodynamic solution for the internal absorption of a small dielectric sphere in a lossy background medium is analyzed below.\nThe definition of the spherical vector waves, the spherical Bessel and Hankel functions and the related Lommel integrals are given in Appendix \\ref{sect:spherical}, see also \\cite{Nordebo+etal2017a}.\n\nConsider the scattering of the electromagnetic field due to a homogeneous dielectric sphere of radius $a$, complex-valued permittivity $\\epsilon$, \nand wavenumber $k=k_0\\sqrt{\\epsilon}$.\nThe medium surrounding the sphere is characterized by the permittivity $\\epsilon_\\mrm{b}$ and the wavenumber $k_\\mrm{b}=k_0\\sqrt{\\epsilon_\\mrm{b}}$.\nThe incident and the scattered fields for $r>a$ are expressed as in \\eqref{eq:Esphdef} with multipole coefficients $a_{\\tau ml}$ and $b_{\\tau ml}$, respectively,\nand the interior field is similarly expressed using regular spherical vector waves for $r0$, this factor becomes\n\\begin{equation}\\label{eq:factorgovconv2}\nF_2=\\frac{A{\\epsilon_\\mrm{b}^{\\prime\\prime}}^{\\alpha+1}}{{\\epsilon_\\mrm{b}^{\\prime\\prime}}^2+A^4{\\epsilon_\\mrm{b}^{\\prime\\prime}}^{4\\alpha}C_0^2\/16}.\n\\end{equation}\nA detailed study of the expression \\eqref{eq:factorgovconv2} for small $\\epsilon_\\mrm{b}^{\\prime\\prime}>0$ \nreveals the condition for convergence of \\eqref{eq:CabssphTMdipnorm}, which can be summarized as\n\\begin{equation}\\label{eq:convdiv}\n\\left\\{\\begin{array}{ll}\n\\mrm{Convergence} & 0<\\alpha<\\frac{1}{3}, \\vspace{0.2cm} \\\\\n\\mrm{Divergence} & \\frac{1}{3}\\leq \\alpha\\leq 1, \\vspace{0.2cm} \\\\\n\\mrm{Convergence} & \\alpha>1, \n\\end{array}\\right.\n\\end{equation}\nwhere $k_0a=A{ \\epsilon_\\mrm{b}^{\\prime\\prime}}^\\alpha$, $\\epsilon_\\mrm{b}^\\prime$ is fixed and $\\epsilon=-2\\epsilon_\\mrm{b}^*$ (the quasistatic optimal conjugate match).\n\nFor a fixed background loss parameter $\\epsilon_\\mrm{b}^{\\prime\\prime}>0$, the factor \\eqref{eq:factorgovconv1} can furthermore be maximized\nwith respect to the electrical size $k_0a$, yielding\n\\begin{equation}\\label{eq:k0aoptasymptotics}\nk_0a=\\frac{2}{3^{1\/4}C_0^{1\/2}}{\\epsilon_\\mrm{b}^{\\prime\\prime}}^{1\/2}.\n\\end{equation}\nFor fixed $\\epsilon_\\mrm{b}^{\\prime\\prime}$, the relation \\eqref{eq:k0aoptasymptotics} expresses a stationary point\nfor \\eqref{eq:CabssphTMdipnormapprox2} regarded as a function of $k_0a$, and hence an indicator of the domain of validity of \nthe quasistatic approximation \\eqref{eq:Cabssphere1o}.\n\nIn conclusion, it has been shown that the normalized absorption cross section area\n\\eqref{eq:CabssphTMdipnorm} has a subtle limiting behavior and is in fact unbounded as \n$(k_0a,\\epsilon^{\\prime\\prime},\\epsilon_\\mrm{b}^{\\prime\\prime})\\rightarrow (0,0,0)$ for fixed $\\epsilon^{\\prime}$ and $\\epsilon_\\mrm{b}^{\\prime}$.\nIt is also shown that if $k_0a$ is sufficiently small, the optimal conjugate match $\\epsilon=-2\\epsilon_\\mrm{b}^*$ defined in \\eqref{eq:epsilonodef} yields\nan accurate quasistatic approximation \\eqref{eq:Cabssphere1o} provided that \n\\begin{equation}\\label{eq:k0aoptasymptotics2}\n\\epsilon_\\mrm{b}^{\\prime\\prime}>\\frac{3\\sqrt{3}}{5}\\left(k_0a\\right)^2{\\epsilon_\\mrm{b}^\\prime}^2,\n\\end{equation}\nwhere \\eqref{eq:k0aoptasymptotics} and $C_0=12{\\epsilon_\\mrm{b}^\\prime}^2\/5$ have been used.\nIt should be noted that a direct maximization of \\eqref{eq:CabssphTMdipnorm} with respect to $\\epsilon$ would be possible by numerical optimization techniques,\nbut is not necessary if the quasistatic approximation is valid. The validity of the quasistatic approximation can be assessed by checking\nthe criteria \\eqref{eq:k0aoptasymptotics2} as well as by a direct comparison of \\eqref{eq:Cabssphere1} and \\eqref{eq:Cabssphere1o} with\nthe electrodynamic solution \\eqref{eq:CabssphTMdipnorm}.\n\n\\subsection{Optimal absorption of the small dielectric sphere in a lossy media}\\label{sect:optimalabsorptionepsilonsol}\nFinally, the asymptotic analysis above with \\eqref{eq:r21as} and \\eqref{eq:CDdef} makes it possible to analyze the pole structure of\n\\eqref{eq:CabssphTMdipnorm} in full dynamics. By making the following Ansatz for the pole\n$\\epsilon_\\mrm{p}=a_0+a_1 k_0a +a_2(k_0a)^2+a_3(k_0a)^3$,\nand identifying terms up to the third order in the denominator of \\eqref{eq:r21as},\nit is found that \\eqref{eq:CabssphTMdipnorm} can be approximated for small $k_0a$ as\n\\begin{equation}\\label{eq:Qabsdynpoleapprox}\nQ_\\mrm{abs}^\\mrm{dyn}\\sim 12 k_0a \\frac{\\left| \\epsilon_\\mrm{b} \\right|^2}{\\Re\\{\\sqrt{\\epsilon_\\mrm{b}}\\}} \\frac{\\Im\\{\\epsilon\\}}{\\left|\\epsilon-\\epsilon_\\mrm{p} \\right|^2},\n\\end{equation}\nand where the pole is given by \n\\begin{equation}\\label{eq:epsilonpolespherethirdorder}\n\\epsilon_\\mrm{p}=-2\\epsilon_\\mrm{b}-\\frac{12}{5}\\epsilon_\\mrm{b}(k_\\mrm{b}a)^2-\\iu 2 \\epsilon_\\mrm{b}(k_\\mrm{b}a)^3+ {\\cal O}\\{(k_\\mrm{b}a)^4\\},\n\\end{equation}\nsee also \\cite{Tretyakov2014}, and \\cite[Eq.~(11) on p.~3]{Tzarouchis+etal2016}.\nThe expression \\eqref{eq:Qabsdynpoleapprox} is of the form \\eqref{eq:fofepsilondef} where $\\Im\\{\\epsilon_\\mrm{p}\\}<0$ for small $\\epsilon_\\mrm{b}^{\\prime\\prime}$, \nand hence it can be concluded that the maximal absorption is approximately (asymptotically) achieved at $\\epsilon_\\mrm{opt}=\\epsilon_\\mrm{p}^*$, yielding\n\\begin{equation}\\label{eq:epsilonospherethirdorder}\n\\epsilon_\\mrm{opt}=-2\\epsilon_\\mrm{b}^*-\\frac{12}{5}\\epsilon_\\mrm{b}^*(k_\\mrm{b}^*a)^2+\\iu 2 \\epsilon_\\mrm{b}^*(k_\\mrm{b}^*a)^3+ {\\cal O}\\{(k_\\mrm{b}^*a)^4\\}.\n\\end{equation}\nNote that the result \\eqref{eq:epsilonospherethirdorder} generalizes the previous quasistatic result $\\epsilon_\\mrm{opt}=-2\\epsilon_\\mrm{b}^*$ given by \\eqref{eq:epsilonodef}.\nThe expression is valid for small $k_0a$ as well as for small loss factors $\\epsilon_\\mrm{b}^{\\prime\\prime}$.\nIn particular, the optimal absorption of the sphere is given by\n\\begin{equation}\\label{eq:Qabsoptasymptotic}\nQ_\\mrm{abs}^\\mrm{dyn,opt}\\sim 3k_0a\\frac{\\left| \\epsilon_\\mrm{b} \\right|^2}{\\Re\\{\\sqrt{\\epsilon_\\mrm{b}}\\}}\\frac{1}{\\Im\\{\\epsilon_\\mrm{opt}\\}},\n\\end{equation}\nand in the limit as the exterior losses vanish, \nwe have that $\\Im\\{\\epsilon_\\mrm{opt}\\}\\rightarrow 2{\\epsilon_\\mrm{b}^\\prime}^2\\sqrt{\\epsilon_\\mrm{b}^\\prime}(k_0a)^3$ and\n\\begin{equation}\\label{eq:Qabsoptasymptotic2}\n\\lim_{\\epsilon_\\mrm{b}^{\\prime\\prime}\\rightarrow 0}Q_\\mrm{abs}^\\mrm{dyn,opt}=\\frac{3}{2}\\frac{1}{(k_\\mrm{b} a)^2},\n\\end{equation}\nwhere $k_\\mrm{b}=k_0\\sqrt{\\epsilon_\\mrm{b}^\\prime}$, and which is in agreement with the upper bound \\eqref{eq:Cabsdip} given in \\cite{Tretyakov2014}.\n\nFinally, it should be noted that there are major discrepancies in the asymptotic analysis \nof the full dynamic solution performed in this section in comparison to the analysis of the quasistatic\napproximation as in \\eqref{eq:CabssphTMdipnormapprox1} and \\eqref{eq:CabssphTMdipnormapprox2} where $\\epsilon^{\\prime}$ and $\\epsilon_\\mrm{b}^{\\prime}$ are fixed \nand $\\epsilon^{\\prime}=-2\\epsilon_\\mrm{b}^{\\prime}$.\nIn essence, the quasistatic approximation \\eqref{eq:epsilonodef} is missing the second-order term giving a shift in the resonance frequency as well as the\nthird-order term taking the scattering loss into account. \nThese factors are negligible only when the external losses are large enough to make the second-order term redundant as expressed in \\eqref{eq:k0aoptasymptotics2},\nat the same time as the electrical size of the sphere is small enough to make the scattering loss insignificant. \nHence, when the external losses are too small in relation to the electrical size of the sphere, \nthe quasistatic solution $\\epsilon_\\mrm{opt}=-2\\epsilon_\\mrm{b}^*$ is not the correct choice to maximize the absorption and\none should instead use \\eqref{eq:epsilonospherethirdorder}.\n\n\n\\section{Numerical examples}\\label{sect:numexamples}\n\nIn Figures \\ref{fig:matfig3} through \\ref{fig:matfig4b} we show the normalized absorption cross section area $Q_\\mrm{abs}$\nand its behavior for $(k_0a,\\epsilon_\\mrm{b}^{\\prime\\prime})$ close to $(0,0)$.\nHere, $Q_\\mrm{abs}^\\mrm{dyn}$ denotes the electrodynamic solution given by \\eqref{eq:CabssphTMdipnorm} and $Q_\\mrm{abs}^\\mrm{qs,opt}$ corresponds to the optimal quasistatic solution\ngiven by \\eqref{eq:Cabssphere1o}, both of which are calculated for the background permittivity $\\epsilon_\\mrm{b}=\\epsilon_\\mrm{b}^{\\prime}+\\iu\\epsilon_\\mrm{b}^{\\prime\\prime}$\nand the quasistatic optimal conjugate match $\\epsilon=-2\\epsilon_\\mrm{b}^*$, yielding $\\epsilon+2\\epsilon_\\mrm{b}=\\iu 4\\epsilon_\\mrm{b}^{\\prime\\prime}$. \nIn Figure \\ref{fig:matfig3} we show also the normalized absorption cross section $Q_\\mrm{abs}^\\mrm{dyn,opt}$ corresponding\nto the dynamic optimal solution \\eqref{eq:epsilonospherethirdorder}, as well as the break points for the quasistatic approximation given by \\eqref{eq:k0aoptasymptotics}.\nAs a comparison with the case with a lossless background $(\\epsilon_\\mrm{b}^{\\prime\\prime}=0)$, the optimal dipole absorption cross section \n$Q_\\mrm{abs}^\\mrm{dip}$ corresponding to \\eqref{eq:Cabsdip} is also shown in Figure \\ref{fig:matfig3}.\nIn Figures \\ref{fig:matfig3} through \\ref{fig:matfig8} the background\nis defined by $\\epsilon_\\mrm{b}^{\\prime}=1$, and in Figure \\ref{fig:matfig4b} the background corresponds to a saline water with $\\epsilon_\\mrm{b}^{\\prime}=80$.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=0.48\\textwidth]{matfig3b.pdf}\n\\end{center}\n\\vspace{-5mm}\n\\caption{The quasistatic optimal normalized absorption cross section $Q_\\mrm{abs}^\\mrm{qs,opt}$ and the corresponding dynamic cross section $Q_\\mrm{abs}^\\mrm{dyn}$ \nas functions of the electrical size $k_0a$. The break points are given by the asymptotic analysis\n\\eqref{eq:k0aoptasymptotics}. For comparison, the optimal dynamic $Q_\\mrm{abs}^\\mrm{dyn,opt}$ is calculated based on \\eqref{eq:epsilonospherethirdorder}.\nHere, $\\epsilon_\\mrm{b}^{\\prime}=1$.}\n\\label{fig:matfig3}\n\\end{figure}\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=0.48\\textwidth]{matfig4.pdf}\n\\end{center}\n\\vspace{-5mm}\n\\caption{The quasistatic optimal normalized absorption cross section $Q_\\mrm{abs}^\\mrm{qs,opt}$ and the corresponding dynamic cross section $Q_\\mrm{abs}^\\mrm{dyn}$ \nas functions of the background loss $\\epsilon_\\mrm{b}^{\\prime\\prime}$. Here, $\\epsilon_\\mrm{b}^{\\prime}=1$.}\n\\label{fig:matfig4}\n\\end{figure}\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=0.48\\textwidth]{matfig8.pdf}\n\\end{center}\n\\vspace{-5mm}\n\\caption{The normalized dynamic absorption cross section $Q_\\mrm{abs}^\\mrm{dyn}$ as a function of the electrical size $k_0a$ \nand the background loss $\\epsilon_\\mrm{b}^{\\prime\\prime}$, and where $\\epsilon_\\mrm{b}^{\\prime}=1$. \nThe blue dashed line shows the stationary (break) points given by the asymptotic analysis\n\\eqref{eq:k0aoptasymptotics} indicating the region of validity \\eqref{eq:k0aoptasymptotics2} of the quasistatic approximation.\nNote also the asymptotes for divergence given by $\\log k_0a=\\log A+\\alpha\\log { \\epsilon_\\mrm{b}^{\\prime\\prime}}$ with $\\alpha=1\/3$ and $\\alpha=1$ as in \\eqref{eq:convdiv}.}\n\\label{fig:matfig8}\n\\end{figure}\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=0.48\\textwidth]{matfig4b.pdf}\n\\end{center}\n\\vspace{-5mm}\n\\caption{The quasistatic optimal normalized absorption cross section $Q_\\mrm{abs}^\\mrm{qs,opt}$ and the corresponding dynamic cross section $Q_\\mrm{abs}^\\mrm{dyn}$ \nas functions of the background loss $\\epsilon_\\mrm{b}^{\\prime\\prime}$. Here, $\\epsilon_\\mrm{b}^{\\prime}=80$, as with a biological tissue in the lower GHz region.}\n\\label{fig:matfig4b}\n\\end{figure}\n\nAs can be seen in the Figures \\ref{fig:matfig3} through \\ref{fig:matfig8}, the normalized absorption cross section $Q_\\mrm{abs}^\\mrm{dyn}$\nhas a very subtle behavior for $(k_0a,\\epsilon_\\mrm{b}^{\\prime\\prime})$ close to the plasmonic singularity at $(0,0)$ where $\\epsilon=-2$.\nNote that $Q_\\mrm{abs}^\\mrm{dyn}\\rightarrow 0$ as $k_0a\\rightarrow 0$\n ($\\epsilon_\\mrm{b}^{\\prime\\prime}>0$ fixed) and $Q_\\mrm{abs}^\\mrm{dyn}\\rightarrow 0$ as $\\epsilon_\\mrm{b}^{\\prime\\prime}\\rightarrow 0$\n ($k_0a>0$ fixed), \\textit{cf.}\\\/, also the approximate expression in \\eqref{eq:CabssphTMdipnormapprox2}.\n However, as illustrated in Figure \\ref{fig:matfig8}, $Q_\\mrm{abs}^\\mrm{dyn}$ is unbounded in\n any neighborhood where $(k_0a,\\epsilon_\\mrm{b}^{\\prime\\prime})$ is close to $(0,0)$ in accordance with the analysis given in Section \\ref{sect:asymptoticanalys}.\n It should be noted that this behavior is perfectly consistent with the bound for a lossless background $C_\\mrm{abs}\\leq 3\\lambda^2\/8\\pi$ given in \\cite{Tretyakov2014}\n since $C_\\mrm{abs}\/a^2\\leq (3\/8\\pi)\/(a\/\\lambda)^2$ is unbounded as $a\/\\lambda$ approaches zero.\n \nFigures \\ref{fig:matfig3} through \\ref{fig:matfig8} illustrate how the approximation $Q_\\mrm{abs}^\\mrm{qs,opt}\\sim Q_\\mrm{abs}^\\mrm{dyn}$\nbecomes valid when $k_0a$ is sufficiently small \nat the same time as $\\epsilon_\\mrm{b}^{\\prime\\prime}$ is sufficiently large. \nIn Figure \\ref{fig:matfig3} is also illustrated that the break points given by \\eqref{eq:k0aoptasymptotics} (or \\eqref{eq:k0aoptasymptotics2})\ngive a very accurate estimate for the validity of the quasistatic approximation \\eqref{eq:Cabssphere1o}.\n\nThe whole feasibility investigation regarding the quasistatic approximation above can \nreadily be executed similarly for any other background permittivity $\\epsilon_\\mrm{b}^{\\prime}$.\nAs \\textit{e.g.}\\\/, Figure \\ref{fig:matfig4b} shows a comparison between $Q_\\mrm{abs}^\\mrm{dyn}$ and $Q_\\mrm{abs}^\\mrm{qs,opt}$ \nwith $\\epsilon_\\mrm{b}^{\\prime}=80$ corresponding to the permittivity of biological tissue for frequencies \nin the lower GHz region, \\textit{cf.}\\\/, \\cite{Gabriel+etal1996b}. In this frequency region the corresponding\ndielectric losses will be at least $\\epsilon_\\mrm{b}^{\\prime\\prime}>10$. Hence, the evaluation shown in Figure \\ref{fig:matfig4b}\nverifies that the previous investigations made in \\cite{Nordebo+etal2017a,Dalarsson+etal2017a} are safely in the quasistatic regime\nif \\textit{e.g.}\\\/, $k_0a<10^{-2}$, and which is certainly the case with cellular ($\\mu$m) structures in the GHz frequency range.\n\nFinally, to illustrate the theory based on an application in plasmonics, we investigate the absorption in a silver \nnanosphere embedded in a slightly lossy medium.\nFigure \\ref{fig:matfig21} shows the dielectric function of silver according to the Brendel-Bormann (BB) model fitted to experimental data as presented in \n\\cite[the dielectric model in Eq. (11) with parameter values from Table 1 and Table 3]{Rakic+etal1998}.\nThe frequency axis is given here in terms of the photon energy $h\\nu$ in units of electron volts\\unit{(eV)} where $h$ is Planck's constant and $\\nu$ the frequency.\nFigures \\ref{fig:matfig24} and \\ref{fig:matfig24b} present the normalized absorption cross section areas of a sphere with radius $a=10$\\unit{nm} \nand where the quasistatic optimal $Q_\\mrm{abs}^\\mrm{qs,opt}$ is given by \\eqref{eq:Cabssphere1o}, \nthe dynamic optimal $Q_\\mrm{abs}^\\mrm{dyn,opt}$ is given by \\eqref{eq:CabssphTMdipnorm} \ntogether with \\eqref{eq:epsilonospherethirdorder} and $Q_\\mrm{abs}^\\mrm{Ag}$ is given by \\eqref{eq:CabssphTMdipnorm} together with the dielectric model of silver as illustrated\nin Figure \\ref{fig:matfig21}. The background medium is defined by $\\epsilon_\\mrm{b}=1+\\iu\\epsilon_\\mrm{b}^{\\prime\\prime}$, where \n$\\epsilon_\\mrm{b}^{\\prime\\prime}=10^{-1}$ and $\\epsilon_\\mrm{b}^{\\prime\\prime}=10^{-3}$, respectively.\nWhat is interesting to observe here is that the absorption of silver is in fact rather close to being optimal (the peak at $h\\nu= 3.4$\\unit{eV}) \nwith the larger loss factor $\\epsilon_\\mrm{b}^{\\prime\\prime}=10^{-1}$,\nand where the quasistatic and dynamic optimal solutions also agree rather well. With the smaller loss factor $\\epsilon_\\mrm{b}^{\\prime\\prime}=10^{-3}$,\nthe absorption is no longer close to being optimal and the quasistatic and dynamic optimal solutions diverge. \nHere, we have chosen the loss factors so that $10^{-3}<\\epsilon_\\mrm{b,break}^{\\prime\\prime}<10^{-1}$ where\n$\\epsilon_\\mrm{b,break}^{\\prime\\prime}=0.03$ corresponds to the break point defined by \\eqref{eq:k0aoptasymptotics2} for resonance at $h\\nu=3.4$\\unit{eV}. \n\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=0.48\\textwidth]{matfig21.pdf}\n\\end{center}\n\\vspace{-5mm}\n\\caption{Dielectric function of silver according to the Brendel-Bormann model fitted to experimental data \\cite{Rakic+etal1998}.}\n\\label{fig:matfig21}\n\\end{figure}\n\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=0.48\\textwidth]{matfig24.pdf}\n\\end{center}\n\\vspace{-5mm}\n\\caption{Normalized absorption cross sections for a sphere with radius $a=10$\\unit{nm} \nwith $Q_\\mrm{abs}^\\mrm{dyn,opt}$ denoting the dynamic optimal absorption, $Q_\\mrm{abs}^\\mrm{qs,opt}$ the quasistatic optimal absorption and \n$Q_{\\mrm{abs}}^\\mrm{Ag}$ the full dynamic dipole absorption corresponding to the Brendel-Bormann \nmodel for the dielectric function of silver \\cite{Rakic+etal1998}, \\textit{cf.}\\\/, also Fig.~\\ref{fig:matfig21}.\nHere, the background loss is given by $\\epsilon_\\mrm{b}=1+\\iu 10^{-1}$.\n}\n\\label{fig:matfig24}\n\\end{figure}\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=0.48\\textwidth]{matfig24b.pdf}\n\\end{center}\n\\vspace{-5mm}\n\\caption{Same plot as in Fig.~\\ref{fig:matfig24}, but here with the background loss given by $\\epsilon_\\mrm{b}=1+\\iu 10^{-3}$.}\n\\label{fig:matfig24b}\n\\end{figure}\n\n\n\n\n\n\\section{Summary and conclusions}\n\nIt has been demonstrated that the maximal dipole absorption of a small dielectric or conducting sphere is \nunbounded under the quasistatic approximation if the losses in the surrounding medium can be made arbitrarily small. \nThis deviation has been rectified by using the general Mie theory and an asymptotic analysis to give explicit formulas to assess the validity of the quasistatic approximation.\nIn particular, it turns out that the quasistatic theory is valid provided that the electrical size of the sphere is sufficiently small\nat the same time as the exterior losses are sufficiently large. Moreover, it has been shown that the optimal normalized absorption cross section area of \nthe small dielectric sphere has a very subtle limiting behavior, and is in fact unbounded when both the electrical size as well as the exterior losses tend to zero.\nFinally, an improved asymptotic formula based on full dynamics has been given for the optimal plasmonic dipole absorption of the sphere, which is \nvalid for small spheres as well as for small losses. Numerical examples have been included to illustrate the asymptotic results.\n\n\n\\begin{acknowledgments}\nThis work has been partly supported by the Swedish Foundation for Strategic Research (SSF) under the programme \nApplied Mathematics and the project Complex Analysis and Convex Optimization for EM Design.\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{Introduction}\n\n\\begin{figure*}[ht]\n \\centering\n \\includegraphics[scale=0.67]{figures\/new_main_figure_3.pdf}\n \\caption{An example from the SciERC dataset~\\cite{luan2018multi}, where a system is expected to identify that \\ts{MORPA} and \\ts{parser} are entities of type \\ts{Method}, \\textsc{text-to-speech} is a \\ts{Task}, as well as \\textsc{MORPA} is a \\ti{hyponym of} \\textsc{parser} and \\textsc{MORPA} is \\ti{used for} \\textsc{text-to-speech}. Our entity model (a) predicts all the entities at once and our relation model (b) considers every pair of entities independently by inserting typed entity markers (e.g., \\textsc{[S:Md]} = the subject is a \\ts{Method}, \\textsc{[O:Tk]} = the object is a \\ts{Task}). We also proposed an approximation relation model (c) which supports batch computations. The tokens of the same color in (c) share the positional embeddings. See text for more details. }\n \\label{fig:mainfig}\n\\end{figure*}\n\nExtracting entities and their relations from unstructured text is a fundamental problem in information extraction. This problem can be decomposed into two subtasks: named entity recognition~\\cite{sang2003introduction,ratinov2009design} and relation extraction~\\cite{zelenko2002kernel,bunescu2005shortest}. Early work employed a pipelined approach, training one model to extract entities and another to classify relations between them. More recently, however, end-to-end evaluations have been dominated by systems that model these two tasks jointly~\\cite{li2014incremental,miwa2016end,katiyar2017going,zhang2017end,li2019entity,luan2018multi,luan2019general,wadden2019entity,lin2020joint,wang2020two}. It is commonly thought that joint models can better capture the interactions between entities and relations and help mitigate error propagation issues.\n\nIn this work, we review this problem and present a very simple approach which learns \\ti{two} encoders built on top of deep pre-trained language models~\\cite{devlin2019bert,beltagy2019scibert,lan2020albert}. The two models --- which we refer them as to an \\ti{entity model} and a \\ti{relation model} throughout the paper --- are trained independently and the relation model only relies on the entity model as input features. Our entity model builds on span-level representations and our relation model builds on contextual representations specific to a given pair of spans. Despite its simplicity, we find this pipelined approach to be extremely effective: using the same pre-trained encoders, our model outperforms all previous joint models on three standard benchmarks (ACE04, ACE05, SciERC).\n\nTo understand why this approach performs so well, we carry out a series of careful analyses. We observe that, (1) the contextual representations for the entity and relation models essentially capture distinct information, so sharing their representations hurts performance; (2) it is crucial to fuse the entity information (both boundary and type) at the \\ti{input layer} of the relation model; (3) leveraging cross-sentence information is useful in both tasks; (4) stronger pre-trained language models can bring further gains. Hence, we expect that this simple model will serve as a very strong baseline and make us rethink the value of joint training in end-to-end relation extraction.\n\nFinally, one possible shortcoming of our approach is that we need to run our relation model once for every pair of entities. To alleviate this issue, we present a novel and efficient alternative by approximating and batching the computations of different groups of entity pairs at inference time. This approximation achieves a 8-16$\\times$ speedup with only a very small accuracy drop (e.g., 0.5-0.9\\% F1 drop on ACE05), which makes our model fast and accurate to use in practice.\n\nWe summarize our contributions as follows:\n\\begin{itemize}\n \\item We present a very simple and effective approach for end-to-end relation extraction, which learns two independent encoders for entity recognition and relation extraction. Our model establishes the new state-of-the-art on three standard benchmarks and surpasses all previous joint models using the same pre-trained models.\n \\item We conduct careful analyses to understand why our approach performs so well and how different factors affect final performance. We conclude that it is more effective to learn distinct contextual representations for entities and relations than to learn them jointly.\n \\item To speed up the inference time of our model, we also propose a novel and efficient approximation, which achieves a 8-16$\\times$ runtime improvement with only a small accuracy drop.\n\\end{itemize}\n\n\\label{sec:intro}\n\n\\section{Related Work}\n\\label{sec:related}\n\nTraditionally, extracting relations between entities in text has been studied as two separate tasks: named entity recognition and relation extraction. In the last several years, there has been a surge of interest in developing models for joint extraction of entities and relations~\\cite{li2014incremental,miwa2016end}. We group existing work of end-to-end relation extraction into two main categories: \\ti{structured prediction} and \\ti{multi-task learning}.\n\n\n\n\n\\vspace{-0.4em} \\paragraph{Structured prediction} The first category casts the two tasks into one structured prediction framework, although it can be formulated in various ways. \\newcite{li2014incremental} propose an action-based system which identifies new entities as well as links to previous entities, \\newcite{zhang2017end,wang2020two} adopt a table-filling approach proposed in \\cite{miwa2014modeling};\n\\newcite{katiyar2017going} and \\newcite{zheng2017joint} employ sequence tagging-based approaches; \\newcite{sun2019joint} and \\newcite{fu2019graphrel} propose graph-based approaches to jointly predict entity and relation types;\nand, \\newcite{li2019entity} convert the task into a multi-turn question answering problem. All of these approaches need to tackle a global optimization problem and perform joint decoding at inference time, using beam search or reinforcement learning.\n\n\n\\vspace{-0.4em} \\paragraph{Multi-task learning} This family of models essentially builds two separate models for entity recognition and relation extraction and optimizes them together through parameter sharing. \\newcite{miwa2016end} propose to use a sequence tagging model for entity prediction and a tree-based LSTM model for relation extraction.\nThe two models share one LSTM layer for contextualized word representations and they find sharing parameters improves performance for both models. The approach of \\newcite{bekoulis2018adversarial} is similar except that they model relation classification as a multi-label head selection problem. Note that these approaches still perform pipelined decoding: entities are first extracted and the relation model is applied on the predicted entities.\n\n\nThe closest work to ours is probably DYGIE and DYGIE++~\\cite{luan2018multi,luan2019general,wadden2019entity} which builds on recent span-based models for coreference resolution~\\cite{lee2017end} and semantic role labeling~\\cite{he2018jointly}. The key idea is to share span representations between the two tasks with joint optimization. It is later improved by incorporating relation and coreference propagation layers to update span representations~\\cite{luan2019general} and replacing LSTM encodings with BERT-based representations~\\cite{wadden2019entity}. A more recent work~\\cite{lin2020joint} further extends~\\newcite{wadden2019entity} by incorporating global features based on cross-substask and cross-instance constraints.\\footnote{This is an orthogonal contribution to ours and we will explore it for future work.} Compared to DYGIE++, our approach is much simpler: we use two independent encoders and do not use beam search or graph propagation layers. We will detail the differences in Section~\\ref{sec:our-model} and argue why our simpler model performs better.\n\n\n\n\n\n\\paragraph{BERT for relation extraction} Earlier work explored a variety of neural network architectures for relation extraction, including convolutional neural networks~\\cite{zeng2014relation}, recurrent neural networks~\\cite{xu2015classifying,zhang2017tacred}, and graph neural networks~\\cite{zhang2018graph}. More recent work uses pre-trained language models (LMs) such as BERT, built on deep Transformer encoders~\\cite{shi2019simple,soares2019matching}. We follow this trend and also study the impact of different pre-trained LMs on final performance.\n\n\\section{Method}\n\\label{sec:method}\n\nIn this section, we first formally define the problem of end-to-end relation extraction in Section~\\ref{sec:problem-definition} and then detail our approach in Section~\\ref{sec:our-model}. Finally, we present our approximation solution in Section~\\ref{sec:method-approx} which improves the efficiency of our approach during inference considerably.\n\n\\subsection{Problem Definition}\n\\label{sec:problem-definition}\nThe input of the problem is a sentence $X$ consisting of $n$ tokens $x_1, \\dots, x_n$. Let $S=\\{s_1, \\dots, s_m\\}$ be all the possible spans in $X$ of up to length $L$ and $\\ts{START}(i)$ and $\\ts{END}(i)$ denote start and end indices of $s_i$. Optionally, we can incorporate cross-sentence context to help improve predictions, which we will elaborate in the next section. The problem can be decomposed into two sub-tasks:\n\\paragraph{Named entity recognition} Let $\\mathcal{E}$ denote a set of pre-defined entity types. The named entity recognition task is, for each span $s_i \\in S$, to predict an entity type $y_e(s_i) \\in \\mathcal{E}$ or $y_e(s_i) = \\epsilon$ representing span $s_i$ is not an entity. The output of the task is $Y_e = \\{(s_i, e), s_i \\in S, e \\in \\mathcal{E}\\}$.\n\\paragraph{Relation extraction} Let $\\mathcal{R}$ denote a set of pre-defined relation types. The task is, for every pair of spans $s_i \\in S, s_j\\in S$, to predict a relation type $y_r(s_i, s_j) \\in \\mathcal{R}$, or there is no relation between them: $y_r(s_i, s_j) = \\epsilon$. The output of the task is $Y_r = \\{(s_i, s_j, r), s_i, s_j \\in S, r \\in \\mathcal{R}\\}$.\n\nWe aim to build a model which takes $X$ as input and outputs $Y_e$ and $Y_r$ at the same time. During evaluation, $Y_e$ and $Y_r$ are compared against the ground truth $Y^*_e$ and $Y^*_r$ and entity and relation F1 will be reported respectively.\\footnote{The strict evaluation of relation F1 also takes the entity type into account. See Section~\\ref{sec:exp-setup} for more details.}\n\n\\subsection{Our Model}\n\\label{sec:our-model}\nIn the following, we will describe our full model which consists of an entity and a relation model. As shown in Figure~\\ref{fig:mainfig}, our entity model first takes the input sentence and predict an entity type (or $\\epsilon$) for each single span. We then process every pair of candidate entities independently in the relation model by inserting extra marker tokens to highlight the subject and object and their types. We will detail each component below. At the end of this section, we will also summarize the main differences of our approach and DYGIE++~\\cite{wadden2019entity}, which is the closest work to ours and serves as a strong baseline in the literature.\n\n\\paragraph{Entity model} Our entity model is a standard span-based model following prior work~\\cite{lee2017end,luan2018multi,luan2019general,wadden2019entity}. We first use a pre-trained language model (e.g., BERT) to obtain contextualized representations $\\mf{x}_t$ for each input token $x_t$. Given a span $s_i \\in S$, the span representation $\\mathbf{h}_e(s_i)$ is defined as:\n\\vspace{-0.3em}\n\\begin{align*}\n \\mathbf{h}_e(s_i) = [\\mathbf{x}_{\\textsc{START}(i)}; \\mathbf{x}_{\\textsc{END}(i)}; \\phi(s_i)],\n\\end{align*}\nwhere $\\phi(s_i) \\in \\mathbb{R}^{d_W}$ represents the learned embeddings of span width features. The span representation $\\mathbf{h}_e(s_i)$ is then used to predict entity types $e \\in \\mathcal{E} \\cup \\{\\epsilon\\}$:\n\\vspace{-0.3em}\n\\begin{align*}\n P_e(e \\mid s_i) = \\mathrm{softmax}(\\mathbf{W}_e \\text{FFNN}(\\mathbf{h}_e(s_i)).\n\\end{align*}\nWe follow~\\newcite{wadden2019entity} and use a 2-layer feedforward neural network with ReLU activations.\n\n\\paragraph{Relation model} The relation model aims to take a pair of spans $s_i, s_j$ (a subject and an object) as input and predicts a relation type or $\\epsilon$, between the two spans. Previous approaches~\\cite{luan2018multi,luan2019general,wadden2019entity} re-use span representations $\\mf{h}_e(s_i), \\mf{h}_e(s_j)$ to predict their relation. We observe that these representations only capture contextual information around each individual entity and might fail to capture the dependencies between a specific pair of spans. We also hypothesize that sharing the contextual representations for different pairs of spans may be suboptimal. For example, the words \\ti{is a} in ~Figure~\\ref{fig:mainfig} are crucial in classifiying the relationship between \\ts{MORPA} and \\ts{parser} but not for \\ts{MORPA} and \\ts{text-to-speech}.\n\n\nOur relation model instead processes each pair of spans independently and inserts typed markers at the input layer to highlight the subject and object and their types. Specifically, given an input sentence $X$ and a pair of spans $s_i, s_j$, where $s_i$, $s_j$ have a type of $e_i, e_j \\in \\mathcal{E} \\cup \\{\\epsilon\\}$ respectively.\nWe define text markers as $\\langle \\textsc{S:}e_i\\rangle$, $\\langle \\textsc{\/S:}e_i\\rangle$, $\\langle \\textsc{O:}e_j\\rangle$, and $\\langle \\textsc{\/O:}e_j\\rangle$,\nand insert them into the input sentence before and after the subject and object spans (Figure~\\ref{fig:mainfig} (b)).\\footnote{Our final model indeed only considers $e_i, e_j \\neq \\epsilon$. We have explored using spans which are predicted as $\\epsilon$ for the relation model but didn't find improvement. See Section~\\ref{sec:error-prop} for more discussion.}\nLet $\\widehat{X}$ denote this modified sequence with text markers inserted:\n\\vspace{-0.3em}\n\\begin{align*}\n &\\widehat{X} = \\dots \\langle \\textsc{S:}e_i\\rangle, x_{\\textsc{START}(i)}, \\dots, x_{\\textsc{END}(i)}, \\langle \\textsc{\/S:}e_i\\rangle, \\\\\n &\\dots \\langle \\textsc{O:}e_j\\rangle, x_{\\textsc{START}(j)}, \\dots, x_{\\textsc{END}(j)}, \\langle \\textsc{\/O:}e_j\\rangle, \\dots.\n\\end{align*}\n\nWe then apply another pre-trained encoder on $\\widehat{X}$ and denote the output representations by $\\mathbf{\\widehat{x}}_t$. We concatenate the output representations of two start positions and obtain the span pair representation:\n\\begin{align*}\n \\mathbf{h}_r(s_i, s_j)=[\\mathbf{\\widehat{x}}_{\\widehat{\\textsc{START}}(i)}; \\mathbf{\\widehat{x}}_{\\widehat{\\textsc{START}}(j)}],\n\\end{align*}\nwhere $\\widehat{\\textsc{START}(i)}$ and $\\widehat{\\textsc{START}(j)}$ are the indices of $\\langle \\textsc{S:}e_i\\rangle$ and $\\langle \\textsc{O:}e_j\\rangle$ in $\\widehat{X}$. Finally, the representation $\\mathbf{h}_r(s_i, s_j)$ will be used to predict the relation type $r \\in \\mathcal{R} \\cup \\{\\epsilon\\}$:\n\\begin{align*}\n P_r(r|s_i, s_j) = \\mathrm{softmax}(\\mathbf{W}_r \\mathbf{h}_r(s_i, s_j)),\n\\end{align*}\n\n\n\n\nThis idea of using additional markers to highlight the subject and object is not entirely new as it has been studied recently in relation classification tasks~\\cite{zhang2019ernie,soares2019matching}. However, most relation classification tasks (e.g., TACRED~\\cite{zhang2017tacred}) only focus on a given pair of subject and object in an input sentence and its effectiveness has not been evaluated in the end-to-end setting in which we need to classify the relationships between multiple entity mentions. We observed a large improvement in our experiments (Section~\\ref{sec:input-features}) and this strengthens our hypothesis that modeling the relationship between different entity pairs in one sentence require different contextual representations. Furthermore, \\newcite{zhang2019ernie,soares2019matching} only considered untyped markers (e.g., $\\langle\\textsc{S}\\rangle$, $\\langle\\textsc{\/S}\\rangle$) and previous end-to-end models e.g., \\cite{wadden2019entity} inject the entity type information into the relation model through auxiliary losses. Using \\ti{typed} entity markers hasn't been explored before. We find that injecting type information at the input layer is very helpful in distinguishing entity types --- for example, whether ``Disney'' refers to a \\ti{person} or an \\ti{organization}--- before trying to understand the relations between them.\n\n\n\\paragraph{Cross-sentence context}\nCross-sentence information can be used to help predict entity types and relations, especially for pronominal mentions.\n\\newcite{luan2019general,wadden2019entity} employ a propagation mechanism through coreference and relation links to incorporate cross-sentence context. \\newcite{wadden2019entity} also add a 3-sentence context window which is shown to improve performance.\nWe also evaluate the importance of leveraging cross-sentence context in end-to-end relation extraction. As we expect that pre-trained language models to be able to capture long-range dependencies already, we simply incorporate cross-sentence context by extending the sentence to a fixed window size $W$ for both the entity and relation model. Specifically, given an input sentence with $n$ words, we augment the input with $(W-n) \/ 2$ words from the left context and right context respectively ($W = 100$ in our default model).\n\n\\paragraph{Training \\& inference} For both entity model and relation model, we employ two pre-trained language models and fine-tune them using task-specific losses. We use cross-entropy loss for both models:\n\\begin{align*}\n \\mathcal{L}_e &= -\\sum_{s_i \\in S} \\log P_e(e_i^* | s_i) \\\\\n \\mathcal{L}_r &= -\\sum_{s_i, s_j \\in S_G, s_i \\neq s_j} \\log P_r(r_{i,j}^* \\mid s_i, s_j),\n\\end{align*}\nwhere $e_i^*$ represents the gold entity type of $s_i$ and $r_{i,j}^*$ represents the gold relation type of span pair $s_i, s_j$ in the training data. For training the relation model, we only consider the gold entities $S_G \\subset S$ in the training set and use the gold entity labels as the input of the relation model. We considered training on all spans $S$ (with pruning) as well as predicted entity types but none of them led to meaningful improvements compared to this simple pipeline training (see more details in Section~\\ref{sec:error-prop}).\n\nDuring inference, we first predict the entities by taking $y_e(s_i) = \\argmax_{e\\in \\mathcal{E} \\cup \\{\\epsilon\\}}P_e(e | s_i)$. Denote $S_{\\text{pred}} = \\{s_i: y_e(s_i) \\neq \\epsilon\\}$, we enumerate all the spans $s_i, s_j \\in S_{\\text{pred}}$ and use $y_e(s_i), y_e(s_j)$ as the inputs for the relation model $P_r(r \\mid s_i, s_j)$.\n\n\n\\paragraph{Differences from DYGIE++} Our model differs from DYGIE++ in the following ways: (1) We use separate encoders for the entity and relation model, without any multi-task learning; the predicted entity labels are used directly as the input features of the relation model. (2) The contextual representations in the relation model are specific to each pair of spans. (3) We incorporate cross-sentence information by extending the input with additional context. (4) We do not use beam search or graph propagation layers. As a result, our model is much simpler. Moreover, we will show that it also achieves large gains in all the benchmarks, using the same pre-trained encoders.\n\n\\subsection{Efficient Batch Computations}\n\\label{sec:method-approx}\nDespite the simplicity and effectiveness of our approach (which we will demonstrate in our experiments), one possible shortcoming is that we need to run our relation model once for every pair of entities.\nTo alleviate this issue, we propose a novel and efficient alternative for our relation model. The key problem is that we would like to re-use computations for different span pairs in the same sentence. This is impossible in our original model because we must insert special entity markers for each pair of spans independently. Thus we propose an approximation model by making two major changes to the original relation model. First, instead of directly inserting entity markers into the original sentence, we tie the position embeddings of the markers with the start and end tokens of the corresponding span:\n\\begin{align*}\n & \\textsc{pos}(\\langle \\textsc{S:}e_i\\rangle) = \\textsc{pos}(x_{\\textsc{START}(i)}) \\\\\n & \\textsc{pos}(\\langle \\textsc{\/S:}e_i\\rangle) = \\textsc{pos}(x_{\\textsc{END}(i)}) \\\\\n & \\textsc{pos}(\\langle \\textsc{O:}e_j\\rangle) = \\textsc{pos}(x_{\\textsc{START}(j)}) \\\\\n & \\textsc{pos}(\\langle \\textsc{\/O:}e_j\\rangle) = \\textsc{pos}(x_{\\textsc{END}(j)}),\n\\end{align*}\nwhere $\\textsc{pos}(\\cdot)$ denotes the position id of a token. As the example shown in Figure~\\ref{fig:mainfig}, if we want to classify the relationship between \\ts{MORPA} and \\ts{parser}, the first entity marker \\ts{$\\langle S$$:\\ts{Method}\\rangle$} will share the positional embedding with the token \\ts{mor}. To do this, the positional embeddings of the original tokens will not be changed.\n\nSecond, we add a constraint to the attention layers: We enforce the text tokens to only attend to text tokens and not attend to the marker tokens while an entity marker token can attend to all the text tokens and all the 4 marker tokens associated with the same span pair. These two modifications allow us to re-use the computations of all text tokens, because text tokens are independent of the entity marker tokens.\nThus, we can batch multiple pairs of spans from the same sentence in one run of the relation model. In practice, we add all marker tokens to the end of the sentence to form an input that batches a set of span pairs (Figure~\\ref{fig:mainfig} (c)). This leads to a large speedup at inference time and only a small drop in performance (Section~\\ref{sec:approx}).\n\n\n\\section{Experiments}\n\\label{sec:exp}\n\n\\subsection{Experimental Setup}\n\\label{sec:exp-setup}\n\n\\paragraph{Datasets} We evaluate our approach on three end-to-end relation extraction datasets: ACE04, ACE05\\footnote{\\url{catalog.ldc.upenn.edu\/LDC2005T09} \\url{catalog.ldc.upenn.edu\/LDC2006T06}}, and SciERC~\\cite{luan2018multi}.\nWe follow \\newcite{luan2019general}'s preprocessing steps and split ACE04 into 5 folds, and split ACE05 and SciERC into train, development, and test sets.\nThe detailed data statistics are given in Table~\\ref{tab:data_stats}.\n\n\n\\input{tables\/data_stat.tex}\n\\input{tables\/main_result}\n\\input{tables\/approx.tex}\n\n\\paragraph{Evaluation metrics}\nWe follow the standard evaluation protocol and use F1 scores as the evaluation metric. For the named entity recognition task, a predicted entity is considered as a correct prediction if its span boundaries and the predicted entity type are both correct.\nFor the relation extraction task, a predicted relation is considered as a correct prediction if the boundaries of two spans are correct and the predicted relation type is correct.\nWe also report the strict relation F1 score (denoted by Rel+), where a predicted relation is considered as a correct prediction if the entity types and boundaries of two spans are correct, as well as the predicted relation type is correct. More discussion of the evaluation settings can be found in \\newcite{bekoulis2018adversarial,taille2020sincere}.\n\n\\paragraph{Implementation details}\nFor the entity model, we follow \\newcite{wadden2019entity} and set the width embedding size as $d_W = 150$ and use a 2-layer FFNN with $150$ hidden units. For our approximation model (Section~\\ref{sec:approx}), we batch candidate pairs by adding $4$ markers for each pair to the end of the sentence, until the total number of tokens exceeds $250$.\nWe use a context window size of $W=100$ in our default setting using cross-sentence context and we will study the effect of different context sizes in Section~\\ref{sec:context}. We consider spans up to $L = 8$ words. For all the experiments, we report the averaged F1 scores of 5 runs.\n\nWe implement our models based on HuggingFace's \\ti{Transformers} library~\\cite{Wolf2019HuggingFacesTS}. We use \\ttt{bert-base-uncased}~\\cite{devlin2019bert} and \\ttt{albert-xxlarge-v1}~\\cite{lan2020albert} as the base encoders for ACE04 and ACE05, for a fair comparison with previous work and an investigation of the impact of small vs large pre-trained models.\\footnote{ As detailed in Table~\\ref{tab:main-results}, some previous work used BERT-large models. We are not able to do a comprehensive study of all the pre-trained models and our BERT-base results are generally higher than most published results using larger models.} We also use \\ttt{scibert-scivocab-uncased}~\\cite{beltagy2019scibert} as the base encoder for SciERC, as this in-domain pre-trained model is shown to be more effective than BERT~\\cite{wadden2019entity}.\nWe train our models with Adam optimizer of a linear scheduler with a warmup ratio of $0.1$. For all the experiments, we train the entity model for $100$ epochs, and a learning rate of 1e-5 for weights in pre-trained LMs, 5e-4 for others and a batch size of 16. We train the relation model for $10$ epochs with a learning rate of 2e-5 and a batch size of 32.\n\n\\subsection{Main Results}\n\nTable~\\ref{tab:main-results} compares our approach to all the previous results.\nWe report the F1 scores in both single-sentence (no cross-sentence context) and cross-sentence (a context window size of $W=100$) settings for a fair comparison with previous work. As is shown, our single-sentence models achieve strong performance and incorporating cross-sentence context further improves the results consistently.\nOur BERT-base (or SciBERT) models achieve similar or better results compared to all the previous work including models built on top of larger pre-trained LMs, and the performance is further improved by using a larger encoder, i.e., ALBERT.\n\n\nFor entity recognition, our best model achieves an absolute F1 improvement of $+1.4\\%$, $+1.7\\%$, $+0.7\\%$ on ACE05, ACE04, and SciERC respectively.\nThis shows that cross-sentence information is useful for the entity model and that pre-trained Transformer encoders are able to capture long-range dependencies from a large context. For relation extraction, our approach outperforms the best previous methods by an absolute F1 of $+2.6\\%$, $+2.8\\%$, $+1.7\\%$ on ACE05, ACE04, and SciERC respectively. We also obtained a $4.3\\%$ higher relation F1 on ACE05 compared to DYGIE++~\\cite{wadden2019entity} using the same BERT-base pre-trained model. All these large improvements demonstrate the effectiveness of learning distinct representations for entities and relations of different entity pairs, as well as fusing entity information at the input layer of the relation model.\n\nWe also noticed that compared to the previous state-of-the-art model~\\cite{wang2020two} based on ALBERT, our model achieves a similar entity F1 (89.5 vs 89.7) but a substantially better relation F1 (67.6 vs 69.0) without using cross-sentence context. This clearly demonstrates the superiority of our relation model.\n\n\\subsection{Batch Computations and Speedup}\n\\label{sec:approx}\n\nIn Section~\\ref{sec:method-approx}, we proposed an efficient approximation solution for the relation model, which enables us to re-use the computations of text tokens and batch multiple span pairs in one input sentence.\nWe evaluate this approximation model on ACE05 and SciERC.\nTable~\\ref{tab:approx} shows the relation F1 scores and the inference speed of the full relation model and the approximation model.\nOn both datasets, our approximation model significantly improves the efficiency of the inference process. For example, in the single-sentence setting, we obtain a $11.9\\times$ speedup on ACE05 and a $8.7\\times$ speedup on SciERC.\nBy re-using a large part of computations, we are able to make predictions on the full ACE05 test set (2k sentences) in less than $10$ seconds on a single NVIDIA GeForce 2080 Ti GPU. On the other hand, this approximation only brings a small performance drop --- compared to the full model, the F1 score drops $0.5\\%$ and $1.0\\%$ on ACE05 and SciERC respectively in the single-sentence setting. Considering the accuracy and efficiency of this approximation model, we expect it to be very effective to use in practice.\n\n\n\\section{Analysis}\n\\label{sec:analysis}\n\nDespite its simple design and training paradigm, we have shown that our approach outperforms all previous joint models. In this section, we aim to take a deeper look and understand why this model performs so well and what contributes to its final performance.\n\n\\subsection{Importance of Typed Text Markers}\n\\label{sec:input-features}\n\nOur first argument is that it is important to build different contextual representations for different pairs of spans and an early fusion of entity type information can further improve performance. To validate the importance of typed text markers, we experiment the following variants on both ACE05 and SciERC when the gold entities are given:\n\n\\vspace{0.5em}\n\\noindent\\textbf{\\textsc{Text}}: We use the span representations defined in the entity model (Section~\\ref{sec:our-model}) and concatenate the hidden representations for the subject and the object, as well as their element-wise multiplication: $[\\mathbf{h}_e(s_i), \\mathbf{h}_e(s_j), \\mathbf{h}_e(s_i) \\odot \\mathbf{h}_e(s_j)]$. This is similar to the relation model in \\newcite{luan2018multi,luan2019general}.\n\n\\noindent\\textbf{\\textsc{TextEType}}: In addition to \\ts{Text}, we concatenate the span-pair representations with entity type embeddings $\\psi(e_i), \\psi(e_j) \\in \\mathbb{R}^{d_E}$ ($d_E$ = 150).\n\n\\noindent\\textbf{\\textsc{Markers}}: We use untyped entity types ($\\langle\\ts{S}\\rangle$, $\\langle\\ts{\/S}\\rangle$, $\\langle\\ts{O}\\rangle$, $\\langle\\ts{\/O}\\rangle$) at the input layer and concatenate the representations of two spans' starting points.\n\n\\noindent\\textbf{\\textsc{MarkersEType}}: In addition to \\ts{Markers}, we concatenate the span-pair representations with entity type embeddings $\\psi(e_i), \\psi(e_j) \\in \\mathbb{R}^{d_E}$ ($d_E$ = 150).\n\n\n\\noindent\\textbf{\\textsc{MarkersELoss}}: We also consider a variant which uses untyped markers but add another FFNN to predict the entity types of subject and object through auxiliary losses. This is similar to how the entity information is used in multi-task learning~\\cite{luan2019general,wadden2019entity}.\n\n\\noindent\\textbf{\\textsc{TypedMarkers}}: This is our final model described in Section~\\ref{sec:our-model}. We use typed markers at the input layer.\n\n\\input{tables\/input_features}\n\nTable~\\ref{tab:input} shows the performance of all the variants and it clearly indicates that different input representations make a real difference in the relation accuracy. Compared to \\ts{Text}, \\ts{TypedMarkers} improved the F1 scores largely by $+5.5\\%$ and $+7.4\\%$ absolute. All the variants of using marker tokens are significantly better than the standard text representations and this suggests the importance of learning different representations with respect to different subject-object pairs. Finally, entity type is useful in improving the relation performance and an early fusion of entity information is particularly effective (\\ts{TypedMarkers} vs \\ts{MarkersEType} and \\ts{MarkersELoss}). We also find that \\ts{MarkersEType} to perform even better than \\ts{MarkersEloss} which suggests that using entity types directly as features is better than using them to provide training signals through auxiliary losses.\n\n\\subsection{Modeling Interactions between Entities and Relations}\nOne main argument for joint models is that modeling the interactions between the two tasks can contribute to each other. In this section, we aim to validate if it is the case in our approach. We first study whether sharing the two representation encoders can improve the performance or not. We train the entity and relation models together by jointly optimizing $\\mathcal{L}_e + \\mathcal{L}_r$. As shown in Table~\\ref{tab:shared-encoder}, we find that simply sharing the encoders hurts both the entity and relation F1. We think this is because the two tasks have different input formats and require different features for predicting entity types and relations, thus using separate encoders indeed learns better task-specific features.\n\\input{tables\/shared_encoder}\n\nIn the previous section, we have already shown that the entity information is useful in the relation model (either entity embeddings, auxiliary loss or input features) and the best way to use it is through typed markers. Next, we aim to investigate whether the relation information can improve the entity performance. To do so, we add an auxiliary loss to our entity model, which concatenates the two span representations as well as their element-wise multiplication (see the \\textsc{Text} variant in Section~\\ref{sec:input-features}) and predicts the relation type between the two spans ($r \\in \\mathcal{R}$ or $\\epsilon$). Through joint training with this auxiliary relation loss, we observe a negligible improvement ($<0.1\\%$) on averaged entity F1 over $5$ runs on the ACE05 development set. Hence, we conclude that relation information does not improve the entity model substantially.\n\nTo summarize our findings, (1) entity information is clearly helpful in predicting relations. However, we don't find enough evidence in our experiments that relation information can improve the entity performance substantially.\\footnote{\\newcite{miwa2016end} observed a slight improvement on entity F1 by sharing the parameters (80.8 $\\rightarrow$ 81.8 F1) on the ACE05 development data. \\newcite{wadden2019entity} observed that their relation propagation layers improved the entity F1 slightly on SciERC but it hurts performance on ACE05.} (2) Simply sharing the encoders does not provide benefits to our approach.\n\n\\input{tables\/error_propagation}\n\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[scale=0.47]{figures\/context_size.pdf}\n \\vspace{-0.6em}\n \\caption{Effect of different context window sizes, measured on the ACE05 development set with the BERT-base model. We use the same entity model (an entity model with $W=100$) to report the relation F1 scores.}\n \\label{fig:context_window}\n \\vspace{-0.4em}\n\\end{figure}\n\\subsection{Mitigating Error Propagation}\n\\label{sec:error-prop}\n\nOne well-known drawback of pipeline training is the error propagation issue. In our final model, we use gold entities (and their types) to train the relation model and the predicted entities during inference and this may lead to a discrepancy between training and testing. In the following, we describe several attempts we made to address this issue.\n\nWe first study whether using predicted entities --- instead of gold entities --- during training can mitigate this issue. We adopt a 10-way jackknifing method, which is a standard technique in many NLP tasks such as dependency parsing~\\cite{agic2017not}. Specifically, we divide the data into $10$ folds and predict the entities in the $k$-th fold using an entity model trained on the remainder. As shown in Table~\\ref{tab:error-propagation}, we find that jackknifing strategy hurts the final relation performance surprisingly. We hypothesize that it is because it introduced additional noise during training.\n\nSecond, we consider using more pairs of spans for the relation model at both training and testing time. The main reason is that in the current pipeline approach, if a gold entity is missed out by the entity model during inference, the relation model will not be able to predict any relations associated with that entity. Following the beam search strategy used in the previous work~\\cite{luan2019general,wadden2019entity}, we consider using $\\lambda n$ ($\\lambda = 0.4$ and $n$ is the sentence length)\\footnote{This pruning strategy achieves a recall of $96.7\\%$ of gold relations on the development set of ACE05.} top spans scored by the entity model. We explored several different strategies for encoding the top-scoring spans for the relation model: (1) typed markers: the same as our main model except that we now have markers e.g., $\\langle \\textsc{S:}\\epsilon\\rangle$, $\\langle \\textsc{\/S:}\\epsilon\\rangle$ as input tokens; (2) untyped markers: in this case, the relation model is unaware of a span is an entity or not; (3) untyped markers trained with an auxiliary entity loss ($e \\in \\mathcal{E}$ or $\\epsilon$). As Table~\\ref{tab:error-propagation} shows, none of these changes led to significant improvements and using untyped markers is especially worse because the relation model struggles to identify whether a span is an entity or not.\n\nIn sum, we don't find any of these attempts improved performance significantly and our simple pipeline training turns out to be a surprisingly effective strategy. We do not argue that this error propagation issue does not exist or cannot be solved, while we will need to explore better solutions to address this issue.\n\n\n\\subsection{Effect of Cross-sentence Context}\n\\label{sec:context}\n\nIn Table~\\ref{tab:main-results}, we demonstrated the improvements from using cross-sentence context on both the entity and relation performance. Finally, we explore the effect of different context sizes $W$ in Figure~\\ref{fig:context_window}. We find that using cross-sentence context clearly improves both entity and relation F1. However, the results don't further increase from $W = 100$ to $W = 300$. In our final models, we use $W=100$ for both the entity model and relation model.\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nIn this paper, we present a very simple and effective approach for end-to-end relation extraction. Our model learns two encoders for entity recognition and relation extraction independently and our experiments show that it outperforms previous state-of-the-art on three standard benchmarks considerably. We conduct extensive analysis to undertand the superior performance of our approach and we validate the importance of learning distinct contextual representations for entities and relations and using entity information as input features for the relation model. We also propose an efficient approximation, obtaining a large speedup at inference time with a small accuracy drop. We hope that this simple model will serve as a very strong baseline and make us rethink the value of joint training in end-to-end relation extraction.\n\n\\section{Training Details}\n\\label{sec:appendix}\n\nWe train our models with BertAdam optimizer. For the entity model, we train the model for $100$ epochs, with a learning rate of 1e-5 for parameters in the pre-trained language models, 5e-4 for other parameters, and a batch size of 16.\nFor the relation model, we follow~\\newcite{joshi2020spanbert} to train the model for $10$ epochs, with a learning rate of 2e-5 and a batch size of 32.\nDuring training, we use a linear scheduler with a warm-up ratio of $0.1$.\nWe implement our models based on HuggingFace's \\ti{Transformers} library~\\cite{Wolf2019HuggingFacesTS}.\n\nWe train our models using a single GPU (NVIDIA GeForce RTX 2080 Ti). It takes about $7$ hours to train the entity model on ACE04 and ACE05 datasets.\nIt takes about $13$ hours to train the relation model on the ACE04 dataset and $16$ hours to train on the ACE05 dataset.\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nA considerable interest in actual research of superconductivity (SC) with high critical temperature \nis focused on the family of doped ferropnictide compounds \\cite{kamihara1, kamihara2} and one \nof their notable distinctions from \"old\" BCS superconductors and more recent doped perovskite \nsystems consists in possibility for a peculiar, so-called extended s-wave symmetry of superconducting\norder parameter which changes its sign between electron and hole segments of the Fermi surface\n\\cite{mazin}. This additional property permits to avoid the fundamental limitation by the Anderson's \ntheorem \\cite{and} for non-magnetic impurities to produce localized impurity levels within the \nsuperconducting band gap \\cite{dzhang, zhang}. At finite, but low enough, impurity concentration, \nsuch levels are expected to give rise to some resonance effects like those well studied in \nsemiconductors at low doping concentrations \\cite{shklo}. Analogous effects in superconductors \nwere theoretically predicted and experimentally discovered for magnetic impurities, either in BCS \nsystems \\cite{shiba,rus,maki} and in the two-band MgB$_2$ system \\cite{con, moca}. In all those \ncases, the breakdown of the Anderson's theorem is only due to the breakdown of the spin-singlet \nsymmetry of an $s$-wave Cooper pair by a spin-polarized impurity, and the main physical interest \nof the considered case of SC iron pnictides from the point of view of disorder in general is the \npossibility for pair-breaking even on non-magnetic impurity scatterers. The latter theoretical prediction \nwas confirmed by the observations of various effects from localized impurity states, for instance, in the \nsuperfluid density (observed through the London penetration length) \\cite{gordon, kitagawa}, transition \ncritical temperature \\cite{guo, li} and electronic specific heat \\cite{hardy}, all mainly due to an emerging \nspike of electronic density of states against its zero value in the initial band gap.\n\nBut it is also known that indirect interactions between random impurity centers of certain type (the \nso-called deep levels at high enough concentrations) in doped semiconductors can lead to formation \nof collective band-like states \\cite{iv, yama}. This corresponds to the Anderson transition in a general \ndisordered system \\cite{and1}, and the emerging new band of quasiparticles in the spectrum can \nessentially change thermodynamics and transport in the doped material \\cite{mott}. An intriguing \npossibility for similar banding of impurity levels within the SC gap \\cite{bal, lokt} was recently discussed \nfor the doped ferropnictides \\cite{pog}. The present work is aimed on a more detailed analysis of the \nband-like impurity states, focused on their observable effects that cannot be produced by localized \nimpurity states. We use the specific form of Green functions for superconducting quasiparticles \nderived in the previous work \\cite{pog} in the general Kubo-Greenwood formalism \\cite{kubo} to obtain \nthe temperature and frequency dependences of optical and thermal conductivity and also of thermoelectric \ncoefficients. These results are compared with the available experimental data and some suggestions are \ndone on possible practical applications of such impurity effects.\n\n\\section{Green functions for disordered SC ferropnictide}\nWe begin from a brief summary of the Green function description of electronic spectrum in\nLaOFeAs with impurities (not necessarily dopants) using the minimal coupling model \n\\cite{dag,tsai} for the non-perturbed Hamiltonian. It considers only 2 types of local Fe \norbitals, $d_{xz}$ (or $x$) and $d_{yz}$ (or $y$), on sites of square lattice with lattice \nparameter $a$ and 4 hopping parameters between nearest neighbors (NNs) and next nearest \nneighbors (NNNs): i) $t_1$ for $xx$ or $yy$ NNs along their orientations, and $t_2$ across \nthem, and ii) $t_3$ for $xx$ or $yy$ NNNs, and $t_4$ for $xy$ NNNs. The resulting band \nHamiltonian is diagonal in quasimomentum ${\\bf k}$ and spin $\\sigma$, but non-diagonal with respect \nto the orbital indices of the 2-spinors $\\psi^\\dagger({\\bf k},\\sigma) = (x^\\dagger_{{\\bf k},\\sigma},\ny^\\dagger_{{\\bf k},\\sigma})$:\n\\begin{equation}\n H_t = \\sum_{{\\bf k},\\sigma} \\psi^\\dagger({\\bf k},\\sigma)\\hat h({\\bf k})\\psi({\\bf k},\\sigma).\n \\label{eq1}\n \\end{equation}\nHere the energy matrix in orbital basis is expanded in Pauli matrices $\\hat\\sigma_i$: \n$\\hat h({\\bf k}) = \\varepsilon_{+,{\\bf k}}\\hat\\sigma_0 + \\varepsilon_{-,{\\bf k}}\\hat\\sigma_3 + \\varepsilon_{xy,{\\bf k}}\n\\hat\\sigma_1$ with the energy factors $\\varepsilon_{\\pm,{\\bf k}} = (\\varepsilon_{x,{\\bf k}} \\pm \\varepsilon_{x,{\\bf k}})\/2$, \nand\n\\begin{eqnarray}\n \\varepsilon_{x,{\\bf k}} & = & - 2t_1 \\cos ak_x - 2t_2 \\cos ak_y - 4t_3 \\cos ak_x \\cos ak_y,\\nonumber\\\\\n \\varepsilon_{y,{\\bf k}} & = & - 2t_1 \\cos ak_y - 2t_2 \\cos ak_x - 4t_3 \\cos ak_x \\cos ak_y,\\nonumber\\\\\n \\varepsilon_{xy,{\\bf k}} & = & - 4t_4 \\sin ak_x \\sin ak_y.\\nonumber\n \\end{eqnarray}\nIt is readily diagonalized at passing from the orbital to subband basis: $ \\hat h_b({\\bf k})\n= \\hat U({\\bf k})\\hat h({\\bf k})\\hat U({\\bf k})^\\dagger$, with the unitary matrix $\\hat U({\\bf k}) =\n\\exp(-i\\hat\\sigma_2\\theta_{\\bf k}\/2)$ and $\\theta_{\\bf k} = \\arctan \\left(\\varepsilon_{xy,{\\bf k}}\/\\varepsilon_{-,{\\bf k}}\\right)$.\nThe resulting eigen-energies for electron and hole subbands are:\n\\begin{equation}\n \\varepsilon_{h,e}({\\bf k}) = \\varepsilon_{+,{\\bf k}} \\pm \\sqrt{\\varepsilon_{xy,{\\bf k}}^2 + \\varepsilon_{-,{\\bf k}}^2},\n \\label{eq2}\n \\end{equation}\nand respective electron and hole segments of the Fermi surface are defined by the equations\n$\\varepsilon_{e,h}({\\bf k}) = \\varepsilon_{\\rm F}$. A reasonable fit to the LaOFeAs band structure by the more\ndetailed LDA calculations \\cite{xu} is attained with the parameter choice (in $|t_1|$ units)\nof $t_1 = -1$, $t_2 = 1.3$, $t_3 = t_4 = -0.85$ \\cite{raghu}.\n\nThe SC state of such multiband electronic system is suitably described in terms of \"multiband\n-Nambu\" 4-spinors $\\Psi_{\\bf k}^\\dagger = \\left(\\alpha_{{\\bf k},\\uparrow}^\\dagger,\\alpha_{-{\\bf k},\\downarrow}, \\beta_{{\\bf k},\\uparrow}\n^\\dagger,\\beta_{-{\\bf k},\\downarrow}\\right)$ with the multiband spinor $\\left(\\alpha_{{\\bf k},\\sigma}^\\dagger,\\beta_{{\\bf k},\\sigma}\n^\\dagger\\right) = \\psi^\\dagger({\\bf k},\\sigma)\\hat U^\\dagger({\\bf k})$, by a 4$\\times$4 extension of the\nHamiltonian Eq. \\ref{eq1} in the form:\n\\begin{equation}\n H_s = \\sum_{{\\bf k},\\sigma} \\Psi_{\\bf k}^\\dagger\\hat h_s({\\bf k})\\Psi_{\\bf k},\n \\label{eq3}\n \\end{equation}\nwhere the 4$\\times$4 matrix $\\hat h_s({\\bf k}) = \\hat h_b({\\bf k})\\otimes\\hat\\tau_3 + \\Delta_{{\\bf k}}\\hat\\sigma_0\n\\otimes\\hat\\tau_1$, includes the Pauli matrices $\\hat\\tau_i$ acting on the Nambu (particle-antiparticle)\nindices in $\\Psi$-spinors. The simplified form for the extended \\emph{s}-wave gap function takes\nconstant values, $\\Delta_{\\bf k} = \\Delta$ on the electron segments and $\\Delta_{\\bf k} = -\\Delta$ on the hole segments.\n\nThe observable values result from the (Fourier transformed) GF 4$\\times$4 matrices $\\hat\nG_{{\\bf k},{\\bf k}'} = \\langle\\langle\\Psi_{{\\bf k}}|\\Psi_{{\\bf k}'}^\\dagger\\rangle\\rangle$, and for the\nnon-perturbed system, Eq. \\ref{eq1}, they are diagonal in quasimomentum: $\\hat G_{{\\bf k},{\\bf k}'}\n= \\delta_{{\\bf k},{\\bf k}'} \\hat g_{\\bf k}$ with\n\\begin{eqnarray}\n \\hat g_{\\bf k} & = & \\frac{\\varepsilon\\hat\\tau_0 + \\varepsilon_e({\\bf k})\n \\hat\\tau_3 + \\Delta\\hat\\tau_1}{2d_{e,{\\bf k}}}\\otimes\\hat\\sigma_e\\nonumber\\\\\n & + & \\frac{\\varepsilon\\hat\\tau_0 + \\varepsilon_h({\\bf k})\\hat\\tau_3 - \\Delta\\hat\\tau_1}{2d_{h,{\\bf k}}}\\otimes\\hat\\sigma_h,\n \\label{eq4}\n \\end{eqnarray}\n$\\hat\\sigma_{e,h} = \\left(\\hat\\sigma_0 \\pm \\hat\\sigma_3\\right)\/2$, $d_{i,{\\bf k}} = \\varepsilon^2 - \\varepsilon_i^2({\\bf k}) - \\Delta^2$.\n\nTo simplify the treatment of impurity perturbations, the band structure is approximated to \nidentical circular electron and hole Fermi segments of radius $k_{\\rm F}$ around respective \npoints ${\\bf K}_{e,h}$ in the Brillouin zone and to similar linear dispersion of normal state \nquasiparticles near the Fermi level $\\varepsilon_{\\rm F}$: $\\varepsilon_e({\\bf k}) - \\varepsilon_{\\rm F} = \\hbar v_{\\rm F}\n\\left(|{\\bf k} - {\\bf K}_e| - k_{\\rm F}\\right)$ and $\\varepsilon_h({\\bf k}) - \\varepsilon_{\\rm F} = -\\hbar v_{\\rm F} \n\\left(|{\\bf k} - {\\bf K}_h| - k_{\\rm F}\\right)$. Moreover, we shall describe the contributions of \nboth segments to overall electronic properties by a single quasimomentum variable $\\xi$ that \nidentifies electron $\\xi_e = \\varepsilon_e({\\bf k}) - \\varepsilon_{\\rm F}$ and hole $\\xi_h = \\varepsilon_h({\\bf k}) - \\varepsilon_{\\rm F}$ \nones.\n\nNext, the Hamiltonian of the disordered SC system is chosen as $H = H_s + H_{imp}$ including \nbesides $H_s $, Eq. \\ref{eq3}, the term due to non-magnetic impurities \\cite{dzhang} on random \nsites ${\\bf p}$ in Fe square lattice with an on-site energy shift $V$ (supposed positive without loss \nof generality). It is written in the multiband-Nambu spinor form as:\n\\begin{equation}\nH_{imp} = {1 \\over N} \\sum_{{\\bf p},{\\bf k},{\\bf k}'} {\\rm e}^{i({\\bf k}' - {\\bf k})\\cdot{\\bf p}}\\Psi_{\\bf k}^\\dagger \n\\hat V_{{\\bf k},{\\bf k}'}\\Psi_{{\\bf k}'},\n \\label{eq5}\n \\end{equation}\nwith the number $N$ of unit cells in the crystal and the 4$\\times$4 scattering matrix \n$\\hat V_{{\\bf k},{\\bf k}'} = V\\hat U^\\dagger({\\bf k})\\hat U({\\bf k}')\\otimes\\hat\\tau_3$. In presence of this \nperturbation, the GFs can be expressed in specific forms depending on whether the considered \nquasiparticle energy falls into the range of band-like or localized states. Namely, for band-like \nstates, the momentum diagonal GF:\n\\begin{equation}\n \\hat G_{\\bf k} = \\hat G_{{\\bf k},{\\bf k}} = (\\hat g_{\\bf k}^{-1} - \\hat \\Sigma_{\\bf k})^{-1},\n \\label{eq6}\n \\end{equation}\ninvolves the self-energy matrix $\\hat \\Sigma_{\\bf k}$ in the form of the so-called renormalized \ngroup expansion \\cite{ilp}:\n\\begin{equation}\n \\hat \\Sigma_{\\bf k} = c\\hat T \\left(1 + c \\hat B_{\\bf k} + \\dots\\right).\n \\label{eq7}\n \\end{equation} \nThis series in powers of impurity concentration $c$ begins from the (k-independent) T-matrix, \n$\\hat T = \\hat V \\left(1 - \\hat G \\hat V\\right)^{-1}$. From the matrices $\\hat V = \n\\hat V_{{\\bf k},{\\bf k}} = V\\hat\\tau_3$ and $\\hat G = N^{-1} \\sum_{\\bf k} \\hat g_{\\bf k} = \\pi \\varepsilon \\rho_{\\rm F}\n\\hat\\tau_0\/\\sqrt{\\Delta^2 - \\varepsilon^2}$ (with the Fermi density of states $\\rho_{\\rm F}$ and the henceforth \nomitted trivial factor $\\hat\\sigma_0$), the T-matrix explicit form is:\n\\begin{equation}\n \\hat T = \\frac{V}{1 + v^2}\\frac{v\\varepsilon\\sqrt{\\Delta^2 - \\varepsilon^2}\\hat\\tau_0 - \\left(\\Delta^2 - \\varepsilon^2\\right)\n \\hat\\tau_3}{\\varepsilon^2 - \\varepsilon_0^2}.\n \\label{eq8}\n \\end{equation} \nwhere $\\varepsilon_0 = \\Delta\/\\sqrt{1 + v^2}$ defines the in-gap impurity levels \\cite{tsai} through the \ndimensionless impurity perturbation parameter $v = \\pi\\rho_{\\rm F}V$. Inside the gap, the \nT-matrix, Eq. \\ref{eq8}, is a real function which can be approximated near the impurity \nlevels $\\pm\\varepsilon_0$ as: $\\hat T \\approx \\gamma^2 \\left(\\varepsilon \\hat\\tau_0 - \\varepsilon_0 \\hat\\tau_3\\right)\/\\left(\\varepsilon^2 \n- \\varepsilon_0^2\\right)$, with the effective coupling constant $\\gamma^2 = V\\varepsilon_0(v\\varepsilon_0\/\\Delta)^2$. In contrary, \noutside the gap it is dominated by its imaginary part: Im$\\hat T = (\\gamma^2\/v\\varepsilon_0) \\varepsilon \\sqrt{\\varepsilon^2 - \n\\Delta^2}\/\\left(\\varepsilon^2 - \\varepsilon_0^2\\right)$.\n\nThe next terms besides unity in the brackets of Eq. \\ref{eq7} describe the effects of indirect \ninteractions between impurities, with $\\hat B_{\\bf k} $ related to pairs and the omitted terms to \ngroups of three and more impurities. The series convergence defines the energy ranges of \nband-like states, delimited by the Mott mobility edges $\\varepsilon_c$ \\cite{mott}. Within the band-like \nenergy ranges, the self-energy matrix can be safely approximated by the T-matrix, $\\hat \\Sigma_{\\bf k} \n\\approx c \\hat T$, and the dispersion laws for corresponding bands at given quasimomentum \n${\\bf k}$ are defined from the $\\hat G_{\\bf k}$ denominator:\n\\begin{eqnarray}\n D_{\\bf k}(\\varepsilon) & = & \\det \\hat G_{\\bf k}^{-1}(\\varepsilon) = \\tilde d_{e,{\\bf k}}(\\varepsilon) \\tilde d_{h,{\\bf k}}(\\varepsilon) \\nonumber\\\\\n & = & \\left(\\tilde\\varepsilon^2 - \\tilde\\xi_e^2 - \\Delta^2\\right)\\left(\\tilde\\varepsilon^2 - \\tilde\\xi_h^2 - \n \\Delta^2\\right),\n \\label{eq9}\n \\end{eqnarray}\nwith the renormalized energy and momenta forms:\n\\begin{eqnarray}\n \\tilde\\varepsilon & = & \\varepsilon\\left(1 - \\frac{c V v}{1 + v^2}\\frac{\\sqrt{\\Delta^2 - \\varepsilon^2}}{\\varepsilon^2 - \\varepsilon_0^2}\\right),\\nonumber\\\\\n&& \\tilde\\xi_j = \\xi_j - \\frac{c V}{1 + v^2} \\frac{\\Delta^2 - \\varepsilon^2}{\\varepsilon^2 - \\varepsilon_0^2}.\\nonumber\n \\end{eqnarray}\nThe roots of the dispersion equation Re $D_{\\bf k}(\\varepsilon) = 0$ define up to 8 subbands: 4 of them \nwith energies near the roots of the non-perturbed denominators $d_{j,{\\bf k}}$ in the $e$- and \n$h$-segments can be called \"principal\" or $pr$-bands, they are similar to quasiparticles in \nthe pure crystal; and other 4, \"impurity\" or $imp$-bands, with energies near $\\pm \\varepsilon_0$ in the \nsame segments are only specific for disordered systems. The dispersion law for $p$-bands\nis presented in the $\\xi$-scale as:\n\\begin{equation}\n \\varepsilon_{pr}(\\xi) \\approx \\sqrt{\\xi^2 + \\Delta^2},\n \\label{eq10}\n \\end{equation} \nand it only differs from the non-perturbed one by the finite linewidth $\\Gamma(\\varepsilon) \\approx c{\\rm Im}\\hat T$, \nso that the validity range of Eq. \\ref{eq10} defined from the known Ioffe-Regel-Mott criterion, \n$\\xi d\\varepsilon_b\/d\\xi \\gtrsim \\Gamma(\\varepsilon_b(\\xi))$ \\cite{IR}, \\cite{mott} as $\\xi \\gtrsim c\/(\\pi\\rho_{\\rm F})$. This defines \nthe mobility edge in closeness to the gap edge,\n\\begin{equation}\n \\varepsilon_c - \\Delta \\sim c^2\/c_0^{4\/3}\\Delta.\n \\label{eq11}\n \\end{equation} \nHere $c_0 = (\\pi\\rho_{\\rm F}\\varepsilon_0)^{3\/2}\/\\left(a k_{\\rm F}\\right)\\sqrt{2v\/(1 + v^2)}$ is the characteristic \nimpurity concentration such that the impurity bands emerge just at $c > c_0$ \\cite{pog}. Their dispersion \n(in $\\xi$) for the exemplar case of positive energies and $e$-segment is approximated as:\n\\begin{equation}\n \\varepsilon_{imp}(\\xi) \\approx \\varepsilon_0 + c\\gamma^2\\frac{\\xi - \\varepsilon_0}{\\xi^2 + \\xi_0^2}.\n \\label{eq12}\n \\end{equation} \n \n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=8cm]{Fig1.eps}\n\\caption{Dispersion laws in the modified quasiparticle spectrum of a SC ferropnictide with \nimpurities. The impurity perturbation parameters were chosen as: $v = 0.5$, $c_0 = 1.3\\cdot \n10^{-3}$, $c_1 = 1.7\\cdot 10^{-2}$, $c = 4\\cdot 10^{-3}$. For compactness, the plot superimposes \nthe blue lines for the in-gap impurity subbands near the electron-like pockets of the Fermi surface \nand red lines for those near the hole-like pockets.}\n\\label{fig1}\n\\end{center}\n\\end{figure}\n\nThe formal upper limit energy by Eq. \\ref{eq12}, $\\varepsilon_+ = \\varepsilon_0 + c\\gamma^2\/[2(\\Delta \n+ \\varepsilon_0)]$, is attained at $\\xi = \\xi_+ = \\varepsilon_0 + \\Delta$ and the lower limit $\\varepsilon_- \n= \\varepsilon_0 - c\\gamma^2\/[2(\\Delta - \\varepsilon_0)]$ at $\\xi_- = \\varepsilon_0 - \\Delta$. But in fact, \nthis dispersion law is only valid until the related mobility edges $\\varepsilon_{c,\\pm}$ whose onset \nnear the $i$-band edges is due to the higher terms in the group expansion, Eq. \\ref{eq7}, and amounts \nto:\n\\begin{eqnarray}\n\\varepsilon_+ - \\varepsilon_{c,+} & \\sim & \\left(\\varepsilon_{max} - \\varepsilon_0\\right)\\left(\\frac{c_0}c\\right)^4,\\nonumber\\\\\n \\varepsilon_{c,-} - \\varepsilon_- & \\sim & \\left(\\varepsilon_0 - \\varepsilon_{min}\\right)\\left(\\frac{c_0}c\\right)^4.\n \\label{eq13}\n \\end{eqnarray}\nThese limitations restrict $\\xi$ to beyond some vicinities of the extremal points: $|\\xi - \\xi_\\pm| \\gtrsim \n\\xi_\\pm\\left(c_0\/c\\right)^2$ (narrow enough at $c \\gg c_0$). Another limitation is that $\\xi$ not be too \nfar from these points: $|\\xi - \\xi_\\pm| \\lesssim \\xi_\\pm(c\/c_0)^4$. A symmetric replica of Eq. \\ref{eq12} \nnear $-\\varepsilon_0$ at the $e$-segment is the impurity subband with the dispersion law $-\\varepsilon_i(\\xi)$. Yet two \nmore impurity subbands near the $h$-segment are described in the unified $\\xi$ frame by the inverted \ndispersion laws $\\pm\\varepsilon_{imp}(-\\xi)$. The overall composition of band-like states in this frame is shown in Fig. \n\\ref{fig1}. It is also important to notice that the above described in-gap impurity band structure is only \njustified until it is narrow enough compared to the SC gap $\\Delta$ itself. From Eq. \\ref{eq12}, this requires \nthat the impurity concentration stays well below the upper critical value\n\\[c_1 = \\pi\\rho_{\\rm F}\\Delta\\sqrt{1 + v^2}.\\]\nthat can amount about few percents. In what follows, the condition $c \\ll c_1$ is presumed.\n\nAt least, for $c < c_0$, all the in-gap states are localized and more adequately described by \nan alternative, the so called non-renormalized group expansion of $\\hat G_{\\bf k}$ (though this \ncase is beyond the scope of the present study) while the principal bands are still defined by \nEqs. \\ref{eq10},\\ref{eq11}.\n\nIn-gap impurity states, either localized and band-like, can produce notable resonance effects \non various thermodynamical properties of disordered superconductors, as transition critical \ntemperature, London penetration length, electronic specific heat, etc. \\cite{pog}. But besides that, \nother effects, only specific for new quasiparticle bands, can be expected on kinetic properties of \nthe disordered material, while the localized impurity states should have practically no effect on \nthem. Such phenomena can be naturally described in terms of the above indicated GF matrices as \nseen in what follows.\n\n\\section{Kubo-Greenwood formalism for multiband superconductor}\n\nThe relevant kinetic coefficients for electronic processes in the considered disordered superconductor \nfollow from the general Kubo-Greenwood formulation \\cite{kubo}, adapted here to the specific multiband \nstructure of Green function matrices. Thus, one of the basic transport characteristics, the (frequency and \ntemperature dependent) electrical conductivity is expressed in this approach as:\n\\begin{eqnarray}\n\\sigma(\\omega,T) & = & \\frac{e^2 }{\\pi} \\int d\\varepsilon\\frac{f(\\varepsilon) - f(\\varepsilon')}\\omega \\int d{\\bf k} v_x({\\bf k},\\varepsilon) \n v_x({\\bf k},\\varepsilon') \\nonumber\\\\\n & \\times & {\\rm Tr}\\left[{\\rm Im}\\hat G_{{\\bf k}}(\\varepsilon){\\rm Im}\\hat G_{{\\bf k}}(\\varepsilon')\\right],\n \\label{eq14}\n \\end{eqnarray}\nfor $\\varepsilon' = \\varepsilon - \\hbar\\omega$ and the electric field applied along the $x$-axis. Besides the common Fermi \noccupation function $f(\\varepsilon) = ({\\rm e}^{\\beta\\varepsilon} + 1)^{-1}$ with the inverse temperature $\\beta = 1\/k_{\\rm B}T$, the \nabove formula involves the generalized velocity function:\n\\begin{equation}\n {\\bf v}({\\bf k},\\varepsilon) = \\left(\\hbar\\frac{\\partial {\\rm Re}D_{\\bf k}(\\varepsilon)}{\\partial \\varepsilon}\\right)^{-1}{\\bf \\nabla}_{\\bf k} \n {\\rm Re}D_{\\bf k}(\\varepsilon).\n \\label{eq15}\n \\end{equation} \nThis function is defined in the whole $\\xi,\\varepsilon$ plane in a way to coincide with the physical quasiparticle \nvelocities for each particular band, Eqs. \\ref{eq9}, \\ref{eq12}, along the corresponding dispersion laws: \n${\\bf v}({\\bf k},\\varepsilon_j({\\bf k})) = \\hbar^{-1}{\\bf \\nabla}_{\\bf k} \\varepsilon_j({\\bf k}) = v_{j,{\\bf k}}$, $j = pr,imp$. The conductivity resulting \nfrom Eq. \\ref{eq13} can be then used for calculation of optical reflectivity.\n\nOther relevant quantities are the static (but temperature dependent) transport coefficients, as the heat \nconductivity:\n\\begin{eqnarray}\n\\kappa(T) & = & \\frac{\\hbar}{\\pi } \\int d\\varepsilon \\frac{\\partial f(\\varepsilon)}{\\partial \\varepsilon} \\varepsilon^2 \\int d{\\bf k} \n \\left[v_x({\\bf k},\\varepsilon)\\right]^2 \\nonumber\\\\\n& \\times & {\\rm Tr}\\left[{\\rm Im}\\hat G_{{\\bf k}}(\\varepsilon)\\right]^2,\n \\label{eq16}\n \\end{eqnarray}\nand the thermoelectric coefficients associated with the static electrical conductivity $\\sigma(T) \\equiv \\sigma(0,T)$ \n\\cite{note}, the Peltier coefficient:\n\\begin{eqnarray}\n \\Pi(T) & = & \\frac{\\hbar e}{\\pi \\sigma(0,T)} \\int d\\varepsilon \\frac{\\partial f(\\varepsilon)}{\\partial \\varepsilon} \\varepsilon \\int \n d{\\bf k} \\left[v_x({\\bf k},\\varepsilon)\\right]^2 \\nonumber\\\\\n& \\times & {\\rm Tr}\\left[{\\rm Im}\\hat G_{{\\bf k}}(\\varepsilon)\\right]^2,\n \\label{eq17}\n \\end{eqnarray}\nand the Seebeck coefficient $S(T) = \\Pi(T)\/T$. All these transport characteristics, though being relatively \nmore complicated from the theoretical point of view than the purely thermodynamical quantities as, e.g., \nspecific heat or London penetration length \\cite{pog}, permit an easier and more reliable experimental \nverification and so could be of higher interest for practical applications of the considered impurity effects \nin the multiband superconductors.\n\nNext, we consider the particular calculation algorithms for the expressions, Eqs. \\ref{eq14}, \n\\ref{eq16}, \\ref{eq17}, beginning from the more involved case of dynamical conductivity, Eq. \n\\ref{eq14}, and then reducing it to simpler static quantities, Eqs. \\ref{eq16}, \\ref{eq17}.\n\n\n\\section{Optical conductivity}\n\nThe integral in Eq. \\ref{eq14} is dominated by the contributions from $\\delta$-like peaks of the \n${\\rm Im}\\hat G_{{\\bf k}}(\\varepsilon)$ and ${\\rm Im}\\hat G_{{\\bf k}}(\\varepsilon ')$ matrix elements. \nThese peaks arise from the above dispersion laws, Eqs. \\ref{eq9}, \\ref{eq11}, thus restricting the \nenergy integration to the band-like ranges: $|\\varepsilon| > \\varepsilon_c$ for the $pr$-bands and \n$\\varepsilon_{c,-} < |\\varepsilon| < \\varepsilon_{c,+}$ for the $imp$-bands. Regarding the occupation \nnumbers $f(\\varepsilon)$ and $ f(\\varepsilon')$ at reasonably low temperatures $k_{\\rm B}T \\ll \n\\Delta,\\varepsilon_0$, the most effective contributions correspond to positive $\\varepsilon$ values, \neither from $pr$- or $imp$-bands, and to negative $\\varepsilon'$ values from their negative counterparts, \n$pr'$ or $imp'$. There are three general kinds of such contributions: i) $pr-pr'$, due to transitions between \nthe principal bands, similar to those in optical conductivity by the pure crystal (but with a slightly shifted \nfrequency threshold: $\\hbar\\omega \\geq 2\\varepsilon_c$), ii) $pr-imp'$ (or $imp-pr'$), due to combined \ntransitions between the principal and impurity bands within the frequency range $\\hbar\\omega \\geq \n\\varepsilon_c + \\varepsilon_{c,-}$, and iii) $imp-imp'$, due to transitions between the impurity bands within \na narrow frequency range of $2\\varepsilon_{c,-} < \\hbar\\omega < 2\\varepsilon_{c,+}$. The frequency-momentum \nrelations for these processes and corresponding peaks are displayed in Fig. \\ref{fig2}. The resulting optical \nconductivity reads $\\sigma(\\omega,T) = \\sum_\\nu \\sigma_\\nu(\\omega,T)$ with $\\nu = pr-pr',\\,imp-imp'$, and \n$imp-pr'$.\n\nFor practical calculation of each contribution, the relevant matrix Im$\\hat G_{\\bf k}(\\varepsilon)$ (within \nthe band-like energy ranges) can be presented as Im$\\hat G_{\\bf k}(\\varepsilon) = \\hat N(\\varepsilon,\\xi) {\\rm Im}\n\\left[D_{\\bf k}(\\varepsilon)^{-1}\\right]$ where the numerator matrix:\n\\begin{equation}\n \\hat N(\\varepsilon,\\xi) = {\\rm Re}\\left(\\tilde\\varepsilon + \\tilde\\xi \\hat \\tau_3 + \\Delta \\hat \\tau_1\\right),\n \\label{eq18}\n \\end{equation} \nis a smooth enough function while the above referred peaks result from zeros of Re$D_{\\bf k}(\\varepsilon)$. \nNow, the quasimomentum integration in Eq. \\ref{eq14} under the above chosen symmetry of \nFermi segments spells as $\\int d{\\bf k} = 2(h v_{\\rm F})^{-1}\\int d\\varphi \\int d\\xi$ where the factor 2 \naccounts for identical contributions from $e$- and $h$-segments. The azimuthal integration \ncontributes by the factor of $\\pi$ (from $x$-projections of velocities) and the most important radial \nintegration is readily done after expanding its integrand in particular pole terms:\n\\begin{eqnarray}\n v(\\xi,\\varepsilon)v(\\xi,\\varepsilon') & {\\rm Tr} & \\left[{\\rm Im}\\hat G(\\xi,\\varepsilon){\\rm Im}\\hat G(\\xi,\\varepsilon')\\right] \n \\nonumber\\\\\n& = & \\sum_\\alpha A_\\alpha(\\varepsilon,\\varepsilon') \\delta\\left(\\xi - \\xi_\\alpha\\right),\n \\label{eq19}\n \\end{eqnarray}\n where $v(\\xi,\\varepsilon) = |{\\bf v}({\\bf k},\\varepsilon)|$ and $\\hat G(\\xi,\\varepsilon') \\equiv \\hat G_{\\bf k}(\\varepsilon')$ \n define the respective residues:\n \\begin{equation}\n A_\\alpha(\\varepsilon,\\varepsilon') = \\pi v_\\alpha v'_\\alpha \\frac{\\tilde\\varepsilon\\tilde\\varepsilon' + \\tilde\\xi\\tilde\\xi' + \n \\Delta^2}{\\prod_{\\beta \\neq \\alpha}\\left(\\xi_\\alpha - \\xi_\\beta\\right)}.\n \\label{eq20}\n \\end{equation} \n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=8cm]{Fig2.eps}\n\\caption{Configuration of the poles $\\xi_j$ of GF's contributing to different types of optical conductivity \nprocesses over one part (electronic pocket) of the quasiparticles spectrum by Fig. \\ref{fig1}.}\n\\label{fig2}\n\\end{center}\n\\end{figure}\n\nHere $v_\\alpha \\equiv v\\left(\\varepsilon,\\xi_\\alpha\\right)$, $v'_\\alpha \\equiv v\\left(\\varepsilon',\\xi_\\alpha\\right)$, and \nthe indices $\\alpha,\\beta$ run over all the poles of the two Green functions. As follows from Eqs. \\ref{eq10}, \\ref{eq12} \nand seen in Fig. \\ref{eq2}, there can be two such poles of $\\hat G(\\xi,\\varepsilon)$ related to band-like states with \npositive $\\varepsilon$ and respective quasi-momentum values denoted as $\\xi_{1,2}(\\varepsilon)$. For energies \nwithin the $pr$-band, $\\varepsilon > \\varepsilon_c$, they are symmetrical: \n\\begin{equation}\n \\xi_{1,2}(\\varepsilon) \\approx \\pm \\sqrt{\\varepsilon^2 - \\Delta^2},\n \\label{eq21}\n \\end{equation} \nwhile within the $imp$-band at $\\varepsilon_{c,-} < \\varepsilon < \\varepsilon_{c,+}$, their positions are asymmetrical:\n\\begin{equation}\n \\xi_{1,2}(\\varepsilon) \\approx \\frac{c\\gamma^2 \\mp 2\\varepsilon_0\\sqrt{\\left(\\varepsilon_+ - \\varepsilon\\right)\n \\left(\\varepsilon - \\varepsilon_-\\right)}}{2\\left(\\varepsilon - \\varepsilon_0\\right)}.\n \\label{eq22}\n \\end{equation} \nNotice also that, within the $imp$-band, there is a narrow vicinity of $\\varepsilon_0$ of $\\sim c_0^{1\/3} (c_0\/c)^3 \\varepsilon_0$ \nwidth where only the $\\xi_1$ pole by Eq. \\ref{eq22} is meaningful and the other contradicts to the IRM criterion (so that there is \nno band-like states with that formal $\\xi_2$ values in this energy range). Analogous poles of $\\hat G(\\xi,\\varepsilon')$ at negative \n$\\varepsilon'$ are referred to as $\\xi_{3,4}(\\varepsilon')$ in what follows. Taking into account a non-zero Im$D_{\\bf k}(\\varepsilon)$ \n(for the $imp$-band, it is due to the non-trivial terms in the group expansion, Eq. \\ref{eq7}), each $\\alpha$th pole becomes a \n$\\delta$-like peak with an effective linewidth $\\Gamma_\\alpha$ but this value turns to be essential (and will be specified) only at \ncalculation of static coefficients like Eqs. \\ref{eq16}, \\ref{eq17}.\n\nSince four peaks in Eq. \\ref{eq19} for optical conductivity are typically well separated, the $\\xi$-integration \nis trivially done considering them true $\\delta$-functions, then the particular terms in $\\sigma(\\omega,T)$ follow as the \nenergy integrals:\n\\begin{equation}\n \\sigma_\\nu(\\omega,T) = 2e^2 \\int_{\\varepsilon_{\\nu,-}}^{\\varepsilon_{\\nu,+}} d\\varepsilon\\frac{f(\\varepsilon) - f(\\varepsilon')}\n \\omega \\sum_{\\alpha=1}^4 A_\\alpha(\\varepsilon,\\varepsilon'),\n \\label{eq23}\n \\end{equation} \nwhere $\\nu$ takes the values $pr-pr'$, $imp-pr'$, or $imp-imp'$ and the limits $\\varepsilon_{\\nu,\\pm}$ should assure that both \n$\\varepsilon$ and $\\varepsilon'$ are kept within the respective band-like energy ranges. \n\nThus, in the $pr-pr'$ term, the symmetry of the poles $\\xi_{1,2}(\\varepsilon)$ and $\\xi_{3,4}(\\varepsilon')$ by Eq. \\ref{eq21} and the\nsymmetry of $pr$- and $pr'$-bands themselves defines their equal contributions, then using simplicity of the generalized velocity \nfunction $v(\\xi,\\varepsilon) = \\xi\/\\varepsilon$ and the non-renormalized energy and momentum variables, $\\tilde\\varepsilon \\to \n\\varepsilon$, $\\tilde\\xi \\to \\xi$, the energy integration between the limits $\\varepsilon_{pr-pr',-} = \\varepsilon_c$ and \n$\\varepsilon_{pr-pr',+} = \\hbar\\omega - \\varepsilon_c$ provides its explicit analytic form as $\\sigma_{pr-pr'}(\\omega,T) = \n\\sigma_{pr-pr'}(\\omega,0) - \\sigma_{pr-pr',T}(\\omega)$. Here the zero-temperature limit value is:\n\\begin{widetext}\n\\begin{eqnarray}\n \\sigma_{pr-pr'}(\\omega,0) & \\approx & \\sigma_0\\frac{2\\omega_c}{\\omega^2} \\left\\{\\sqrt{4\\omega^2 - \\omega_c^2} \\ln\\left[ 2 \n \\frac{\\omega(2\\omega - \\omega_c) + \\sqrt{\\omega(\\omega - \\omega_c) (4\\omega^2 - \\omega_c^2)}}{\\omega_c^2} - 1\\right] \n \\right.\\nonumber\\\\\n& + & \\left. 2\\omega \\ln\\left[2\\frac{\\omega - \\sqrt{\\omega(\\omega - \\omega_c)}}{\\omega_c} - 1\\right] - 2\\sqrt{\\omega(\\omega - \n \\omega_c)} \\right\\},\n \\label{eq24}\n \\end{eqnarray}\n\\end{widetext}\nwith the characteristic scale $\\sigma_0 = e^2\/\\Delta^2$ and simple asymptotics: \n\\begin{eqnarray}\n \\sigma_{pr-pr'}(\\omega,0) & \\approx & (2\/3)\\sigma_0(\\omega \/\\omega_c -1 )^{3\/2}, \\quad \\omega - \\omega_c \\ll \\omega_c,\\nonumber\\\\\n \\sigma_{pr-pr'}(\\omega,0) & \\approx & \\sigma_0(32\\omega_c\/\\omega)\\ln(2\\omega\/\\omega_c), \\quad\\quad\\quad \\omega \\gg \n \\omega_c,\\nonumber\n \\end{eqnarray} \nwith respect to the threshold frequency $\\omega_c = 2\\varepsilon_c\/\\hbar$, reaching the maximum value $\\approx 1.19\\sigma_0$ \nat $\\omega \\approx 2.12\\omega_c$ as seen in Fig. 3. The (small) finite-temperature correction to the above value:\n\\begin{widetext}\n\\begin{eqnarray}\n \\sigma_{pr-pr',T}(\\omega) \\approx \\sigma_0\\frac{2\\omega_c^2{\\rm e}^{-\\beta\\Delta}} {\\beta\\hbar(\\omega - \\omega_c)\\omega\\sqrt\\Delta}\n &&\\left[\\frac{\\sqrt{\\hbar\\omega}} \\Delta \\left(1 - \\frac{F(\\sqrt{\\beta\\hbar(\\omega - \\omega_c)})} {\\sqrt{\\beta\\hbar(\\omega - \\omega_c)}}\\right) \n \\right.\\nonumber\\\\\n && \\qquad\\qquad + \\left.\\frac{\\sqrt{2\\Delta}}{\\hbar\\omega - \\Delta}\\left(\\frac{\\sqrt \\pi}2 \\frac{{\\rm erf} (\\sqrt{\\beta\\hbar(\\omega - \\omega_c)})}\n {\\sqrt{\\beta\\hbar(\\omega - \\omega_c)}} - {\\rm e}^{-\\beta\\hbar(\\omega - \\omega_c)}\\right)\\right],\n \\label{eq25}\n \\end{eqnarray}\n\\end{widetext}\ninvolves the Dawson function $F(z) = \\sqrt\\pi {\\rm e}^{-z^2} {\\rm erf}(iz)\/(2i)$ and the error function ${\\rm erf}(z)$ \\cite{abst}. \n\nCalculation of the $imp-pr'$-term is more complicated since asymmetry of the $imp$-band poles $\\xi_{1,2}(\\varepsilon)$ by Eq. \n\\ref{eq22} and their non-equivalence to the symmetric poles $\\xi_{3,4}(\\varepsilon')$ of the $pr'$-band analogous to Eq. \n\\ref{eq21}. More complicated expressions also define the generalized velocity function within the $imp$-band range: \n\\begin{equation}\n \\hbar v(\\xi,\\varepsilon) = \\frac{c\\gamma^2 - \\xi(\\varepsilon - \\varepsilon_0)}{\\varepsilon(\\varepsilon - \\varepsilon_0 - c\\gamma^2\/\n \\varepsilon_0)},\n \\label{eq26}\n \\end{equation} \nand the energy integration limits: $\\varepsilon_{imp-pr',-} = \\varepsilon_{c,-}$ and $\\varepsilon_{imp-pr',+} = \\min[\\varepsilon_{c,+},\n\\hbar\\omega - \\varepsilon_c]$. Then the function $\\sigma_{imp-pr'}(\\omega,T)$ follows from a numerical integration in Eq. \\ref{eq23} and, as \nseen in Fig. 3, it has a lower threshold frequency $\\omega_c' = \\varepsilon_c + \\varepsilon_{c,-}$ than the $pr-pr'$-term. Above this threshold, \nit starts to grow linearly as $\\sim (\\omega\/\\omega_c'-1)c^{5\/2}c_0^{-5\/3}\\sigma_0$ and, for the impurity concentrations within the \"safety range\", \n$c \\ll c_1 \\sim c_0^{2\/3}$, becomes completely dominated by the $pr-pr'$-function, Eq. \\ref{eq24} above its threshold $\\omega_c$.\n\nFinally, the $imp-imp'$-term is obtained with a similar numerical routine on Eq. \\ref{eq23}, using Eq. \\ref{eq22} either for \nthe poles $\\xi_{1,2}(\\varepsilon)$ by the $imp$-band and for the $\\xi_{3,4}(\\varepsilon')$ by the $imp'$-band and Eq. \\ref{eq25} for respective \ngeneralized velocities while the energy integration limits in this case are $\\varepsilon_{imp-imp',-} = \\varepsilon_{c,-}$ and $\\varepsilon_{imp-imp',+} = \n\\min[\\varepsilon_{c,+}, \\hbar\\omega - \\varepsilon_{c.-}]$. The resulting function $\\sigma_{imp-imp'}(\\omega,T)$ occupies the narrow frequency band from \n$\\omega_{imp-imp',-} = 2 \\varepsilon_{c,-}\/\\hbar$ to $\\omega_{imp-imp',+} = 2\\varepsilon_{c,+}\/\\hbar$ (Fig. \\ref{fig3}) and its asymptotics near these \nthresholds and in the zero-temperature limit are obtained analytically as:\n\\begin{equation}\n \\sigma_{imp-imp'}(\\omega,0) \\approx \\sigma_0\\frac{16c^{7\/2}\\gamma^7}{3\\sqrt 2 \\xi_-^7}\\left(\\frac{\\omega - \\omega_{-}}{\\omega_{-}}\\right)^{3\/2},\n \\label{eq27}\n \\end{equation} \nat $0 < \\omega - \\omega_{-} \\ll \\omega_{-}$ and a similar formula for $0 < \\omega_+ - \\omega \\ll \\omega_+$ only differs from it by the change: \n$\\xi_-\\to\\xi_+$ and $\\omega_-\\to\\omega_+$. \n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=8cm]{Fig3.eps}\n\\caption{General picture of the optical conductivity showing three types of contributions for the impurity perturbation parameters \nas chosen in Fig. \\ref{fig1}.}\n\\label{fig3}\n\\end{center}\n\\end{figure}\n\nThen the maximum contribution by the $imp-imp'$-term is estimated by extrapolation of the above asymptotics to the \ncenter of the impurity band: $|\\omega - \\omega_\\pm| \\sim |\\omega_0 - \\omega_\\pm|$, resulting in: $\\sigma_{imp-imp',max} \n\\sim \\sigma_0 c^5 c_0^{-10\/3} (\\xi_+\/\\xi_-)^{7\/2}$. This estimate shows that the narrow $imp-imp'$-peak of optical conductivity \naround $\\omega \\approx 2\\epsilon_0\/\\hbar$ can, unlike the \"combined\" $imp-pr'$-term, can become as intense or even more \nthan the maximum of \"principal\" $pr-pr'$ intensity, Eq. \\ref{eq24}, if the small factor $\\sim (c\/c_1)^5$ be overweighted by the \nnext factor $(\\xi_+\/\\xi_-)^{7\/2}$. The latter is only possible if the impurity perturbation is {\\it weak} enough: $v \\ll 1$. Then the \nratio $\\xi_+\/\\xi_-$ turns $\\approx (2\/v)^2 \\gg 1$ and can really overweight the concentration factor if the impurity concentration \n$c$ reaches $\\sim c_1 (v\/2)^{7\/5} \\ll c_1$, that is quite realistic within the \"safety\" range $c \\ll 1$. The overall picture of optical \nconductivity for an example of weakly coupled, $v = 0.5$, impurities at high enough concentration $c = 4c_0$ is \nshown in Fig. \\ref{fig3}. The expressed effect of \"giant\" optical conductivity by the in-gap impurity excitations could be \ncompared with the well known Rashba enhancement of optical luminescence by impurity levels at closeness to \nthe edge of excitonic band \\cite{rashba} or with the huge impurity spin resonances in magnetic crystals \\cite{ilp}, \nbut with a distinction that it appears here in a two-particle process instead of the above mentioned single-particle \nones.\n\n\\section{Static kinetic coefficients}\n\nNow we can pass to the relatively simpler calculation of the kinetic coefficients in the static limit of $\\omega \\to 0$. \nTo begin with, consider the heat conductivity, Eq. \\ref{eq16}, where the momentum integration at coincidence \nof the above mentioned poles $\\xi_{1.3}$ and $\\xi_{2.4}$ is readily done using the general convolution formula:\n\\begin{equation}\n \\int L_{\\Gamma_j}\\left(\\xi - \\xi_j\\right) L_{\\Gamma_k'}\\left(\\xi- \\xi_{k}'\\right)d\\xi = L_{\\Gamma_j + \\Gamma_k'}\\left(\\xi_j - \\xi_k'\\right),\n \\label{eq28}\n \\end{equation} \nfor two Lorentzian fuctions $L_\\Gamma(\\xi) = \\Gamma\/(\\xi^2 + \\Gamma^2)$, and in the limit of $\\xi_i = \\xi_k'$ and $\\Gamma_j = \\Gamma_k'$ \nobtaining simply $(2\\Gamma_j)^{-1}$, a \"combined lifetime\". This immediately leads to a Drude-like formula for heat \nconductivity as a sum of principal and impurity terms, $\\kappa(T) = \\kappa_{pr}(T) + \\kappa_{imp}(T)$, each of them given by:\n\\begin{eqnarray}\n \\kappa_{pr}(T) & = & \\frac{\\hbar(1 + v^2)}{\\pi cVv} \\int_{\\varepsilon_c}^{\\infty} d\\varepsilon \\frac{\\partial f(\\varepsilon)}{\\partial \\varepsilon} \\frac{\\varepsilon\\left(\\varepsilon^2 \n - \\varepsilon_0^2\\right)}{\\sqrt{\\varepsilon^2 - \\Delta^2}}\\nonumber\\\\\n& \\approx & \\frac{\\hbar\\rho_{\\rm F}\\Delta^2}{c} \\sqrt{ \\frac{\\pi\\beta\\Delta}{2}}\\exp (-\\beta\\Delta),\n \\label{eq29}\n \\end{eqnarray}\n and:\n \\begin{eqnarray}\n \\kappa_{imp}(T) & \\approx & \\frac{\\hbar}{\\pi\\left(\\varepsilon_{c,+} - \\varepsilon_{c,-}\\right)} \\left(\\frac{c}{c_0}\\right)^4\\int_{\\varepsilon_{c,-}}^{\\varepsilon_{c,+}} \n d\\varepsilon \\frac{\\partial f(\\varepsilon)}{\\partial \\varepsilon}\\varepsilon^2\\nonumber\\\\\n& \\approx & \\frac{\\hbar}{\\pi} \\left(\\frac{c}{c_0}\\right)^4 \\beta\\varepsilon_0^2\\exp (-\\beta\\varepsilon_0).\n \\label{eq29}\n \\end{eqnarray}\nThen the comparison of Eqs. \\ref{eq28} and \\ref{eq29} shows that the impurity contribution to the heat conductance \n$\\kappa_{imp}$ for impurity concentrations $c$ above the critical value $c_0$ turns to dominate over the principal contribution \n$\\kappa_{pr}$ at all the temperatures (of course, below the critical transition temperature). Such strong impurity effect is \ncombined from enhanced thermal occupation of impurity states and from their growing lifetime as $\\sim c^3$ against \nthe decreasing as $\\sim 1\/c$ lifetime in the principal band. \n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=8cm]{Fig4.eps}\n\\caption{Logarithmic plots for two contributions to the heat conductivity shows domination of the impurity term at all the \ntemperatures where SC itself exists. The same impurity perturbation and concentration parameters are used as in Fig. \\ref{fig1}}\n\\label{fig4}\n\\end{center}\n\\end{figure}\n\nSimilar strong impurity effects should also follow for the static electric conductivity $\\sigma(0,T)$ (see \\cite{note}) and for the \nthermoelectric Peltier and Seebeck coefficients, Eq. \\ref{eq17}. All of them can be considered as fully due to the \ncorresponding impurity contributions and the temperature dependencies of thermoelectric coefficients should be \nnon-exponential: $\\Pi(T) \\approx \\Pi(0) = $ const, and $S(T) \\approx \\Pi(0)\/T$, alike the non-perturbed case but at much \nhigher level. Finally, it is important to underline that the above predictions are only for impurity concentrations above \nthe critical value, $c \\gtrsim c_0$, while the system transport properties should stay almost non-affected by impurities \nbelow this concentration, $c < c_0$. Fig. \\ref{fig4} demonstrates these differences between temperature dependencies of static \nconductivities and of thermoelectric coefficients for low and high concentrations of impurities at the choice of perturbation \nparameter as $v = 1$. Such drastic changes of transport behavior are of interest for experimental verification in properly \nprepared samples of SC ferropnictides with controlled concentration of specific impurities. \n\n\n\\section{Conclusive remarks}\nIn conclusion, the essential modification of quasiparticle spectra in a SC ferropnictide with impurities of simplest \n(local and non-magnetic) perturbation type is expected, consisting in formation of localized in-gap impurity states \nand their development into specific narrow bands of impurity quasiparticles at impurity concentration above a certain \n(quite low) critical value $c_0$ and leading to a number of effects in the system observable properties. Besides the \npreviously discussed thermodynamical effects, expected to appear at all impurity concentrations, that is either due \nto localized or band-like impurity states, a special interest is seen in studying the impurity effects on electronic transport \nproperties of such systems, only affected by the impurity band-like states. It was shown above that the latter effects can \nbe very strongly pronounced, either for high-frequency transport and for static transport processes. In the first case, the \nimpurity effect is expected to most strongly reveal in a narrow peak of optical conductance at its closeness to the edge \nof conductance band in non-perturbed crystal, resembling the known resonance enhancement of impurity absorption \n(or emission) processes near the edge of main quasiparticle band in normal systems, here it would be possible if the \nimpurity perturbation be weak enough. The static transport coefficients at overcritical impurity concentrations are also \nexpected to be strongly enhanced compared to those in a non-perturbed system, including the thermoelectric Peltier \nand Seebeck coefficients. The experimental verifications of these predictions would be of evident interest, since they \ncan open perspectives for important practical applications, e.g., in narrow-band microwave devices or advanced \nlow-temperature sensors, but this would impose rather hard requirements on the quality and composition of the necessary \nsamples, they should be extremely pure aside the extremely low (by common standards) and well controlled contents \nof specially chosen and uniformly distributed impurity centers within the SC iron-arsenic planes of a ferropnictide \ncompound. This situation can be compared to the requirements on doped semiconductor devices and hopefully should \nnot be a real problem for modern lab technologies.\n\n\\section{Acknowledgements}\nY.G.P. and M.C.S. acknowledge the support of this work through the Portuguese FCT project PTDC\/FIS\/101126\/2008.\nV.M.L. is grateful to the EU Research Programs for a partial support through the grant FP7-SIMTECH No 246937.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}