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+{"text":"\n\\section{Introduction} \\label{sec:intro}\nDense prediction tasks such as semantic segmentation~\\cite{cordts2016cityscapes,neuhold2017mapillary,yu2020bdd100k} and boundary detection~\\cite{silberman2012indoor,arbelaez2010contour} are fundamental enablers for many computer vision applications. \nSemantic segmentation requires predictors to absorb intra-class variability while establishing inter-class decision boundaries.\nBoundary detection also requires an understanding of fine-grained scene details and object-level boundaries.\nA popular solution is to exploit the multi-scale representations to balance preserving spatial details from shallower features and maintaining relevant semantic context in deeper ones.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.47\\textwidth]{images\/Cropped_Teaser.pdf}\n\t\\caption{ Attentive Feature Aggregation. $F_s$ is the shallower input feature and $F_d$ is the deeper one. We use attention to aggregate different scale or level information and gain aggregated feature $F_\\mathtt{agg}$ with rich representation.}\n\t\\label{fig:teaser}\n\\end{figure}\n\nThere are two major approaches to obtaining effective multi-scale representations.\nDilated convolutions~\\cite{yu2015multi} can aggregate context information while preserving spatial information.\nMost of the top performing segmentation methods adopt this approach~\\cite{chen2018encoder,zhao2017pyramid,yu2017dilated,wang2020axial} to extract contextual pixel-wise information.\nThe drawback is the extensive usage of layer memory for storing high-resolution feature maps.\nAn alternative approach is to progressively downsample the layer resolution as in image classification networks and then upsample the resolution by aggregating information from different layer scales with layer concatenations~\\cite{long2015fully,yu2018deep,ronneberger2015u}.\nMethods using this approach achieve state-of-the-art results with reduced computational efforts and fewer parameters~\\cite{wang2020deep}.\nEven though many works design new network architectures to effectively aggregate multi-scale information, the predominant aggregation operations are still feature concatenation or summation~\\cite{ronneberger2015u,long2015fully,wang2020deep,yu2018deep,zhao2018icnet}.\nThese linear operations do not consider feature interactions or selections between different levels or scales.\n\nWe propose \\emph{Attentive Feature Aggregation} (AFA) as a nonlinear feature fusion operation to replace the prevailing tensor concatenation or summation strategies.\nOur attention module uses both spatial and channel attention to learn and predict the importance of each input signal during fusion.\nAggregation is accomplished by computing a linear combination of the input features at each spatial location, weighted by their relevance.\nCompared to linear fusion operations, our AFA module can attend to different feature levels depending on their importance.\nAFA introduces negligible computation and parameter overhead, and can be easily used to replace fusion operations in existing methods such as skip connections.\nFig.~\\ref{fig:teaser} illustrates the concept of AFA.\n\nAnother challenge of dense prediction tasks is that the fine detail is better handled in scaled-up pictures while the coarse one is contrary.\nThe multi-scale inference~\\cite{chen2017deeplab,chen2016attention,tao2020hierarchical} has become a common approach to eliminate this trade-off, and using an attention mechanism is now a best practice.\nInspired by neural volume rendering~\\cite{drebin1988volume,mildenhall2020nerf}, we extend our AFA to \\emph{Scale-Space Rendering} (SSR) as a novel attention mechanism to fuse multi-scale predictions.\nWe treat the prediction from each scale as sampled data in the scale-space and leverage the volume-rendering formulation to design coarse-to-fine attention and render the final results.\nOur SSR is proved to being robust against the gradient vanishing and saving resource during training and thus achieves higher performance.\n\nWe demonstrate the effectiveness of AFA when applied to a wide range of existing networks on both semantic segmentation and boundary detection benchmarks.\nWe plug our AFA module into various popular segmentation models: FCN~\\cite{long2015fully}, U-Net~\\cite{ronneberger2015u}, HRNet~\\cite{wang2020deep}, and Deep Layer Aggregation (DLA)~\\cite{yu2018deep}.\nExperiments on several challenging semantic segmentation datasets including Cityscapes~\\cite{cordts2016cityscapes} and BDD100K~\\cite{yu2020bdd100k} show that AFA can significantly improve the segmentation performance of each representative model.\nAdditionally, \\emph{AFA-DLA} has competitive results compared to the state-of-the-art models despite having fewer parameters and using less computation.\n\nFurthermore, on boundary detection datasets NYUDv2~\\cite{silberman2012indoor} and BSDS500~\\cite{arbelaez2010contour}, AFA-DLA achieves the new state-of-the-art performances.\nWe conduct comprehensive ablation studies to validate the advantages of each component of our AFA module.\nOur source code will be released.\n\n\\section{Related Work} \\label{related_work}\nRecent semantic segmentation and boundary detection methods are based on fully convolutional networks~\\cite{long2015fully} and focus on incorporating different components to the architecture to improve the performance.\n\n\\parsection{Multi-Scale Context.}\nTo better handle fine details, segmentation models use convolutional trunks with low output strides.\nHowever, this limits the receptive field and the semantic information contained in the final feature.\nSome works utilize dilated backbones~\\cite{yu2015multi} and multi-scale context~\\cite{yang2018denseaspp,lin2019zigzagnet,fu2019adaptive} to address this problem.\nPSPNet~\\cite{zhao2017pyramid} uses a Pyramid Pooling Module~(PPM) to generate multi-scale context and fuse them as the final feature.\nThe DeepLab models~\\cite{chen2018encoder} use Atrous Spatial Pyramid Pooling (ASPP) to assemble context from multiple scales, yielding denser and wider features.\nIn contrast, our AFA-DLA architecture extensively uses attention to conduct multi-scale feature fusion to increase the receptive field without using expensive dilated convolutions.\nThus, our model can achieve comparable or even better performance with much less computation and fewer parameters.\n\n\\parsection{Feature Aggregation.}\nAggregation is widely used in the form of skip connections or feature fusion nodes in most deep learning models~\\cite{long2015fully,ronneberger2015u,wang2020deep}.\nThe Deep Layer Aggregation~\\cite{yu2018deep} network shows that higher connectivity inside the model can enable better performance with fewer parameters, but its aggregation is still limited to linear operators.\nRecently, some works have explored better aggregation of feature pyramid multi-scale features.\nGFF~\\cite{li2020gated} uses gate maps to fuse multi-level features and dense feature pyramid to generate the final representation.\nFaPN~\\cite{huang2021fapn} improves the original FPN architecture with feature alignment and selection during fusion.\nIn contrast, our AFA module leverages extracted spatial and channel information during aggregation to efficiently select the essential features and increase the receptive field simultaneously.\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[width=0.85\\textwidth]{images\/Overview.pdf}\n\t\\caption{ \n        (a) Overview of AFA-DLA with SSR. We use two scales [0.5, 1.0] during training and more scales during inference to pursue higher performance. (b) Binary fusion module for input feature $F_s$ and $F_d$. SA denotes our spatial attention module and generates spatial attention $a_s$. CA stands for our channel attention module and responsible for channel attention $a_c$. (c) Multiple feature fusion module for three input features $F_1$, $F_2$, and $F_3$. \\emph{SA $\\times$ CA} represents computing spatial attention $a_s$ and channel attention $a_c$ first and then using element-wise multiplication to get the attention $a_i$ for $F_i$.\n\t}\n\t\\label{fig:overview}\n\\end{figure*}\n\n\\parsection{Attention Module.}\nAttention mechanisms are commonly used to improve the expressiveness of features for downstream tasks.\nSpatial attention captures the inter-spatial relationship of features, while channel attention associates different semantic information across input features.\nCBAM~\\cite{woo2018cbam} and SCA-CNN~\\cite{chen2016sca} uses channel and special self-attention to perform adaptive feature refinement and improves convolutional networks in image classification, object detection and image captioning.\nDANet~\\cite{fu2019dual} appends two separate branches with the proposed spatial and channel attention module on top of dilated FCN.\nThey refine the features in spatial and channel dimension respectively and fuse them through the summation operator as the final representation and boost the performance in the semantic segmentation task.\nInspired by the aforementioned works, we propose attentive feature aggregation, AFA, to leverage attention mechanism to perform feature aggregation from different levels.\nOur AFA module generates attention with respect to the property of input features and can be directly adapted to in the popular architectures.\nFurthermore, without using heavy self-attention like DANet, AFA can be extensively used at negligible computational and parameter overhead.\n\n\\parsection{Multi-Scale Inference.}\nMany computer vision tasks leverage multi-scale inference to get higher performance.\nThe most common way to fuse the multi-scale results is to use average pooling~\\cite{chen2017deeplab,yu2018deep,li2020improving}, but it applies the same weighting to each scale.\nSome approaches use an explicit attention model~\\cite{chen2016attention,yang2018attention} to learn a suitable weighting for each scale.\nHowever, the main drawback is the increased computational requirement for evaluating multiple scales.\nTo overcome this problem, HMA~\\cite{tao2020hierarchical} proposes a hierarchical attention mechanism that only needs two scales during training but can utilize more scales during inference.\nIn this work, we propose scale-space rendering (SSR), a more robust multi-scale attention mechanism that generalizes the aforementioned hierarchical approach and exploits feature relationships in scale-space to improve the performance further.\n\n\\section{Method} \\label{sec:method}\nIn this section, we first introduce our attentive feature aggregation (AFA) module and extend our AFA to scale-space rendering (SSR) attention for multi-scale inference. The overview of the complete architecture is shown in Fig.~\\ref{fig:overview}.\n\n\\subsection{Attentive Feature Aggregation}\nOur attentive feature aggregation (AFA) module computes both spatial and channel attention based on the relation of the input feature maps.\nThe attention values are then used to modulate the input activation scales and produce one merged feature map.\nThe operation is nonlinear in contrast to feature concatenation or summation.\nAs shown in Fig.~\\ref{fig:Attention_comparision}, unlike previous attention methods~\\cite{chen2016sca,woo2018cbam,fu2019dual}, AFA focuses on aggregating feature maps of different network layers to obtain more expressive representations.\nWithout using the Transformer~\\cite{vaswani2017attention} self-attention mechanism in DANet~\\cite{fu2019dual}, our AFA consumes only $1\/2$ FLOPs and model parameters with $1\/4$ GPU memory under the same input features, and can be extensively used.\nWe use two different basic self-attention mechanisms to generate spatial and channel attention maps and resemble them concerning the relation between the input features.\n\nFor the input feature $F_s \\in \\mathbb{R}^{C \\times H \\times W}$, the spatial attention uses a convolutional block $\\omega_s$ consists of two $3 \\times 3$ convolutions to encode $F_s$. It is defined as\n\\begin{equation}\n    a_s\\triangleq\\sigma(\\omega_s(F_s))\\,,\n\\end{equation}\nwhere $a_s \\in \\mathbb{R}^{1 \\times H \\times W}$ and $\\sigma$ is the sigmoid activation.\n\nFor computing the channel attention of input feature $F_d \\in \\mathbb{R}^{C \\times H \\times W}$, we first apply average pooling to get $F^{\\mathtt{avg}}_d$ and max pooling to get $F^{\\mathtt{max}}_d$.\nThen, we use another convolutional block $\\omega_c$ to further transform features to $F^{\\mathtt{avg}}_c$ and $F^{\\mathtt{max}}_c$, which consists of two $1 \\times 1$ convolutions with a bottleneck input-output channel design.\nBy summing them up with equal weighting and using sigmoid $\\sigma$ as the activation function, we can generate channel attention $a_c \\in \\mathbb{R}^{C \\times 1 \\times 1}$, i.e., \n\\begin{equation}\n    a_c\\triangleq\\sigma(\\omega_c(\\mathtt{AvgPool}(F_d)) + \\omega_c(\\mathtt{MaxPool}(F_d)))\\,.\n\\end{equation}\n\nWith the basis of the above attention mechanisms, we design two types of AFA for different aggregation scenarios and enable the network to model complex feature interactions and attend to different features.\n\n\\parsection{Binary Fusion.}\nWe employ a simple attention-based aggregation mechanism using our spatial and channel attentions to replace standard binary fusion nodes.\nWhen merging two input feature maps, we apply channel and spatial attention hierarchically to capture the relation of input features.\nAs shown in Fig.~\\ref{fig:overview}~(b), when two features are aggregated, we denote the shallower feature map as $F_s$ and the other as $F_d$.\n$F_s$ is used to compute $a_s$ and $F_d$ is responsible for $a_c$.\nBecause the shallower layers will contain richer spatial information, and the deeper ones will have more complex channel features.\nThen, we define our attentive binary fusion to aggregate the inputs and obtain the aggregated feature $F_{\\mathtt{agg}}$ as\n\\begin{equation}\n    F_\\mathtt{agg}\\triangleq a_s\\odot{(1-a_c)\\odot{F_s}}+(1-a_s)\\odot{a_c\\odot{F_d}}\\,,\n\\end{equation}\nwhere $\\odot$ denotes element-wise multiplication (with broadcasted unit dimensions).\nBy leveraging the input features properties, our binary fusion is simple yet effective.\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.47\\textwidth]{images\/Attention_Comparison.pdf}\n    \\caption{(a) CBAM~\\cite{woo2018cbam} module. CA and SA are their channel attention and spatial attention blocks. (b) DANet~\\cite{fu2019dual} architecture. PAM stands for their proposed Position Attention Module while CAM denotes channel attention module.}\n    \\label{fig:Attention_comparision}\n\\end{figure}\n\n\\parsection{Multiple Feature Fusion.}\nWe extend the binary fusion node to further fuse together multiple multi-scale features.\nRecent works~\\cite{ronneberger2015u,wang2020deep,yu2018deep} iteratively aggregate features across the model, but only exploit the final feature for downstream tasks, neglecting intermediate features computed during the aggregation process.\nBy applying our AFA module on these intermediate features, we give the model more flexibility to select the most relevant features.\n\nGiven $k$ multi-scale features $F_i$ for $i \\in \\{1,\\ldots,k\\}$, we first order them based on the amount of aggregated information they contain, i.e. , a feature with higher priority will have gone through a higher number of aggregations.\nThen, we compute both spatial and channel attention for each feature and take the product as the new attention.\nThe combined attention $a_i$ is defined as\n\\begin{equation}\n    a_i\\triangleq\\mathtt{SA}(F_i)\\odot{\\mathtt{CA}(F_i)}\\,,\n\\end{equation}\nwhere $\\mathtt{SA}$ denotes our spatial attention function and $\\mathtt{CA}$ our channel attention function.\nFor fusing the multi-scale features, we perform hierarchical attentive fusion by progressively aggregating features starting from $F_1$ to $F_k$ to obtain the final representation $F_\\mathtt{final}$ as\n\\begin{equation}\n    F_\\mathtt{final}\\triangleq\\sum_{i=1}^k \\left[a_i\\odot{F_i\\odot{\\prod_{j=i+1}^{k} (1-a_j)}}\\right]\\,.\n\\end{equation}\nIn Fig.~\\ref{fig:overview} (c), we show an example with $k=3$.\nThe new final representation $F_\\mathtt{final}$ is an aggregation of features at multiple scales, combining information from shallow to deep levels.\n\nWe can apply our AFA module to widely used segmentation models, as shown in Fig.~\\ref{fig:AFA-Seg}.\nIn U-Net~\\cite{ronneberger2015u} and HRNet~\\cite{wang2020deep}, we add our multiple feature fusion module to fully utilize the previously unused aggregated multi-scale features.\nIn FCN~\\cite{long2015fully}, we replace the original linear aggregation node in the decoder with our attentive binary fusion.\nFor DLA~\\cite{yu2018deep}, we not only substitute the original aggregation nodes but also add our multiple feature fusion module.\nDue to higher connectivity of its nodes, the DLA network can benefit more from our improved feature aggregation scheme, and thus we use AFA-DLA as our final model.\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[width=0.9\\textwidth]{images\/AFA_Seg.pdf}\n\t\\caption{Segmentation models with our AFA module. We show parts of the original models related to feature aggregation and our modifications. Red blocks represent auxiliary segmentation head added during training.}\n\t\\label{fig:AFA-Seg}\n\\end{figure*}\n\n\\subsection{Scale-Space Rendering}\nEven though multi-scale attention~\\cite{chen2016attention,tao2020hierarchical} can eliminate the trade-off in fine and coarse detail performance in dense prediction tasks, repeated use of attention layers may lead to numerical instability or vanishing gradients.\nTo resolve the drawback, we extend the attention mentioned above mechanism and resemble a volume rendering scheme applied to the scale space.\nBy treating the multi-scale predictions regarded as samples in a scale-space representation, this scheme provides a hierarchical, coarse-to-fine strategy to combine features, leveraging a scale-specific attention mechanism.\nWe will also show that our approach generalizes the hierarchical multi-scale attention method~\\cite{tao2020hierarchical}.\n\nWithout loss of generality, we focus on a single pixel and assume that our model provides a dense prediction for the target pixel at $k$ different scales. The prediction for the $i$th scale is denoted by $P_i\\in\\mathbb R^d$. Accordingly, $P\\triangleq(P_1,\\ldots,P_k)$ denotes the feature representation of the target pixel in our scale-space. Furthermore, we assume that $i<j$ implies that scale $i$ is coarser than scale $j$.\n\nOur target pixel can be imagined as a ray moving through scale-space, starting from scale $1$ towards scale $k$. We redesign the original hierarchical attention in the proposed multiple feature fusion mechanism and mimic the volume-rendering equation, where the volume is implicitly given by the scale-space. To this end, besides the feature representation $P_i$ at scale $i$, we assume our model to predict for the target pixel also a scalar $y_i\\in\\mathbb R$ so that $e^{-\\phi(y_i)}$ represents the probability that the particle will cross scale $i$, given some non-negative scalar function $\\phi:\\mathbb R\\to\\mathbb R_+$. We can then express the scale attention $\\alpha_i$ as the probability of the particle to reach scale $i$ and stop there, i.e. ,\n\\setlength{\\belowdisplayskip}{3pt}\n\\begin{equation}\n    \\alpha_i(y)\\triangleq\\left[1-e^{-\\phi(y_i)}\\right]\\prod_{j=1}^{i-1}e^{-\\phi(y_j)}\\,,\n\\end{equation}\nwhere $y\\triangleq(y_1,\\ldots,y_k)$.\n\n\\begin{figure*}[t]\n    \\small\n    \\centering\n    \\footnotesize\n    \\setlength{\\tabcolsep}{2.5pt}\n    \\begin{tabular}{cccccc}\n        \\toprule\n        \\multicolumn{2}{c}{scale = 0.5} &\n        \\multicolumn{2}{c}{scale = 1.0} &\n        \\multicolumn{2}{c}{scale = 1.5} \\\\\n        \\midrule\n        \\includegraphics[width=0.15\\linewidth]{images\/SSR\/pred_1_05.png} & \\includegraphics[width=0.15\\linewidth]{images\/SSR\/attn_1_05.pdf} &\n        \\includegraphics[width=0.15\\linewidth]{images\/SSR\/pred_1_10.png} & \\includegraphics[width=0.15\\linewidth]{images\/SSR\/attn_1_10.pdf} &\n        \\includegraphics[width=0.15\\linewidth]{images\/SSR\/pred_1_15.png} & \\includegraphics[width=0.15\\linewidth]{images\/SSR\/attn_1_15.pdf} \\\\\n        \\includegraphics[width=0.15\\linewidth]{images\/SSR\/pred_2_05.png} & \\includegraphics[width=0.15\\linewidth]{images\/SSR\/attn_2_05.pdf} &\n        \\includegraphics[width=0.15\\linewidth]{images\/SSR\/pred_2_10.png} & \\includegraphics[width=0.15\\linewidth]{images\/SSR\/attn_2_10.pdf} &\n        \\includegraphics[width=0.15\\linewidth]{images\/SSR\/pred_2_15.png} & \\includegraphics[width=0.15\\linewidth]{images\/SSR\/attn_2_15.pdf} \\\\ \n        \\bottomrule\n    \\end{tabular}\n    \\caption{Visualization of attention maps generated by scale-space rendering (SSR) and the corresponding predictions. Whiter regions denote higher attention. SSR learns to focus on detailed regions in larger scale images and on lower frequency information in smaller scale images.}\n\t\\label{fig:ms_results}\n\\end{figure*}\n\nFinally, the fused multi-scale prediction for the target pixel can be regarded as the ``rendered'' pixel, where the pixel features at the different scales $P_i$ are averaged by the attention coefficients $\\alpha_i$ following the volume rendering equations. Accordingly,\n\\begin{equation}\n    P_\\mathtt{final}\\triangleq \\sum_{i=1}^k P_i\\alpha_i(y)\\,\n\\end{equation}\nrepresents the feature for the target pixel that we obtain after fusing $P$ across all scales with attention driven by $y$.\n\nThe proposed scale-space rendering (SSR) mechanism can be regarded as a generalization of the hierarchical multi-scale attention proposed in~\\cite{tao2020hierarchical}, for the latter can be obtained from our formulation by simply setting $\\phi(y_i)\\triangleq\\log(1+e^{y_i})$, i.e., $\\phi$ is the soft-plus function, and by fixing $\\phi(y_k)\\triangleq\\infty$.\n\n\\parsection{Choice of $\\phi$.}\nIn our experiments, we use the absolute value function as our $\\phi$, i.e. $\\phi(y_i)\\triangleq|y_i|$. This is motivated by a better preservation of the gradient flow through the attention mechanism, as we found existing attention mechanisms to suffer from vanishing gradient issues. Consider the Jacobian of the attention coefficients, which takes the following form:\n\\begin{equation}\n    J_{i\\ell} \\triangleq \\frac{\\partial \\alpha_i(y)}{\\partial y_\\ell}=\n    \\begin{cases}\n    \\phi'(y_i)\\prod_{j=1}^i e^{-\\phi(y_j)} & \\text{if $\\ell=i$}\\\\\n    0& \\text{if $\\ell>i$}\\\\\n    -\\phi'(y_\\ell)\\alpha_i(y)& \\text{if $\\ell<i$}\\,.\n    \\end{cases}\n\\end{equation}\nIn the presence of two scales, this becomes:\n\\begin{equation}\n    J=\\begin{bmatrix}\n    \\phi'(y_1)a_1&0\\\\\n    -\\phi'(y_1)a_1(1-a_2)&\\phi'(y_2)a_1a_2\\,\n    \\end{bmatrix},\n\\end{equation}\nwhere $a_i\\triangleq e^{-\\phi(y_i)}$.\nAs $a_1\\to 0$, the gradient vanishes, for $J$ tends to a null matrix.\nOtherwise, irrespective of the value of $a_2$, the gradient will vanish only depending on the choice of $\\phi$.\nIn particular, by taking the absolute value as $\\phi$ we have that the Jacobian will not vanish for $a_1>0$ and $(y_1,y_2)\\neq(0,0)$, thus motivating our choice of using the absolute value as $\\phi$.\nIf we consider instead the setting in HMA~\\cite{tao2020hierarchical}, we have that $a_2=0$ and $\\phi'(y_i)=1-a_i$.\nIt follows that the Jacobian vanishes also as $a_1\\to 1$.\nThe conclusion is that the choice of $\\phi$ plays a role in determining the amount of gradient that flows through the predicted attention and that the approach in HMA~\\cite{tao2020hierarchical} is more subject to vanishing gradient issues than our proposed solution.\n\nTo understand which parts of the image SSR attends to at each scale, we visualize generated attention maps in Fig.~\\ref{fig:ms_results}.\nDetailed regions are processed more effectively in larger scale images due to the higher resolution, while the prediction of lower frequency region is often better in smaller scales. SSR learns to focus on the right region for different scales and boosts the final performance.\n\nWe combine our AFA-DLA model with SSR to produce the final predictions.\nAs shown in Fig.~\\ref{fig:overview}, AFA-DLA propagates information from different scales to the SSR module, which then generates attention masks $\\alpha_i$ used to fuse the predictions $P_i$ to get our final prediction $P_{\\mathtt{final}}$.\n\n\\parsection{Training Details.}\nFor fair comparison with other methods~\\cite{yuan2020object,tao2020hierarchical}, we reduce the number of filters from 256 to 128 in the OCR~\\cite{yuan2020object} module and add it after our AFA-DLA network to refine our final predictions.\nOur final model can be trained at $k$ different scales. Due to the limitation of computational resources, we use $k=2$ for training and RMI~\\cite{zhao2019region} to be the primary loss function $L_{\\mathtt{primary}}$ for our final prediction $P_{\\mathtt{final}}$.\nWe add three different types of auxiliary cross-entropy losses to stabilize the training.\nFirst, we use the generated SSR attention to fuse the auxiliary per-scale predictions from OCR, yielding $P^{\\mathtt{aux}}_{\\mathtt{ocr}}$ and the loss $L_{\\mathtt{ocr}}$.\nSecond, we compute and sum up cross-entropy losses for each scale prediction $P_i$ yielding $L_{\\mathtt{scale}}$.\nLastly, we add auxiliary segmentation heads inside AFA-DLA as in Fig.~\\ref{fig:AFA-Seg} (a) and have predictions for each scale. We fuse them with SSR across scales and get $P^{\\mathtt{aux}}_j$, where $1\\leq j\\leq 4$. We compute the auxiliary loss for each and sum then up as $L_{\\mathtt{dla}}$.\nAccordingly, the total loss function is the weighted sum as\n\\begin{equation}\n    L_{\\mathtt{all}}\\triangleq L_{\\mathtt{primary}}+\\beta_o L_{\\mathtt{ocr}}+\\beta_s L_{\\mathtt{scale}}+\\beta_d L_{\\mathtt{dla}},\n\\end{equation}\nwhere we set $\\beta_o \\triangleq 0.4$, $\\beta_s \\triangleq 0.05$ and $\\beta_d \\triangleq 0.05$.\nWe illustrate more details of each loss function in the supplementary materials.\n\n\\section{Experiments} \\label{sec:exp}\nWe conduct experiments on several public datasets on both semantic segmentation and boundary detection tasks, and conduct a thorough analysis with a series of ablation studies.\nDue to the space limit, we leave additional implementation details to our supplementary materials.\n\n\\subsection{Results on Cityscapes}\n\\begin{table}[t]\n  \\caption{\n  Resource usage of different models. AFA-DLA uses much fewer operations and parameters when compared to top performing methods.}\n  \\centering\n  \\begin{tabular}{lcc}\n    \\toprule\n    Method & FLOPs (G) & Param. (M) \\\\\n    \\midrule\n    DeepLabV3+~\\cite{chen2018encoder} & 2514 & 54.4 \\\\\n    DecoupleSegNet~\\cite{li2020improving} & 6197 & 138.4 \\\\\n    \\midrule\n    AFA-DLA-X-102 (Ours) & \\textbf{1333} & \\textbf{36.3} \\\\\n    \\bottomrule\n  \\end{tabular}\n  \\label{cityscapes-flops}\n\\end{table}\nThe Cityscapes dataset~\\cite{cordts2016cityscapes} provides high resolution (2048 x 1024) urban street scene images and their corresponding segmentation maps.\nIt contains 5K well annotated images for 19 classes and 20K coarsely labeled image as extra training data.\nIts finely annotated images are split into 2975, 500, and 1525 for training, validation and testing.\nWe use DLA-X-102 as the backbone for our AFA-DLA model with a batch size of 8 and full crop size.\nFollowing~\\cite{tao2020hierarchical}, we train our model with auto-labeled coarse training data with 0.5 probability and otherwise use the fine labeled training set.\nDuring inference, we use multi-scale inference with [0.5, 1.0, 1.5, 1.75, 2.0] scales, image flipping, and Seg-Fix~\\cite{yuan2020segfix} post-processing.\nWe detail the effect of each post-processing technique in the supplementary material.\n\nThe results on the validation and test set are shown in Table~\\ref{cityscapes-results}.\nWith only using ImageNet~\\cite{deng2009imagenet} pre-training and without using external segmentation datasets, AFA-DLA gains 85.14 mean IoU on the Cityscapes validation set, achieving state-of-the-art performance.\nOn the Cityscapes test set, AFA-DLA also obtains competitive performance with the top performing method, DecoupleSegNet~\\cite{li2020improving}, while using around 75\\% fewer operations and parameters as shown in Table~\\ref{cityscapes-flops}.\n\nWe additionally evaluate the application of AFA to other widely used segmentation models, including FCN, U-Net, and HRNet.\nWe use the same shorter learning schedule and smaller training crop size for all models for fair comparison in Table~\\ref{AFA-other-models}.\nSince we only modify the aggregation operations of each model, we can still use the original ImageNet~\\cite{deng2009imagenet} pre-training weights as initialization.\n\nCombined with our AFA module, the segmentation models can each obtain at least 2.5\\% improvement in mIoU, with only a small computational and parameter overhead.\nIn particular, we even lighten HRNet by replacing its concatenation in the last layer with our multiple feature fusion and still achieve 2.5\\% improvement.\nThis demonstrates AFA as a lightweight module that can be readily applied to existing models for segmentation.\n\n\\begin{table}[t]\n    \\small\n    \\caption{Combining AFA with other widely used segmentation models on the Cityscapes validation set. With our AFA module, each model can obtain at least 2.5\\% improvement in mIoU, with only a small computational and parameter overhead.}\n    \\begin{tabular}{l|cc|c|c}\n        \\toprule\n        Method & FLOP (G) & Param. (M) & mIoU & $\\Delta$ ($\\%$) \\\\\n        \\midrule\n        FCN & 1581.8 & 49.5 & 75.52 & - \\\\\n        AFA-FCN & 1659.2 & 51.9 & \\textbf{77.88} & 3.1 \\\\\n        \\midrule\n        U-Net-S5-D16 & 1622.8 & 29.1 & 62.73 & - \\\\\n        AFA-U-Net & 2146.7 & 29.4 & \\textbf{64.42} & 2.7 \\\\\n        \\midrule\n        HRNet-W48 & 748.7 & 65.9 & 78.48 & - \\\\\n        AFA-HRNet & 701.4 & 65.4 & \\textbf{80.41} & 2.5 \\\\\n        \\bottomrule\n    \\end{tabular}\n    \\label{AFA-other-models}\n\\end{table}\n\n\\subsection{Results on BDD100K}\n\\begin{table}[t]\n    \\caption{Segmentation results on BDD100K validation and testing set. $\\dagger$~denotes using Cityscapes data for pre-training. AFA-DLA achieves the new state-of-the-art performance on both sets.}\n    \\label{bdd100k-results}\n    \\centering\n    \\begin{tabular}{lcc}\n        \\toprule\n        Method & mIoU (val) & mIoU (test) \\\\\n        \\midrule\n        DLA~\\cite{yu2018deep} & 57.84 & \\textit{N\/A} \\\\\n        CCNet~\\cite{huang2019ccnet} & 64.03 & 55.93 \\\\\n        DNL~\\cite{yin2020disentangled} & \\textit{N\/A} & 56.31 \\\\\n        PSPNet~\\cite{zhao2017pyramid} & \\textit{N\/A} & 56.32 \\\\\n        Deeplabv3+~\\cite{chen2018encoder} & 64.49 & 57.00 \\\\\n        $\\text{DecoupleSegNet}\\dagger$~\\cite{li2020improving} & 66.90 & \\textit{N\/A} \\\\\n        \\midrule\n        AFA-DLA (Ours) & \\textbf{67.46} & \\textbf{58.47} \\\\\n        \\bottomrule\n    \\end{tabular}\n\\end{table}\nBDD100K~\\cite{yu2020bdd100k} is a diverse driving video dataset for multitask learning.\nFor the semantic segmentation task, it provides 10K images with same categories as Cityscapes at 1280 x 720 resolution.\nThe dataset consists 7K, 1K, and 2K images for training, validation, and testing.\nConsidering the amount of training data is twice as Cityscapes, we use DLA-169 as the backbone with full image crop and 16 training batch size for 200 epochs.\nDuring inference, we use multi-scale inference with [0.5, 1.0, 1.5, 2.0] scales and image flipping.\n\nThe results on validation and test sets are shown in Table~\\ref{bdd100k-results}.\nAFA-DLA achieves new state-of-the-art performances on both sets despite using fewer operations and parameters when compared to the top performing methods as shown in Table~\\ref{cityscapes-flops}.\nOur method achieves 67.46 mean IoU on the validation set and is even higher than DecoupleSegNet~\\cite{li2020improving}, which uses Cityscapes pre-trained weights.\nMoreover, AFA-DLA obtains 58.47 mean IoU on the test set, which outperforms all the strong official baselines.\n\n\\begin{table}\n    \\caption{Boundary detection results on NYUDv2 test set using three different types of inputs. AFA-DLA achieves state-of-the-art results across all three settings.}\n    \\label{tab:nyud-test-results}\n    \\centering\n    \\begin{tabular}{l|c|cc}\n    \\toprule\n        Method & Input & ODS & OIS \\\\\n        \\midrule\n        AMH-Net~\\cite{liu2017richer} & \\multirow{4}{*}{RGB} & 0.744 & 0.758 \\\\\n        BDCN~\\cite{he2019bi} &  & 0.748 & 0.763 \\\\\n        PiDiNet~\\cite{su2021pixel} &  & 0.733 & 0.747 \\\\\n        AFA-DLA-34 (Ours) &  & \\textbf{0.762} & \\textbf{0.775} \\\\ \\midrule\n        AMH-Net~\\cite{liu2017richer} & \\multirow{4}{*}{HHA} & 0.716 & 0.729 \\\\\n        BDCN~\\cite{he2019bi} &  & 0.707 & 0.719 \\\\\n        PiDiNet~\\cite{su2021pixel} &  & 0.715 & 0.728 \\\\\n        AFA-DLA-34 (Ours) &  & \\textbf{0.718} & \\textbf{0.730} \\\\ \\midrule\n        AMH-Net~\\cite{liu2017richer} & \\multirow{4}{*}{RGB+HHA} & 0.771 & 0.786 \\\\\n        BDCN~\\cite{he2019bi} &  & 0.765 & 0.781 \\\\\n        PiDiNet~\\cite{su2021pixel} &  & 0.756 & 0.773 \\\\\n        AFA-DLA-34 (Ours) &  & \\textbf{0.780} & \\textbf{0.792} \\\\\n        \\bottomrule\n    \\end{tabular}\n\\end{table}\n\n\\begin{table}[t]\n    \\caption{Boundary detection results on BSDS500 test set. AFA-DLA outperforms all other methods in ODS.}\n    \\label{bsds500-test-results}\n    \\centering\n    \\begin{tabular}{lcc}\n        \\toprule\n        Method & ODS & OIS \\\\\n        \\midrule\n        DLA-102~\\cite{yu2018deep} & 0.803 & 0.813 \\\\\n        LPCB~\\cite{deng2018learning} & 0.800 & 0.816 \\\\\n        BDCN~\\cite{he2019bi} & 0.806 & \\textbf{0.826} \\\\\n        AFA-DLA-34 (Ours) & \\textbf{0.812} & \\textbf{0.826} \\\\\n        \\bottomrule\n    \\end{tabular}\n\\end{table}\n\n\\begin{table*}[t]\n  \\caption{Ablation study on Cityscapes validation set with DLA34 as backbone. SA and CA denote using spatial and channel attention for feature fusion, Aux. Head denotes using auxiliary segmentation heads, and MS Inference denotes multi-scale inference. Both stands for using both input feature to generate attention for fusion. Swap denotes switch the input of spatial and channel attention module.}\n  \\label{ablation-results}\n  \\centering\n  \\begin{tabular}{lcc|c|c}\n    \\toprule\n    Binary Fusion & Aux. Head & Multi Feature Fusion & MS Inference & mIoU \\\\\n    \\midrule\n    Original & - & - & Single Scale & 74.43 \\\\\n    Swap & - & - & Single Scale & 75.37 \\\\\n    CA & - & - & Single Scale & 75.54 \\\\\n    CBAM~\\cite{woo2018cbam} & - & - & Single Scale & 75.70 \\\\\n    Both & - & - & Single Scale & 75.77 \\\\\n    SA + CA & - & - & Single Scale & \\textbf{76.14} \\\\\n    \\midrule\n    SA + CA & \\checkmark & - & Single Scale & 76.45 \\\\\n    SA + CA & \\checkmark & \\checkmark & Single Scale & \\textbf{77.08} \\\\\n    \\midrule\n    SA + CA & \\checkmark & \\checkmark & Avg. Pooling & 78.56 \\\\\n    SA + CA & \\checkmark & \\checkmark & HMA~\\cite{tao2020hierarchical} & 80.18 \\\\\n    SA + CA & \\checkmark & \\checkmark & SSR & \\textbf{80.74} \\\\\n    \\bottomrule\n  \\end{tabular}\n\\end{table*}\n\n\\subsection{Boundary Detection}\nWe additionally conduct experiments on boundary detection, which involves predicting a binary segmentation mask indicating the existence of boundaries.\nWe evaluate on two standard boundary detection datasets, NYU Depth Dataset V2 (NYUDv2)~\\cite{silberman2012indoor} and Berkeley Segmentation Data Set and Benchmarks 500 (BSDS500)~\\cite{arbelaez2010contour}.\nFor each dataset, we follow the standard data preprocessing and evaluation protocol in literature \\cite{xie2015holistically,liu2017richer}.\nSpecifically, we augment each dataset by randomly flipping, scaling, and rotating each image.\nWe evaluate using commonly used metrics, which are the F-measure at the Optimal Dataset Scale (ODS) and at the Optimal Image Scale (OIS).\nFollowing~\\cite{yu2018deep}, we also scale the boundary labels by 10 to account for the label imbalance.\nFor simplicity, we do not consider using multi-scale images during inference, so SSR is not used.\n\n\\parsection{Results on NYUDv2.}\nThe NYUDv2 dataset contains both RGB and depth images.\nThere are 381 training images, 414 validation images, and 654 testing images.\nWe follow the same procedure as~\\cite{xie2015holistically,liu2017richer,he2019bi} and train a separate model on RGB and HHA~\\cite{gupta2014learning} images.\nWe additionally evaluate using both RGB and HHA images as input by averaging each model's output during inference.\n\nThe results are shown in Table~\\ref{tab:nyud-test-results}.\nOur AFA-DLA model outperforms all other methods by a large margin across all three types of inputs, achieving state-of-the-art performances.\nIn particular, when using both RGB and HHA as input, AFA-DLA can achieve a high score of 0.780 in ODS and 0.792 in OIS.\nWhen only using RGB images as input, AFA-DLA already outperforms some other methods using both RGB and HHA images.\n\n\\parsection{Results on BSDS500.}\nThe BSDS500 dataset contains 200 training images, 100 validation images, and 200 testing images.\nWe follow standard practice~\\cite{xie2015holistically} and only use boundaries annotated by three or more annotators for supervision.\nWe do not consider augmenting the training set with additional data, so we only utilize the available data in the BSDS500 dataset.\n\nThe results are shown in Table~\\ref{bsds500-test-results}.\nAFA-DLA achieves state-of-the-art performance when comparing to methods only training on the BSDS500 dataset and obtains 0.812 in ODS.\nIn particular, AFA-DLA improves the performance of DLA by 0.09 in ODS and 0.013 in OIS, despite using a smaller backbone.\n\n\\subsection{Ablation Experiments}\nIn this section, we conduct several ablation studies on the Cityscapes validation set to validate each component of AFA-DLA.\nThe main baseline model we compare to is DLA~\\cite{yu2018deep} with DLA-34 as backbone.\nAll the results are listed in Table~\\ref{ablation-results}. We also provide visualizations in order to qualitatively evaluate our model.\n\n\\parsection{Binary Fusion.}\nWe first evaluate our attentive binary fusion module, which can learn and predict the importance of each input signal during fusion.\nCompared to using standard linear fusion operators, introducing nonlinearity and using channel attention during binary fusion achieves around 1.1 mean IoU improvement.\nThis demonstrates that more expressive aggregation can drastically improve the results.\nWhen we additionally use spatial attention, we observe 0.6 points further improvement.\n\n\\parsection{Attention Mechanism.}\nWe validate our AFA design with taking opposite strategy and using both input features to generate each attention. We also apply the CBAM~\\cite{woo2018cbam} module on top of the original DLA linear aggregation node to refine the aggregated feature as our another baseline. As shown in Table~\\ref{ablation-results}, switching the attention module inputs will have even worse performance than only using channel attention. If we generate the the spatial and channel attention by using both two features concatenated together, the performance also decrease to 75.77 mIoU and use more computation. Furthermore, while CBAM improves the original DLA by 1.27 mIoU, our attentive binary fusion further improves by 0.44 mIoU. The results show that the design of the aggregation node should consider the properties of the input features and our AFA is the most effective.\n\n\\begin{figure*}[t]\n    \\centering\n    \\begin{tabular}{cccccc}\n        \\toprule\n        \\multirow{2}{*}{Input Image} & \\multicolumn{2}{c}{Binary Fusion} & \\multicolumn{3}{c}{Multiple Feature Fusion} \\\\\n        \\cmidrule{2-3} \\cmidrule{4-6} & \n        $a_s$ for $F_s$\n        &\n        $a_s$ for $F_d$ \n        &\n        $a_s$ for $F_1$\n        &\n        $a_s$ for $F_2$\n        &\n        $a_s$ for $F_3$ \\\\\n        \\includegraphics[width=0.14\\linewidth]{images\/AFA\/input_1.pdf}\n        &\n        \\includegraphics[width=0.14\\linewidth]{images\/AFA\/L4_L5_4.pdf}\n        &\n        \\includegraphics[width=0.14\\linewidth]{images\/AFA\/L5_L4_4.pdf}\n        &\n        \\includegraphics[width=0.14\\linewidth]{images\/AFA\/L5_4.pdf}\n        &\n        \\includegraphics[width=0.14\\linewidth]{images\/AFA\/L5_3.pdf}\n        &\n        \\includegraphics[width=0.14\\linewidth]{images\/AFA\/L5_2.pdf} \\\\\n        \\includegraphics[width=0.14\\linewidth]{images\/AFA\/input_2.pdf}\n        &\n        \\includegraphics[width=0.14\\linewidth]{images\/AFA\/L4_L5.pdf}\n        &\n        \\includegraphics[width=0.14\\linewidth]{images\/AFA\/L5_L4.pdf}\n        &\n        \\includegraphics[width=0.14\\linewidth]{images\/AFA\/L5_4_2.pdf}\n        &\n        \\includegraphics[width=0.14\\linewidth]{images\/AFA\/L5_3_2.pdf}\n        &\n        \\includegraphics[width=0.14\\linewidth]{images\/AFA\/L5_2_2.pdf} \\\\\n        \\bottomrule\n    \\end{tabular}\n    \\caption{Visualization of spatial attention maps $a_s$ generated by our attentive feature aggregation modules. Whiter regions denote higher attention. Compared to linear fusion operations, our AFA modules provide a more expressive way of combining features.}\n    \\label{fig:AFA_Attention}\n\\end{figure*}\n\n\\begin{figure}[t]\n    \\centering\n    \\setlength\\tabcolsep{2.5pt}\n    \\begin{tabular}{cccc}\n        \\toprule\n        Input & DLA & AFA & GT \\\\\n        \\midrule\n        \\includegraphics[width=0.22\\linewidth]{images\/AFA_comparison\/Input_1.pdf}\n        &\n        \\includegraphics[width=0.22\\linewidth]{images\/AFA_comparison\/DLA_1.pdf}\n        &\n        \\includegraphics[width=0.22\\linewidth]{images\/AFA_comparison\/AFA_1.pdf}\n        &\n        \\includegraphics[width=0.22\\linewidth]{images\/AFA_comparison\/GT_1.pdf} \\\\\n        \\includegraphics[width=0.22\\linewidth]{images\/AFA_comparison\/Input_2.pdf}\n        &\n        \\includegraphics[width=0.22\\linewidth]{images\/AFA_comparison\/DLA_2.pdf}\n        &\n        \\includegraphics[width=0.22\\linewidth]{images\/AFA_comparison\/AFA_2.pdf}\n        &\n        \\includegraphics[width=0.22\\linewidth]{images\/AFA_comparison\/GT_2.pdf} \\\\\n        \\bottomrule\n    \\end{tabular}\n    \\caption{ \n        Comparison of predictions generated by DLA and AFA-DLA. The black pixels are ignored. AFA-DLA can better distinguish object boundaries and correctly classify object classes.\n    }\n\t\\label{fig:DLA_AFA_comparison}\n\\end{figure}\n\n\\parsection{Auxiliary Segmentation Head.}\nWe add several auxiliary segmentation heads into AFA-DLA to stabilize the training of attention mechanism, which is common practice among other popular baseline models.\nThe whole backbone can be supervised by the auxiliary losses efficiently and they can alleviate gradient vanishing issues.\nWe see about 0.3 mean IoU improvement.\n\n\\parsection{Multiple Feature Fusion.}\nWe apply our multiple feature fusion to enable AFA-DLA to fully leverage intermediate features in the network.\nThis gives the network more flexibility in selecting relevant features for computing the final feature.\nBy adding the multiple feature fusion module, we gain another 0.6 mean IoU improvement.\n\n\\parsection{Scale-Space Rendering.}\nWe employ our SSR module to fuse multi-scale predictions.\nAfter applying our SSR with [0.25, 0.5, 1.0, 2.0] inference scales, we gain an impressive improvement of nearly 3.7\\% mean IoU over using only a single scale.\nWe also compare with different multi-scale inference approaches under the same training setting.\nSSR gains 1.2 mean IoU over standard average pooling and further outperforms hierarchical multi-scale attention~\\cite{tao2020hierarchical} by nearly 0.6 mean IoU.\nThis suggests that our scale-space rendering attention can boost the performance by reducing the gradient vanishing problem and still retain the flexibility for selecting different training and inference scales.\nWith both AFA and SSR, our network improves the DLA baseline model performance by nearly 6.3 mean IoU.\n\n\\parsection{Attention Visualization.}\nTo understand which parts of the inputs our AFA fusion modules attends to, we visualize the generated attention maps for a set of input features in Fig.~\\ref{fig:AFA_Attention}.\nAFA learns to attend to different regions of the input features depending on the information they contain.\nBinary fusion module focuses on object boundaries in shallower features $F_s$ and attends to the rest on deeper features $F_d$.\nOur multiple feature fusion module can perform complex selection of features by attending to different regions for each feature level.\n$F_1$ aggregates shallower features and thus the module attends to the boundaries, while the rest attend to objects or the background.\nCompared to linear fusion operations, AFA provides a more expressive way of combining features.\n\n\\parsection{Segmentation Visualization.}\nWe take a deeper look at the segmentations on the Cityscapes validation set produced by AFA-DLA in Fig.~\\ref{fig:DLA_AFA_comparison} and compare them to those produced by DLA~\\cite{yu2018deep}.\nWith our AFA module, the model can better leverage spatial and channel information to distinguish object boundaries more clearly and better classify object classes.\n\n\\section{Conclusion}\nWe propose a novel attention-based feature aggregation module combined with a new multi-scale inference mechanism to build the competitive AFA-DLA model.\nWith spatial and channel attention mechanisms, AFA enlarges the receptive field and fuses different network layer features effectively.\nSSR improves existing multi-scale inference methods by being more robust towards the gradient vanishing problem.\nApplying all of our components, we improve the DLA baseline model performance by nearly 6.3 mean IoU on Cityscapes.\nWhen combining AFA with existing segmentation models, we found consistent improvements of at least 2.5\\% in mean IoU on Cityscapes, with only a small cost in computational and parameter overhead.\nAFA-DLA also establishes new state-of-the-art results on BDD100K and achieves the new best score on Cityscapes when not using external segmentation datasets.\nMoreover, for the boundary detection task, AFA-DLA obtains state-of-the-art results on NYUDv2 and BSDS500.\n\n\\section{Appendix}\nIn this appendix, we provide additional additional results on boundary detection benchmarks, ablation study on post-processing effects on semantic segmentation, training settings and more qualitative results of the attention maps and output predictions.\n\n\\section{Full Results on BSDS500}\nTo complete the results in Table~6 in the main paper, we provide the full evaluation results on BSDS500~\\cite{arbelaez2010contour} in Table~\\ref{tab:bsds500-sup-results}.\nOn BSDS500, we show more evaluation results of using the PASCAL VOC Context dataset (PVC)~\\cite{mottaghi2014role} as additional training data and multi-scale inference.\nWhen using PVC, we double our training epochs to account for the additional data.\nFor multi-scale inference, we use standard average pooling for fair comparison with other methods.\nAFA-DLA achieves state-of-the-art results on single-scale inference when not training with additional data.\nSurprisingly, using PVC does not further improve the results.\nNevertheless, AFA-DLA achieves the same performance with the state-of-the-art method BDCN~\\cite{he2019bi} when using multi-scale inference.\n\n\\section{Ablation Study on Post-processing of Semantic Segmentation}\nTo have a fair competition with other methods, we exploit several post-processing techniques to pursue higher performance.\nWe conduct an ablation study on how each technique affects the final performance on the Cityscapes~\\cite{cordts2016cityscapes} validation set in Table~\\ref{ablation-post-processing}.\nThe main improvement gains are from our Scale Space Rendering (SSR) for multi-scale inference, and the other techniques only bring minor improvements.\n\n\\begin{table}[t]\n    \\caption{Boundary detection results on BSDS500. PVC indicates training with additional PASCAL VOC Context dataset. MS indicates multi-scale inference. AFA-DLA achieves state-of-the-art results on single-scale images without using additional data, and competitive results when using both PVC and MS.\n    }\n    \\label{tab:bsds500-sup-results}\n    \\centering\n    \\begin{tabular}{l|cc|cc}\n        \\toprule\n        Method & PVC & MS & ODS & OIS \\\\\n        \\midrule\n        Human & & & 0.803 & 0.803 \\\\ \\midrule\n        DLA-102~\\cite{yu2018deep} &  &  & 0.803 & 0.813 \\\\\n        LPCB~\\cite{deng2018learning} &  &  & 0.800 & 0.816 \\\\\n        BDCN~\\cite{he2019bi} &  &  & 0.806 & \\textbf{0.826} \\\\\n        AFA-DLA-34 (Ours) &  &  & \\textbf{0.812} & \\textbf{0.826} \\\\\n        \\midrule\n        RCF~\\cite{liu2017richer} & \\checkmark &   & 0.808 & 0.825 \\\\\n        LPCB~\\cite{deng2018learning} & \\checkmark &   & 0.808 & 0.824 \\\\\n        BDCN~\\cite{he2019bi} & \\checkmark &   & \\textbf{0.820} & \\textbf{0.838} \\\\\n        PiDiNet~\\cite{su2021pixel} & \\checkmark &   & 0.807 & 0.823 \\\\\n        AFA-DLA-34 (Ours) & \\checkmark &  & 0.810 & 0.826 \\\\ \\midrule\n        RCF~\\cite{liu2017richer} & \\checkmark & \\checkmark & 0.814 & 0.833 \\\\\n        LPCB~\\cite{deng2018learning} & \\checkmark & \\checkmark & 0.815 & 0.834 \\\\\n        BDCN~\\cite{he2019bi} & \\checkmark & \\checkmark  & 0.828 & 0.844 \\\\\n        AFA-DLA-34 (Ours) & \\checkmark & \\checkmark  & \\textbf{0.828} & \\textbf{0.844} \\\\\n        \\bottomrule\n    \\end{tabular}\n\\end{table}\n\n\\section{Training Losses}\nIn this section, we describe in more detail the formulation of our loss function for AFA-DLA for both semantic segmentation and boundary detection.\n\n\\subsection{Semantic Segmentation}\nWe use $k$ scales for training and RMI~\\cite{zhao2019region} to be the primary loss for our final prediction $P_{\\mathtt{final}}$, i.e.,\n\\begin{equation}\n    L_{\\mathtt{primary}}\\triangleq L_{\\mathtt{rmi}}(\\hat P, P_{\\mathtt{final}})\\,,\n\\end{equation}\n\nwhere $\\hat P$ is the ground truth and $L_\\mathtt{rmi}$ is the RMI loss function.\nThe first auxiliary cross-entropy loss is computed by using the generated scale-space rendering (SSR) attention to fuse the auxiliary per-scale predictions from the OCR~\\cite{yuan2020object} module, yielding\n\\begin{equation}\nL_{\\mathtt{ocr}}\\triangleq L_{\\mathtt{ce}}(\\hat P, P^{\\mathtt{aux}}_{\\mathtt{ocr}})\\,,\n\\end{equation}\nwhere $L_\\mathtt{ce}$ denotes the cross-entropy loss.\nFor the second auxiliary loss, we compute and sum up cross-entropy losses for each scale prediction $P_i$, where $1\\leq i\\leq k$, yielding\n\\begin{equation}\n    L_{\\mathtt{scale}}\\triangleq \\sum_{i=1}^k L_{\\mathtt{ce}}(\\hat P, P_i)\\,.\n\\end{equation}\nLastly, for the auxiliary loss inside AFA-DLA, we fuse the predictions of each auxiliary segmentation head with SSR across scales and get $P^{\\mathtt{aux}}_j$, where $1\\leq j\\leq 4$. We compute the auxiliary loss for each prediction and sum them up as\n\\begin{equation}\n    L_{\\mathtt{dla}}\\triangleq\\sum_{j=1}^4 L_{\\mathtt{ce}}(\\hat P, P^{\\mathtt{aux}}_j))\\,.\n\\end{equation}\nAccordingly, the total loss function is the weighted sum as\n\\begin{equation}\n    L_{\\mathtt{seg}}\\triangleq L_{\\mathtt{primary}}+\\beta_o L_{\\mathtt{ocr}}+\\beta_s L_{\\mathtt{scale}}+\\beta_d L_{\\mathtt{dla}},\n\\end{equation}\nwhere we set $\\beta_o = 0.4$, $\\beta_s = 0.05$, and $\\beta_d = 0.05$.\n\n\\subsection{Boundary Detection}\nFor boundary detection, we opted to using a simpler version of the loss function for semantic segmentation.\nWe use standard binary cross entropy (BCE) to be the primary loss for our final prediction $P_{\\mathtt{final}}$, i.e.,\n\\begin{equation}\n    L_{\\mathtt{primary}}\\triangleq L_{\\mathtt{bce}}(\\hat P, P_{\\mathtt{final}})\\,,\n\\end{equation}\nwhere $\\hat P$ is the ground truth and $L_\\mathtt{bce}$ is the BCE loss function.\n\nWe also use auxiliary segmentation heads to make predictions at each feature level.\nEach prediction $P^{\\mathtt{aux}}_j$, where $1\\leq j\\leq 4$, is upsampled to the original scale and the BCE loss is used to compute the auxiliary loss, i.e.,\n\\begin{equation}\n    L_{\\mathtt{dla}}\\triangleq\\sum_{j=1}^4 L_{\\mathtt{bce}}(\\hat P, P^{\\mathtt{aux}}_j))\\,.\n\\end{equation}\nAccordingly, the total loss function is the weighted sum as\n\\begin{equation}\n    L_{\\mathtt{bd}}\\triangleq L_{\\mathtt{primary}}+\\beta_d L_{\\mathtt{dla}},\n\\end{equation}\nwhere we set $\\beta_d = 0.05$.\n\n\\begin{table}[t]\n  \\caption{Ablation study on Cityscapes validation set with AFA-DLA-X-102 for validating each post-processing technique. SSR indicates our Scale Space Rendering.}\n  \\label{ablation-post-processing}\n  \\centering\n  \\setlength\\tabcolsep{15pt}\n  \\begin{tabular}{ccc|c}\n    \\toprule\n    SSR & Flip & Seg-Fix~\\cite{yuan2020segfix} & mIoU \\\\\n    \\midrule\n    - & - & - & 83.06 \\\\\n    \\checkmark & - & - & 84.81 \\\\\n    \\checkmark & \\checkmark & - &  85.00 \\\\\n    \\checkmark & \\checkmark & \\checkmark & 85.10 \\\\\n    \\bottomrule\n  \\end{tabular}\n\\end{table}\n\n\\begin{table}[t]\n    \\caption{Specific training settings for each dataset. BS stands for training batch size.}\n    \\label{training-setting}\n    \\centering\n    \\begin{tabular}{lccc}\n        \\toprule\n        Dataset & Crop Size & BS & Training Epochs \\\\\n        \\midrule\n        Cityscapes & $2048 \\times 1024$ & 8 & 375 \\\\\n       \n        BDD100K & $1280 \\times 720$ & 16 & 200 \\\\\n        BSDS500 & $416 \\times 416$ & 16 & 10 \\\\\n        NYUDv2 & $480 \\times 480$ & 16 & 54 \\\\\n        \\bottomrule\n    \\end{tabular}\n\\end{table}\n\n\\section{Implementation Details}\nWe provide the general training setting and procedure used for training on Cityscapes~\\cite{cordts2016cityscapes}, BDD100K~\\cite{yu2020bdd100k}, BSDS500~\\cite{arbelaez2010contour}, and NYUDv2~\\cite{silberman2012indoor}.\n\nWe use PyTorch~\\cite{paszke2019pytorch} as our framework and develop based on the NVIDIA semantic segmentation codebase~\\footnote[1]{NVIDIA license: \\url{https:\/\/github.com\/NVIDIA\/semantic-segmentation\/blob\/main\/LICENSE}}. The general training procedure is using SGD~\\cite{robbins1951stochastic} with momentum of 0.9 and weight decay of $10^{-4}$.\nSpecific settings for each dataset are shown in Table~\\ref{training-setting}.\n\n\\subsection{Semantic Segmentation}\nWe use an initial learning rate of $1.0 \\times 10^{-2}$.\nWe use the learning rate warm-up over the initial 1K training iterations and the polynomial decay schedule, which decays the initial learning rate by multiplying $(1 - \\frac{\\mathtt{epoch}}{\\mathtt{max\\_epochs}})^{0.9}$ every epoch.\nWe apply random image horizontal flipping, randomly rotating within 10 degrees, random scales from 0.5 to 2.0, random image color augmentation, and random cropping.\nAs in \\cite{zhu2019improving}, we also use class uniform sampling in the data loader to overcome the data class distribution unbalance problem.\nDue to limitations in computational power, we further use Inplace-ABN~\\cite{bulo2018place} to replace the batch norm and ReLU function to acquire the largest possible training crop size and batch size on 8 Tesla V-100 32G GPUs.\n\n\\subsection{Boundary Detection}\nWe use an initial learning rate of $1.0 \\times 10^{-2}$ and a batch size of 16 for both BSDS500~\\cite{arbelaez2010contour} and NYUDv2~\\cite{silberman2012indoor}.\nWe use the step decay schedule and drop the learning rate by 10 times at around $0.55\\times\\mathtt{max\\_epochs}$ and then again at $0.85\\times\\mathtt{max\\_epochs}$.\nFor augmentation, we follow the standard protocol in literature~\\cite{xie2015holistically,liu2017richer} and apply random flipping, scaling by 0.5 and 1.5, and rotation by 16 different angles.\nWe train all our models on a single GeForce RTX 2080Ti GPU.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.47\\textwidth]{images\/AFA\/AFA-DLA_detail.pdf}\n\t\\caption{ \n        Notation of aggregated features of AFA-DLA.\n\t}\n\t\\label{fig:AFA-DLA_detail}\n\\end{figure}\n\n\\section{Visualization of Attention Maps}\nIn this section, we provide more visualizations of attention maps generated by our proposed AFA module.\nFor reference, we provide a detailed architecture of AFA-DLA and denote the notation of aggregated features of different levels in Figure~\\ref{fig:AFA-DLA_detail}.\n\nWe first look at the attention maps generated by our binary fusion module which aggregates two features in Figure~\\ref{fig:AFA_GATE_1} and Figure~\\ref{fig:AFA_GATE_2}.\nWe provide the spatial attention maps for binary fusion at four different levels.\nWhen the difference of the level information between two input features is larger (e.g., $L^3_4$ and $L^3_5$), our attention mask will become more specific and be able to focus on the right place to be fused.\nTake the fusion of $L^3_4$ and $L^3_5$ as example.\nSince $L^3_4$ contains the information of the $L_1$ feature, our attention focuses on object boundaries on it and attend to the rest on $L^3_5$, which has richer semantic information.\nCompared to linear fusion operations, our AFA module provides a more expressive way of combining features.\n\nWe additionally look at the spatial attention maps generated by our multiple feature fusion module in Figure~\\ref{fig:AFA_Final}.\nOnly using the final aggregated features for prediction may cause our model to overly focus on low level features.\nThus, our multiple feature fusion module provides the model with more flexibility to select between the features that contain different low level information.\nFor input features that contain $L_1$ information like $L^4_5$ and $L^3_5$, the attention focuses more on the object boundaries, similar to our binary fusion module.\nFor other input features like $L^2_5$, the attention can focus on objects or the background.\nWith our multiple feature fusion module, our model can strike a balance between the low-level and the high-level information and perform fusion accordingly.\n\n\\section{Qualitative Results}\nWe provide more qualitative results in this section to visualize AFA-DLA's predictions.\nWe show full predictions of AFA-DLA on Cityscapes in Figure~\\ref{fig:Cityscapes}, BDD100K in Figure~\\ref{fig:BDD}, BSDS500 in Figure~\\ref{fig:BSDS500}, and NYUDv2 in Figure~\\ref{fig:NYUD}.\nThe results on Cityscapes show that our model can handle both fine and coarse details well and is robust towards different input scenes.\nOn BDD100K, the results show the ability of our model to handle more diverse urban scenes, with varying weather conditions and times of the day.\nOn both BSDS500 and NYUDv2, our model can predict both fine-grained scene details as well as object-level boundaries.\nIn particular, on NYUDv2, our model can recover more boundaries than the ground truth.\nWith results across different types of datasets and both semantic segmentation and boundary detection, AFA-DLA demonstrates its strong performance and applicability for dense prediction tasks.\n\n\\clearpage\n\n\\begin{figure*}[t]\n    \\centering\n    \\begin{tabular}{cc}\n        \\toprule\n        \\includegraphics[width=0.4\\linewidth]{images\/AFA\/input_1.pdf}\n        &\n        \\includegraphics[width=0.4\\linewidth]{images\/AFA\/gt_1.png} \\\\\n        Input\n        &\n        Ground Truth \\\\\n        \\includegraphics[width=0.4\\linewidth]{images\/AFA\/L1_L2_1.pdf}\n        &\n        \\includegraphics[width=0.4\\linewidth]{images\/AFA\/L2_L1_1.pdf} \\\\\n        spatial attention for $L_1$\n        &\n        spatial attention for $L_2$ \\\\\n        \\includegraphics[width=0.4\\linewidth]{images\/AFA\/L2_L3_2.pdf}\n        &\n        \\includegraphics[width=0.4\\linewidth]{images\/AFA\/L3_L2_2.pdf} \\\\\n        spatial attention for $L^1_2$ \n        &\n        spatial attention for $L^1_3$ \\\\\n        \\includegraphics[width=0.4\\linewidth]{images\/AFA\/L3_L4_3.pdf}\n        &\n        \\includegraphics[width=0.4\\linewidth]{images\/AFA\/L4_L3_3.pdf} \\\\\n        spatial attention for $L^2_3$ \n        &\n        spatial attention for $L^2_4$ \\\\\n        \\includegraphics[width=0.4\\linewidth]{images\/AFA\/L4_L5_4.pdf}\n        &\n        \\includegraphics[width=0.4\\linewidth]{images\/AFA\/L5_L4_4.pdf} \\\\\n        spatial attention for $L^3_4$ \n        &\n        spatial attention for $L^3_5$ \\\\\n        \\bottomrule\n    \\end{tabular}\n    \\caption{Spatial attention maps at four different levels generated by our binary fusion module which aggregates two features. Whiter regions denote higher attention. Compared to linear fusion operations, our AFA module provides a more expressive way of combining features.}\n    \\label{fig:AFA_GATE_1}\n\\end{figure*}\n\n\\begin{figure*}[t]\n    \\centering\n    \\begin{tabular}{cc}\n        \\toprule\n        \\includegraphics[width=0.4\\linewidth]{images\/AFA\/input_2.pdf}\n        &\n        \\includegraphics[width=0.4\\linewidth]{images\/AFA\/gt_2.png} \\\\\n        Input\n        &\n        Ground Truth \\\\\n        \\includegraphics[width=0.4\\linewidth]{images\/AFA\/L1_L2.pdf}\n        &\n        \\includegraphics[width=0.4\\linewidth]{images\/AFA\/L2_L1.pdf} \\\\\n        spatial attention for $L_1$\n        &\n        spatial attention for $L_2$ \\\\\n        \\includegraphics[width=0.4\\linewidth]{images\/AFA\/L2_L3.pdf}\n        &\n        \\includegraphics[width=0.4\\linewidth]{images\/AFA\/L3_L2.pdf} \\\\\n        spatial attention for $L^1_2$ \n        &\n        spatial attention for $L^1_3$ \\\\\n        \\includegraphics[width=0.4\\linewidth]{images\/AFA\/L3_L4.pdf}\n        &\n        \\includegraphics[width=0.4\\linewidth]{images\/AFA\/L4_L3.pdf} \\\\\n        spatial attention for $L^2_3$ \n        &\n        spatial attention for $L^2_4$ \\\\\n        \\includegraphics[width=0.4\\linewidth]{images\/AFA\/L4_L5.pdf}\n        &\n        \\includegraphics[width=0.4\\linewidth]{images\/AFA\/L5_L4.pdf} \\\\\n        spatial attention for $L^3_4$ \n        &\n        spatial attention for $L^3_5$ \\\\\n        \\bottomrule\n    \\end{tabular}\n    \\caption{Spatial attention maps at four different levels generated by our binary fusion module which aggregates two features. Whiter regions denote higher attention. Compared to linear fusion operations, our AFA module provides a more expressive way of combining features.}\n    \\label{fig:AFA_GATE_2}\n\\end{figure*}\n\n\\begin{figure*}[t]\n    \\centering\n    \\begin{tabular}{ccc}\n        \\toprule\n        \\includegraphics[width=0.3\\linewidth]{images\/AFA\/L5_4.pdf}\n        &\n        \\includegraphics[width=0.3\\linewidth]{images\/AFA\/L5_3.pdf}\n        &\n        \\includegraphics[width=0.3\\linewidth]{images\/AFA\/L5_2.pdf} \\\\\n        spatial attention for $L^4_5$\n        &\n        spatial attention for $L^3_5$\n        & \n        spatial attention for $L^2_5$ \\\\\n        \\includegraphics[width=0.3\\linewidth]{images\/AFA\/L5_4_2.pdf}\n        &\n        \\includegraphics[width=0.3\\linewidth]{images\/AFA\/L5_3_2.pdf}\n        &\n        \\includegraphics[width=0.3\\linewidth]{images\/AFA\/L5_2_2.pdf} \\\\\n        spatial attention for $L^4_5$\n        &\n        spatial attention for $L^3_5$\n        & \n        spatial attention for $L^2_5$ \\\\\n        \\bottomrule\n    \\end{tabular}\n    \\caption{\n        Spatial attention maps generated by our multiple feature fusion module which aggregates multiple features. Whiter regions denote higher attention. With our multiple feature fusion module, our model can strike a balance between the low-level and the high-level information and perform fusion accordingly.\n    }\n\t\\label{fig:AFA_Final}\n\\end{figure*}\n\n\\begin{figure*}[t]\n    \\centering\n    \\begin{tabular}{ccc}\n        \\toprule\n        Input & Ground Truth & Prediction \\\\\n        \\midrule\n        \\includegraphics[width=0.3\\linewidth]{images\/Cityscapes\/input_1.pdf}\n        &\n        \\includegraphics[width=0.3\\linewidth]{images\/Cityscapes\/gt_1.png}\n        &\n        \\includegraphics[width=0.3\\linewidth]{images\/Cityscapes\/prediction_1.png} \\\\\n        \\includegraphics[width=0.3\\linewidth]{images\/Cityscapes\/input_2.pdf}\n        &\n        \\includegraphics[width=0.3\\linewidth]{images\/Cityscapes\/gt_2.png}\n        &\n        \\includegraphics[width=0.3\\linewidth]{images\/Cityscapes\/prediction_2.png} \\\\\n        \\includegraphics[width=0.3\\linewidth]{images\/Cityscapes\/input_3.pdf}\n        &\n        \\includegraphics[width=0.3\\linewidth]{images\/Cityscapes\/gt_3.png}\n        &\n        \\includegraphics[width=0.3\\linewidth]{images\/Cityscapes\/prediction_3.png} \\\\\n        \\includegraphics[width=0.3\\linewidth]{images\/Cityscapes\/input_4.pdf}\n        &\n        \\includegraphics[width=0.3\\linewidth]{images\/Cityscapes\/gt_4.png}\n        &\n        \\includegraphics[width=0.3\\linewidth]{images\/Cityscapes\/prediction_4.png} \\\\\n        \\bottomrule\n    \\end{tabular}\n    \\caption{\n        The qualitative results of AFA-DLA-X-102 on the Cityscapes validation set. Our model can handle both fine and coarse details well and is robust towards different input scenes.\n    }\n\t\\label{fig:Cityscapes}\n\\end{figure*}\n\n\\begin{figure*}\n    \\centering\n    \\begin{tabular}{ccc}\n        \\toprule\n        Input & Ground Truth & Prediction \\\\\n        \\midrule\n        \\includegraphics[width=0.3\\linewidth]{images\/BDD100K\/input_1.pdf}\n        &\n        \\includegraphics[width=0.3\\linewidth]{images\/BDD100K\/gt_1.png}\n        &\n        \\includegraphics[width=0.3\\linewidth]{images\/BDD100K\/prediction_1.png} \\\\\n        \\includegraphics[width=0.3\\linewidth]{images\/BDD100K\/input_2.pdf}\n        &\n        \\includegraphics[width=0.3\\linewidth]{images\/BDD100K\/gt_2.png}\n        &\n        \\includegraphics[width=0.3\\linewidth]{images\/BDD100K\/prediction_2.png} \\\\\n        \\includegraphics[width=0.3\\linewidth]{images\/BDD100K\/input_3.pdf}\n        &\n        \\includegraphics[width=0.3\\linewidth]{images\/BDD100K\/gt_3.png}\n        &\n        \\includegraphics[width=0.3\\linewidth]{images\/BDD100K\/prediction_3.png} \\\\\n        \\includegraphics[width=0.3\\linewidth]{images\/BDD100K\/input_4.pdf}\n        &\n        \\includegraphics[width=0.3\\linewidth]{images\/BDD100K\/gt_4.png}\n        &\n        \\includegraphics[width=0.3\\linewidth]{images\/BDD100K\/prediction_4.png} \\\\\n        \\bottomrule\n    \\end{tabular}\n\t\\caption{\n\t    Qualitative results of AFA-DLA-169 on the BDD100K validation set. Our model can handle diverse urban scenes, with varying weather conditions and times of the day.\n\t}\n\t\\label{fig:BDD}\n\\end{figure*}\n\n\\begin{figure*}[t]\n    \\centering\n    \\begin{tabular}{ccc}\n        \\toprule\n        Input & Ground Truth & Prediction \\\\\n        \\midrule\n        \\includegraphics[width=0.3\\linewidth]{images\/BSDS500\/bsds1_img.jpg}\n        &\n        \\includegraphics[width=0.3\\linewidth]{images\/BSDS500\/bsds1_gt.png}\n        &\n        \\includegraphics[width=0.3\\linewidth]{images\/BSDS500\/bsds1_ours.png} \\\\\n        \\includegraphics[width=0.3\\linewidth]{images\/BSDS500\/bsds2_img.jpg}\n        &\n        \\includegraphics[width=0.3\\linewidth]{images\/BSDS500\/bsds2_gt.png}\n        &\n        \\includegraphics[width=0.3\\linewidth]{images\/BSDS500\/bsds2_ours.png} \\\\\n        \\includegraphics[width=0.3\\linewidth]{images\/BSDS500\/bsds3_img.jpg}\n        &\n        \\includegraphics[width=0.3\\linewidth]{images\/BSDS500\/bsds3_gt.png}\n        &\n        \\includegraphics[width=0.3\\linewidth]{images\/BSDS500\/bsds3_ours.png} \\\\\n        \\includegraphics[width=0.3\\linewidth]{images\/BSDS500\/bsds4_img.jpg}\n        &\n        \\includegraphics[width=0.3\\linewidth]{images\/BSDS500\/bsds4_gt.png}\n        &\n        \\includegraphics[width=0.3\\linewidth]{images\/BSDS500\/bsds4_ours.png} \\\\\n        \\bottomrule\n    \\end{tabular}\n    \\caption{\n        Qualitative results of AFA-DLA-34 on the BSDS500 test set. Results are raw boundary maps obtained using multi-scale inference before Non-Maximum Suppression. Our model can predict both fine-grained scene details and object-level boundaries.\n    }\n\t\\label{fig:BSDS500}\n\\end{figure*}\n\n\\begin{figure*}[t]\n    \\centering\n    \\begin{tabular}{ccc}\n        \\toprule\n        Input & Ground Truth & Prediction \\\\\n        \\midrule\n        \\includegraphics[width=0.3\\linewidth]{images\/NYUD\/nyud1_img.pdf}\n        &\n        \\includegraphics[width=0.3\\linewidth]{images\/NYUD\/nyud1_gt.png}\n        &\n        \\includegraphics[width=0.3\\linewidth]{images\/NYUD\/nyud1_ours.pdf} \\\\\n        \\includegraphics[width=0.3\\linewidth]{images\/NYUD\/nyud2_img.pdf}\n        &\n        \\includegraphics[width=0.3\\linewidth]{images\/NYUD\/nyud2_gt.png}\n        &\n        \\includegraphics[width=0.3\\linewidth]{images\/NYUD\/nyud2_ours.pdf} \\\\\n        \\includegraphics[width=0.3\\linewidth]{images\/NYUD\/nyud3_img.pdf}\n        &\n        \\includegraphics[width=0.3\\linewidth]{images\/NYUD\/nyud3_gt.png}\n        &\n        \\includegraphics[width=0.3\\linewidth]{images\/NYUD\/nyud3_ours.pdf} \\\\\n        \\includegraphics[width=0.3\\linewidth]{images\/NYUD\/nyud4_img.pdf}\n        &\n        \\includegraphics[width=0.3\\linewidth]{images\/NYUD\/nyud4_gt.png}\n        &\n        \\includegraphics[width=0.3\\linewidth]{images\/NYUD\/nyud4_ours.pdf} \\\\\n        \\bottomrule\n    \\end{tabular}\n    \\caption{\n        Qualitative results of AFA-DLA-34 on the NYUDv2 test set. Results are raw boundary maps obtained by averaging predictions on both RGB and HHA images before Non-Maximum Suppression. Our model can extract more boundaries than the ground truth.\n    }\n\t\\label{fig:NYUD}\n\\end{figure*}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nElectromagnetically induced transparency (EIT) \\cite{1,2,3,4,5,6,7}, which is a quantum interference phenomenon that takes place when two laser fields excite resonantly and two different transitions share a common state, provides counter-intuitive ways that how a strong laser field can dress a dielectric medium which leads to the modification of the absorptive and dispersive characteristics of the medium \\cite{8}. This alters the group index of the medium such that group velocity of a weak probe laser injected in the dressed medium can be modified in a controllable fashion, which engenders slow and fast (a.k.a. superluminal) light effects \\cite{8,9,10,11,12,13}.\n\nVarious techniques have been developed to realize slow and fast light in atomic vapors and solid-state media \\cite{8,11}. Techniques related to photon-echo spectroscopy, such as atomic frequency combs \\cite{14} and controllable reversible inhomogeneous broadening \\cite{15}, have been used to efficiently achieve quantum memories via quantum storage and optical buffering. On the other hand, the rapid dephasing rates and inhomogeneous broadening of solid-state electronic resonances, have suggested a plethora of methods and techniques to control the propagation of light. For instance, the techniques based on stimulated Brillouin scattering (SBS) in fibers \\cite{16} and coherent population oscillations (CPO) \\cite{17} have been used to significantly delay classical light fields. In chip-scale photonics, dynamically tunable arrays of cavities in EIT configuration, are an intriguing analogue to ensembles of atoms and provide a route to slowing and even stopping light pulses all-optically \\cite{18,19}.\n\nRecently, the field of cavity optomechanics has cemented present-day photonic technology in view of its promising applications from microscopic to macroscopic domains \\cite{20,21,22,23}. In optomechanical resonators, an optical mode couples to mechanical vibrations via radiation pressure induced by circulating optical fields \\cite{20}. Generally, the combination of additional matter-like systems (for example, mechanical membranes \\cite{24,25,27,26}, single multi-level atoms \\cite{28,29,30,31,32,31} and Bose Einstein Condensate\/Fermions \\cite{33,34,35}) with these opto-mechanical resonators, leads to the emergence of hybrid quantum systems \\cite{22,36,37}, which serve as basic building blocks from quantum state-engineering to the quantum communication networks \\cite{38,39,40}. Due to the ubiquitous nature of the mechanical motion, such resonators couple with many kinds of quantum devices, from atomic systems to the solid-state electronic circuits \\cite{32,Tian}. An excellent example of such a system is quantum electromechanical system (QEMS) \\cite{46,47,Tian}. QEMS is a device \\cite{46,47,Tian,52,522} where, in the observable behavior, the quantum nature of either the electronic or mechanical degrees of freedom becomes important. Such a system was earlier suggested by Wineland {\\it et al.} \\cite{44,45}, and later by Zoller \\& Tian \\cite{46} and Milburn and co-workers \\cite{47}. In QEMS, the two charged cantilevers are coupled electrostatically via Coulomb interaction \\cite{47,44,Bachtold}. In cavity optomechanical and electromechanical systems, the resolved-sideband regime can now be reached with the mechanical frequency far surpassing the cavity bandwidth \\cite{53,54,55,56,57,Bennett}. Moreover, mechanically-induced transparency due to the destructive interference of the probe field with the anti-Stokes field generated by the pump field \\cite{4,6} and normal-mode splitting in the strong coupling limit \\cite{58,59} have been reported.\n\nOur report is based on combining these electromechanically coupled charged cantilevers with optomechanics, that bridges the traditional quantum optics based OMS with electrostatics based QEMS, providing us a new hybrid testbed, viz. quantum opto-electromechanical system (QOEMS) \\cite{Tian,48,50,51,Wang,Tsang,Terry,Regal}. In analogy with multi-level atomic systems, hybrid optomechanics empowers us with a coherent control to the engineered couplings that give rise to Fano-like interferences \\cite{5,6,7,Clerk} and even double EIT phenomenon \\cite{7,60,61}. Here, using the hybrid QOEMS, we demonstrate a scheme that provides us a tunable switch from slow light to fast light in the probe transmission.\nUnlike previous schemes \\cite{63,64,65,66}, our scheme owns some favorable features: {\\it (i)} Two transmission windows appear in our model, due to the additional MR, that causes the high dispersion and leads to the switching mechanism from slow light to fast light in the probe transmission. {\\it (ii)} Slow light propagation can be observed when both the coupling parameters contribute together, which can be adjusted by controlling the Coulomb coupling strength and the pump power. {\\it (iii)} Slow light regime in our model is astonishingly robust to the cavity decay rate. {\\it (iv)} A tunable switch from slow light to fast light can be obtained by turning off the Coulomb coupling strength as $g_c=0$, which can be adjusted by the pump power.  {\\it (v)} Both slow and fast light effects can be observed in the probe transmission in a single experiment.\n\\section{The Model Formulation}\\label{sec2}\nA schematic representation of our proposed model is illustrated in Fig.~\\ref{setup}. We consider a realistic optomechanical system, composed of a high $Q$ Fabry-P\\'{e}rot cavity of length $L$, with a fixed mirror and a movable nano mechanical resonator (MR). Specifically, the mechanical resonator consists of two charged MRs, where the $MR_{1}$ of the optomechanical cavity couples to the cavity field via radiation pressure, as well as electrostatically coupled to another movable oscillator outside the cavity (i.e., MR$_{2}$) through the tunable Coulomb coupling strength, viz. $g_{c}$. Such an eletromechanical subsystem in our QOEMS was earlier suggested by Wineland {\\it et al.} \\cite{44,45}, and later by Milburn and co-workers \\cite{47}, and Zoller \\& Tian \\cite{46} where an ion is trapped between two parallel suspended electrodes represented by MRs (or nanowires or nanotubes). This coupling can be switched on and off using an external bias voltage at an electrode on the oscillators, where MR$_{1}$ is charged by the bias gate voltage $V_{1}$ and subject to the Coulomb force due to another charged MR$_{2}$ with the bias gate voltage $-V_{2}$. Thus, the system has usual optomechanical coupling ($g_{o}$), and the additional tunable Coulomb coupling ($g_c$), forming a hybrid quantum eletro-optomechanical system (QOEMS). The optomechanical cavity is simultaneously driven by an intense pump field of frequency $\\omega_l$, and a weak probe field of frequency $\\omega_p$, along the cavity axis. Hence, under the rotating reference frame at the frequency $\\omega_l$ of the strong driving field, the total Hamiltonian of the system can be written as:\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.5\\textwidth]{fig1.eps}\n\\caption{The schematic of the opto-electromechanical system. A high $Q$ Fabry-P\\'{e}rot cavity of length $L$ consists of a fixed mirror and a movable mirror MR$_{1}$ with small displacement $q_1$. MR$_{1}$ is charged by the bias gate voltage $V_{1}$, and subject to the Coulomb force due to another charged MR$_{2}$ with the bias gate voltage $-V_{2}$. The parameter $g_c$ represents the Coulomb coupling strength between the two mechanical oscillators.}\\label{setup}\n\\end{figure}\n\\begin{eqnarray}\n&&H_T=[\\frac{p_1^2}{2m_1}+\\frac{1}{2}m_1\\omega_1^2q_1^2]\n+[\\frac{p_2^2}{2m_2}+\\frac{1}{2}m_2\\omega_2^2q_2^2] \\notag \\\\ &&+ \\hbar\\Delta_cc^\\dag c -\\hbar g_o c^\\dag cq_1\n+\\hbar g_c q_1q_2  +i\\hbar\\Omega_l(c^\\dag -c) \\notag \\\\\n&&+ i\\hbar(\\varepsilon_p e^{-i\\delta t}c^\\dagger -\\varepsilon_p^* e^{i\\delta t}c), \\label{Ham}\n\\end{eqnarray}\nwhere $\\Delta_c=\\omega_c-\\omega_l$ and $\\delta=\\omega_p-\\omega_l$  are the detunings of the cavity field frequency and the probe field frequency respectively, with strong pump field frequency $\\omega_l$.\nIn Eq.~(\\ref{Ham}), first two terms describe the free Hamiltonian of the moving mirrors (MR$_{1}$ and MR$_{2}$). The operators, $q_i$ and $p_i$, represent the position and momentum operators of both the mirrors respectively, vibration frequency $\\omega_{i}$, and effective mass $m_{i}$. Third term represents single-mode of cavity field with frequency $\\omega_{c}$ and annihilation (creation) operator $c~(c^{\\dag})$. The fourth term expresses optomechanical coupling between MR$_{1}$ and the cavity field with the coupling strength $g_o=\\frac{\\omega_{c}}{L}\\sqrt{\\hbar\/m\\omega_1}$. The fifth term represents the Coulomb coupling between the charged MRs with the Coulomb coupling strength $g_c =\\frac{C_{1}V_{1}C_{2}V_{2}}{2\\pi \\hbar \\epsilon _{0}x_{0}^{3}}$ \\cite{47, 60, 46}; where MR$_{1}$ and MR$_{2}$ take the charges $C_{1}V_{1}$ and $-C_{2}V_{2}$, with $C_{1}(C_{2})$ and $V_{1}(-V_{2})$ being the capacitance and the voltage on the mirrors respectively, and $x_0$ is the equilibrium separation between the two oscillators. Finally, the last two terms correspond to the classical light fields (pump and probe fields) with frequencies $\\omega_l$ and $\\omega_p$, respectively. Furthermore, $\\Omega_l$ and $\\varepsilon_p$ are related to the laser power $P$ by $|\\Omega_l|=\\sqrt{2\\kappa P_{l}\/ \\hbar \\omega_l}$, and $\\varepsilon_p=\\sqrt{2\\kappa P_{p}\/ \\hbar \\omega_p}$.\n\nSince here we deal with the mean response of the coupled system to the probe field, we identify the expectation values of all the operators. By taking the corresponding dissipation and fluctuation terms into account, and using the mean field approximation \\cite{7}, i.e. $\\langle qc\\rangle =\\langle q\\rangle \\langle c\\rangle $, the mean value equations are given by,\n\\begin{eqnarray}\\label{mve}\n&&\\langle \\dot{q_{1}}\\rangle =\\frac{\\langle p_{1}\\rangle }{m_{1}},  \\notag\n~~~ \\langle \\dot{q_{2}}\\rangle =\\frac{\\langle p_{2}\\rangle }{m_{2}},  \\notag \\\\\n&&\\langle \\dot{p_{1}}\\rangle =-m_{1}\\omega _{1}^{2}\\langle\nq_{1}\\rangle -\\hbar g_{c} \\langle q_{2}\\rangle +\\hbar g_o\\langle\nc^{\\dag }\\rangle \\langle\nc\\rangle -\\gamma _{1}\\langle p_{1}\\rangle ,  \\notag \\\\\n&&\\langle \\dot{p_{2}}\\rangle =-m_{2}\\omega _{2}^{2}\\langle\nq_{2}\\rangle -\\hbar g_{c} \\langle q_{1}\\rangle -\\gamma _{2}\\langle\np_{2}\\rangle,\n\\notag \\\\\n&&\\langle \\dot{c}\\rangle =-(\\kappa +i\\Delta _{c})\\langle c\\rangle + ig_o\\langle q_{1}\\rangle \\langle c\\rangle +\\Omega_{l}+\\varepsilon_{p}e^{-i\\delta t}.\n\\end{eqnarray}\n\nHere, $\\gamma _{1}$ and $\\gamma _{2}$ are the decay rates associated with MR$\n_{1}$ and MR$_{2}$, respectively. As we encounter the expectation values of all the operators in Eq.~(\\ref{mve}), therefore, we drop the quantum Brownian noise and input vacuum noise terms which average to zero. In order to acquire the steady-state solutions of the above equations, we make the ansatz~\\cite{7,6}: $\n\\langle h \\rangle = h_s +  h_- e^{-i\\delta t} + h_+ e^{i\\delta t}$,\nwhere, $h_s$ denotes any of the steady-state solutions $c_s$, $q_{is}$ and $p_{is}$. Moreover, $h_\\pm$ are of the same order as $\\varepsilon_p$. In the case of $h_{s}\\gg h_{\\pm }$, Eq. (\\ref{mve}) can be solved by treating\n$h_{\\pm }$ as perturbations. Upon substituting the ansatz into Eqs.~(\\ref{mve}) and upon working to the lowest order in $\\varepsilon_p$, we obtain the following steady-state solutions:\n\\begin{eqnarray}\\label{mean}\n&&\\notag q_{1s}=\\frac{\\hbar g_o|c_{s}|^{2}}{m_{1}\\omega _{1}^{2}-\\frac{\\hbar\n^{2}g_c ^{2}}{m_{2}\\omega _{2}^{2}}}, ~~~\nq_{2s}=\\frac{-\\hbar g_c q_{1s}}{m_{2}\\omega _{2}^{2}}, ~~~    \\\\\n&&c_{s}=\\frac{\\Omega_{l}}{\\kappa+i\\Delta}, \\\\\n&&c_{-}=\\frac{[\\kappa -i(\\Delta +\\delta )][(\\delta ^{2}-\\omega\n_{1}^{2}+i\\delta \\gamma _{1})-\\alpha] - 2i\\omega _{1}\\beta }{[\\Delta\n^{2}-(\\delta +i\\kappa )^{2}][(\\delta\n^{2}-\\omega_{1}^{2}+i\\delta\\gamma_{1}) - \\alpha]+4\\Delta \\omega_{1}\\beta }. \\nonumber\n\\end{eqnarray}\nHere, $\\alpha=\\frac{ \\hbar^2g_c^{2}}{m_1m_2(\\delta\n^{2}-\\omega_{2}^{2}+i\\delta\\gamma_{2})}$, $\\beta =\\frac{\\hbar g_o^{2}|c_{s}|^{2}}{2m\\omega_{1}}$, and $\\Delta=\\Delta_c - gq_{1s}$ is the effective detuning. In order to investigate the optical property of the output field, we use the standard input-output relation \\cite{4}: $c_{out}(t) = c_{in}(t) - \\sqrt{2\\kappa}c(t)$, where $c_{in}$ and $c_{out}$ are the input and output operators, respectively. By using the above input-output relation and the ansatz for $\\langle c(t) \\rangle$, we can obtain the expectation value of the output field as \\cite{13,6},\n\\begin{eqnarray}\n\\langle c_{out}(t) \\rangle &=& (\\Omega_l-\\sqrt{2\\kappa} c_s) + (\\varepsilon_p - \\sqrt{2\\kappa} c_-) e^{-i\\delta t} \\notag \\\\\n&-& \\sqrt{2\\kappa} c_+ e^{i\\delta t}.\n\\end{eqnarray}\n\nNow, the transmission of the probe field, which is ratio of the returned probe field from the coupling system divided by the sent probe field \\cite{6,4}, can be acquired as\n\\begin{equation}\nt_p(\\omega_p) = \\frac{\\varepsilon_p - \\sqrt{2\\kappa} c_-}{\\varepsilon_{p}} = 1- \\frac{\\sqrt{2\\kappa} c_-}{\\varepsilon_p}.\n\\end{equation}\nImportantly, for an optomechanical system, the presence of the pump laser not only induce a strong modification of the transmission of the probe field, but also leads at the same time to a rapid variation of the phase of the transmitted probe field across the transmission window. The rapid phase dispersion \\cite{6}, that is, $\\phi_t(\\omega_p) = arg[t_p(\\omega_p)]$, can\nlead to significant group delays in analogy to that we achieve with EIT in atomic \\cite{69,70} and in solid-state media \\cite{13,6}. The transmission group delay \\cite{4,6} is given as\n\\begin{equation}\n\\tau_g = \\dfrac{d\\phi_t(\\omega_p)}{d\\omega_p}=\\dfrac{d\\{arg[t_p(\\omega_p)]\\}}{d\\omega_p}, \\label{phase}\n\\end{equation}\nwhere, $\\tau_g > 0$ and $\\tau_g < 0$ respectively, correspond to slow and fast light propagation \\cite{8,6,13}. In the following section, we show that two transmission windows occur due to additional mechanical oscillator in the hybrid opto-electromechanical system. The parameter $g_c$ serves as tunable switch that causes the normal dispersion (slow light) when it is present, and yields anomalous dispersion (fast light) when it is zero (or off). Consequently, we expect slow and fast light regimes in the probe transmission in hybrid QOEMS.\n\\section{Results and Discussions}\\label{sec3}\nIn order to quantify the slow and fast light effects of the transmitted probe field, in our numerical simulations, we choose experimentally realizable parametric values of the optomechanical system. We use the parameters from recent experiment \\cite{57}: $L=$25 mm, $\\lambda=$1064 nm, $\\omega_i\/ 2\\pi = 947$ KHz ($i=1,2$), $Q_{i}=\\frac{\\omega_i}{\\gamma_i}=6700$, $m_i=145$ ng, $\\kappa\/2\\pi=215$ KHz, $P_l=6$ $\\mu$W, and $g_c\/2\\pi=8 \\times 10^{6}$ Hz\/m$^{2}$ \\cite{46}. Note that $\\omega_1 > \\kappa$, therefore, the system operates in the resolved-sideband regime \\cite{67,6}. In the following subsections, we explain the occurrence of slow and fast light regimes in the QOEMS, for $g_c \\neq 0$ and $g_c=0$, respectively.\n\\begin{figure}[htb]\n\\includegraphics[width=.48\\textwidth]{fig2}\n\\caption{Plot of the tranmission $|t_p|^2$ versus the normalized probe detuning as a function of the Coulomb interaction $g_c$ (in units of $2\\pi \\times g_c$ MHz\/m${^2}$). The experimental parameters are: $L=25$ mm,  $\\lambda=$1064 nm, $\\omega_{1,2}\/2\\pi=947$ KHz, $Q_{1,2}=\\frac{\\omega_{1,2}}{\\gamma_{1,2}}=6700$, $m_{1,2}=145$ ng, $\\kappa\/2\\pi=215$ KHz, and $\\Omega_l\/2\\pi=2$ MHz \\cite{57}. For $g_c=0$, single transmission window appears, and two trasmission windows occur once the Coulomb coupling is present.}\\label{tr1}\n\\end{figure}\n\\subsection{Slow light regime}\nWe consider Coulomb interaction ($g_{c}$) between the two oscillators together with optomechanical coupling $g_{o}$, in the QOEMS, to control the propagation of the probe field. In Fig.~\\ref{tr1}, we plot the transmission spectrum of the probe field as a function of normalized probe detuning $\\overline{\\delta}\/\\omega_{1}$ for different values of Coulomb coupling $g_{c}$. Here, $\\overline{\\delta}=\\delta-\\omega_1$ is the probe detuning from the line center. In the absence of the Coulomb coupling i.e. $g_c=0$, in Fig.~\\ref{tr1}, we see that a single peak in the transmission spectrum appears, at $\\overline{\\delta} = 0$ ($\\overline{\\delta} = \\omega_1$) in agreement with a previous report for a single MR \\cite{4}.\n\nHowever, the characteristics of the transmitted probe field changes in the presence of the Coulomb coupling between the two MRs and remarkably two windows occur in the probe transmission spectrum with different centers. Figure.~\\ref{tr1} illustrates that increasing the value of the Coulomb coupling broadens the central window, and the two central peaks in the transmission spectrum are further and further apart. Analytically, we note that when there is no Coulomb coupling, $g_c=0$ (i.e. $\\alpha=0$) between the two MRs, see Eq.~(\\ref{mean}), which reduces to Eq.~(5) in Ref.~\\cite{67}. However, different from the output field in Ref.~\\cite{67} which involves a single center frequency for the single transparency window, there are two centers with different frequencies in our model due to the Coulomb interaction. Consequently, under the action of the radiation pressure and the probe field, two transparency windows appear in the transmission profiles. This effect is analogous to the phenomenon of double EIT, that has been extensively reported in various hybrid systems, e.g. in plasmonic nanocavity \\cite{68}, hybrid optomechanics \\cite{7} and optomechanical ring cavity \\cite{62}.\n\n\\begin{figure}[t]\n\\includegraphics[width=0.45\\textwidth]{fig3}\n\\caption{The phase, $\\phi_t$, of the probe field as a function of normalized probe detuning, $\\overline{\\delta}\/\\omega_1$, subject to the Coulomb interaction, for $g_c\/2\\pi=0.1$ MHz\/m$^2$ (dashed-blue curve) and $g_c\/2\\pi=0.2$ MHz\/m$^2$ (solid-black curve). All parameters are the same as in Fig.~\\ref{tr1}.} \\label{ph1}\n\\end{figure}\n\\begin{figure}[t]\n\\includegraphics[width=0.5\\textwidth]{fig4}\n\\caption{(a) Plot of the group delay $\\tau_g$ as a function of the pump power $P_l$ for $\\kappa=2\\pi \\times 215$ KHz, and $g_c\/2\\pi=2,4,8$ MHz\/m$^2$ (from bottom to top). (b) Plot of $\\tau_g$ as a function of the pump power $P_l$, for $g_c\/2\\pi=8$ MHz\/m$^2$, and $\\kappa=\\pi \\times 215$, $2\\pi \\times 215$, $3\\pi \\times 215$ KHz (from bottom to top). All parameters are the same as in Fig.~\\ref{tr1}.} \\label{delay1}\n\\end{figure}\nIn Fig.~\\ref{ph1}, we present the phase of the probe field versus the normalized probe detuning $\\overline{\\delta}\/\\omega_{1}$ for $g_c\/2\\pi=0.1,0.2$ MHz\/m$^2$ (solid curve, dashed curve). Due to the presence of two MRs, we observe a high positive dispersion indicating the occurrence of slow light in the probe transmission. In the control of slow light, it is always advantageous to obtain the high dispersion as it greatly alters the group delay of the pulse injected in the transparent medium \\cite{8,6}. Here, we show that by exploiting the electromechanical coupling $g_c$ in the hybrid QOEMS, high dispersion is acquired.\n\nIn order to examine the quantitative features of the probe propagation with subject to the electromechanical coupling $g_c$, we present the group delay $\\tau_g$ versus the pump power $P_l$ in Fig.~\\ref{delay1}. A positive group delay indicates a slow light in the transmitted probe field in the presence of $g_c$ as expected. We note that: {\\it (i)} slow light occurs when the Coulomb interaction dominates the optomechanical coupling, and the magnitude of the delay can be increased with increase in $g_c$, as shown in Fig.~\\ref{delay1}(a). Here $\\kappa$ is fixed, and $g_c$ takes different values.\n{\\it (ii)} In Fig.~\\ref{delay1}(b), group delay $\\tau_g$ is shown for $g_c\/2\\pi=8$ MHz\/m$^2$, and for the different values of $\\kappa$. Surprisingly, we see that with increase in $\\kappa$, the magnitude of delay increases, which reflects that our slow light regime is robust against the cavity decay rate, which is seldom reported before. This directly confirms the occurrence of slow light regime mentioned in {\\it (i)}, which shall become clear in the later discussion where we obtain the superluminal behavior of the probe field by keeping the parameter, $g_c$, as switched off. At grass-root level, the underlying mechanism behind the robustness of slow light confirms the basis of cavity optomechanics and theory of radiation pressure. This can be understood as follows: Under the constant Coulomb interaction and the constant drive (pump field), the equilibrium position follows the strain of the MR. With the increase in $\\kappa$, the radiation pressure on MR$_{1}$ decreases, while both the MRs will acquire large displacement to provide large strain in order to compensate the reduced radiation pressure \\cite{20,Tian}. The transmission spectrum becomes narrower for the larger displacement of the MRs. In this way, slow light regime in QOEMS is robust against the cavity decay rate $\\kappa$.\n\\subsection{Fast light regime}\nHere, we consider the case when Coulomb coupling is switched off or very small, i.e. $g_c\\simeq 0$. This reduces the hybrid QOEMS to a single-ended OMS. For the present case, in Fig.~\\ref{tr2}, we show the transmission $|t_p|^2$ of the probe field as a function of normalized probe detuning ($\\overline{\\delta}\/\\omega_1$), for different values of optomechanical coupling $g_o$. We see that, in the absence of pump laser, i.e. $g_{o}=0$, a standard Lorentzian curve \\cite{4,13} appears (thick-solid curve) as shown in Fig.~\\ref{tr2}.\n\\begin{figure}[ht]\n\\includegraphics[width=0.45\\textwidth]{fig5}\n\\caption{(a) The transmission $|t_p|^2$ of the probe field as a function of normalized probe detuning $\\overline{\\delta}\/\\omega_1$ in the absence of the Coulomb interaction for $g_{o}=0,0.05,0.1$ MHz, respectively. (b) The phase $\\phi_t$ of the probe field versus the normalized probe detuning for $g\/2\\pi=0.1$ MHz in the absence of the Coulomb interaction, i.e. $g_{c} \\simeq 0$. All parameters are the same as in Fig.~\\ref{tr1}.} \\label{tr2}\n\\end{figure}\n\\begin{figure}[ht]\n\\includegraphics[width=0.48\\textwidth]{fig6.eps}\n\\caption{Plot of the the group delay $\\tau_g$ as a function of the pump power $P_l$, for different values of cavity decay rate $\\kappa$. All parameters are the same as in Fig.~\\ref{tr2}.} \\label{ph2}\n\\end{figure}\nHowever, in the presence of intense pump laser, the transmission spectrum of the probe field shows a prominent transparency window at the resonant region, viz., $\\overline{\\delta} \\sim \\omega_1$. Form Fig.~\\ref{tr2}(a), it can be seen that transparency window broadens and becomes more transparent by increasing the optomechanical coupling, in agreement with previous reports \\cite{13,4,6,63}. Nevertheless, it reflects that our approach is an extension of previously reported single-ended cavities, here our motivation is to acquire a tunable switch that permits us to explicitly control both slow light and fast light propagation in the probe transmission.\n\nIn Fig.~\\ref{tr2}(b), we plot the phase as a function of normalized probe detuning. Note that, the phase of the probe field suffers a steep negative dispersion around $\\overline{\\delta}=0$, as shown in Fig.~\\ref{tr2}(b). This rapid phase change leads to the change in group delay of the probe field such that fast light effect takes place. Figure.~\\ref{ph2} displays the group delay for the different values of cavity decay rate $\\kappa$. The group delay is negative, which reveals that we observe the fast light propagation (pulse advancement) in the transmitted probe field, when $g_c$ is switched off. From figure~\\ref{ph2}, it can be seen that for decrease in $\\kappa$, magnitude of pulse advancement increases. Thus, longer the lifetime of the resonator, more obvious to see the superluminal behavior.\n\nWe emphasis that we observed both the phenomena, slow and fast light behavior of the probe field through the transmission. However previously it is reported that the slow light occurs in the probe transmission and fast light occurs in the reflected field \\cite{6,63,4}. Thus, we report a tunable switch based on a transition from slow light to fast light as a function of $g_c$, and the robustness of slow light against the dissipation. This clarifies the advantage of our scheme in contrast to the earlier reports \\cite{13,63,65,64,66}.\n\\section{Conclusion}\\label{sec4}\nThe following conclusions apply to our reported work: (i) By utilizing the hybridization of optomechanical and electromechanical systems, we report that both slow and fast light behaviors can be achieved in a single hybrid experimental setup. (ii) In contrast to the previous schemes, two tunable transmission windows occur in our model due to the additional MR, that provides us a tunable switching mechanism from slow light to fast light. (iii) We observed slow light in the transmitted probe field in the presence of Coulomb coupling strength, and witnessed that slow light regime in our model is robust to the cavity decay rate. (iv) A tunable switch from slow light to fast light can be obtained in our model by adjusting the Coulomb coupling strength as $g_{c}=0$. (v) Both pulse delay and advancement can be observed in the probe transmission in a single experiment. Slow and fast light characteristics demonstrate great potential in many applications, including integrated quantum optomechanical memory, designing novel quantum information processing gates, classical signal processing, optical buffers for telecommunication systems, delay lines and interferometry.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{\\refname}}{}{}{}\n\\usepackage{amsmath}\n\\usepackage[T1]{fontenc}\n\\usepackage[latin1]{inputenc}\n\\usepackage{amssymb}\n\\usepackage{setspace}\n\\usepackage{color}\n\\usepackage{float}\n\\usepackage[english]{babel}\n\\usepackage[squaren]{SIunits}\n\\usepackage{dcolumn\n\\usepackage{bm\n\\usepackage{txfonts}              \n\\usepackage{graphicx}   \n\\newcommand{\\cre}{\\color{black}}\n\\newcommand{\\cmre}{\\color{myred}}\n\\newcommand{\\cmbr}{\\color{mybrown}}\n\\newcommand{\\cbu}{\\color{black}}\n\\newcommand{\\cbk}{\\color{black}}\n\\newcommand{\\cmdb}{\\color{mydarkblue}}\n\\newcommand{\\cgr}{\\color{green}}\n\\newcommand{\\cmo}{\\color{myolive}}\n\\newcommand{\\cma}{\\color{magenta}}\n\\newcommand{\\cmpur}{\\color{mypurple}}\n\\newcommand{\\ccy}{\\color{cyan}}\n\\newcommand{\\comment}{\\textbf}\n\\newcommand{\\iotabar}       {\\mbox{$\\,\\iota\\!\\!$-}}\n\\renewcommand{\\comment}{\\noindent \\textbf}\n\n\n\n\\newcommand {\\ie}           {{i.e.}}\n\\newcommand {\\eg}           {{e.g.}}\n\\newcommand {\\etal}         {\\textit{et al. }}\n\\newcommand {\\bJ}           {\\mathbf{J}}\n\\newcommand {\\bk}           {\\mathbf{k}}\n\\newcommand {\\bB}           {\\mathbf{B}}\n\\newcommand {\\bA}           {\\mathbf{A}}\n\\newcommand {\\bdS}          {\\mathbf{dS}}\n\\newcommand {\\bdl}          {\\mathbf{dl}}\n\\newcommand {\\br}           {\\mathbf{r}}\n\n\\newcommand {\\bI}           {\\mathbf{\\mathcal{I}}}\n\\newcommand {\\bD}           {\\mathbf{\\mathcal{D}}}\n\\newcommand {\\bM}           {\\mathbf{\\mathcal{M}}}\n\\newcommand {\\bC}           {\\mathbf{\\mathcal{C}}}\n\\newcommand {\\bF}           {\\mathbf{\\mathcal{F}}}\n\n\\newcommand {\\bII}           {\\mathbf{I}}\n\\newcommand {\\bPI}           {\\mathbf{P}}\n\\newcommand {\\bDI}           {\\mathbf{D}}\n\\newcommand {\\bMI}           {\\mathbf{M}}\n\\newcommand {\\bCI}           {\\mathbf{C}}\n\\newcommand {\\bQI}           {\\mathbf{Q}}\n\\newcommand {\\bWI}           {\\mathbf{W}}\n\n\\newcommand {\\bone}         {\\mathbf{1}}\n\\newcommand {\\bzero}        {\\mathbf{0}}\n\\newcommand {\\bW}           {\\mathbf{W}}\n\\newcommand {\\bsigma}       {\\mathbf{\\Sigma}}\n\\newcommand {\\bjpsi}        {\\mathbf{j_\\phi}}\n\\newcommand {\\bppsi}        {\\mathbf{p}}\n\\newcommand {\\bfpsi}        {\\mathbf{f}}\n\\newcommand {\\grad}         {\\nabla}\n\\newcommand {\\curl}         {\\nabla \\times}\n\n\\newcommand{\\mody}{\\color{blue}}\n\\newcommand{\\norm}{\\color{black}}\n\n\n\\providecommand{\\tabularnewline}{\\\\}\n\n\n\\begin{document}\n\n \n\\title{Sandpile modelling of dual location fuelling in fusion plasmas}\n\n\\author{C. A. Bowie}\n\\email{craig.bowie@anu.edu.au}\n\\affiliation{Australian National University, Canberra, ACT 0200, Australia}\n\n\\author{M. J. Hole}\n\\affiliation{Australian National University, Canberra, ACT 0200, Australia}\n\n\\begin{abstract}\n\nWe modify the Chapman sandpile model (Chapman \\textit{et al} \\textit{Physical Review Letters} 86, 2814 (2001)) to form comparisons with pellet pacing, which is used to reduce or eliminate ELMs in a fusion plasma. We employ a variation of that model in which a pedestal with feedback is introduced (Bowie and Hole \\textit{Phys. Plasmas} 25, 012511 (2018)), which we further modify to provide for dual fuelling - sand is added both at the centre of the sandpile, and near the edge. We observe that when the additional sand is added at the top of the pedestal, MLEs  are largely suppressed. While this suppression comes at a cost by way of reduction in total energy confinement, that reduction is lower than the reduction in MLE size. The trade-off between MLE suppression and reduction in energy confinement depends not only on the amount of extra sand, but also on its precise location relative to the top of the pedestal. We suggest that the approach of constant dual fuelling may be equally applicable to plasmas, and may suggest a strategy for ELM suppression in fusion plasmas. We observe that when the proposed amount of extra sand is added in 'pellets', using frequencies and amounts based on those proposed for ELM suppression for ITER, MLEs are similarly suppressed, although MLEs are not significantly suppressed when the pellet rate does not substantially exceed the MLE frequency. This suggests that pellet injection at the top of the pedestal at small pellet size and high frequency may represent a reasonable physical proxy for our proposed scheme. However, our results suggest that it is not the synchronisation of pellets to ELM frequencies which is the key factor for ELM suppression in this regime, but rather the introduction of additional fuelling at the top of the pedestal.\n\n\\end{abstract}\n\n\\maketitle\n\n\\section{Introduction \\label{sec:Introduction}}\n\nNuclear fusion, if it can be effectively controlled, may be critical to our future energy needs. The primary method of seeking to achieve fusion power is via a plasma which is magnetically confined in a torus known as a tokamak. The goal of fusion research is to increase the fusion triple product of temperature, plasma density, and particle confinement time. A step towards this goal, known as H-mode, occurs when the plasma enters into a higher confinement mode, via a mechanism which is not yet fully understood, but which results in the production of a `pedestal' at the edge of the plasma, in which energy confinement rises sharply over a distance of approx 3\\% of the toroidal minor radius\\cite{Beurskens2009}. However, with H-mode comes a plasma instability known as an edge localised mode, or ELM, which triggers a loss of confinement~\\cite{ASDEX1989}. A large ELM may result in a loss of confinement of 5\\%~\\cite{ASDEX1989}, or from 10-40\\% of the pedestal energy~\\cite{Beurskens2009} and can cause damage to the first wall of the tokamak\\cite{Igitkhanov2013}. For ITER, an upper tolerable limit for ELMs of $\\sim$1\\% of the pedestal energy has been suggested\\cite{Beurskens2009,Zhitlukhin2007}. Controlling ELMs in H-mode is therefore a key objective of fusion plasma research.\n\nInjection of fuel `pellets' has been extensively used as a candidate for ELM control and reduction in fusion plasmas, using pellets to trigger ELMs to increase ELM frequency ($f_{ELM}$), and consequently decrease their maximum size ($W_{ELM}$), on the basis that $f_{ELM}*W_{ELM}=constant$.~\\cite{Hong1999,Baylor2005,Baylor2007,Baylor2013,Baylor2015,Lang2004,Lang2014,Lang2015,Pegourie2007,Rhee2012} Pellet size, frequency, and location have all been tested experimentally on ASDEX Upgrade~\\cite{Lang2004,Lang2015, Lang2018}, DIII-D~\\cite{Baylor2005,Baylor2013}, JET~\\cite{Baylor2015, Lang2011, Lang2015}, and EAST~\\cite{Li2014,Yao2017} and ELM control using pellets is being considered for use in ITER~\\cite{Doyle2007,Baylor2015}.\n\nInjection of pellets to the top of the pedestal has been suggested to produce ELM pacing with reduced energy loss in modelling by Hayashi~\\cite{Hayashi2013}, using the code TOPICS-IB. That modelling suggested that pellets with $\\sim$1\\% of the pedestal particle content, with speed sufficient to reach the pedestal top, will reduce energy loss significantly. The penetration depth of the pellet depends both on its size and speed, as smaller pellets do not penetrate as far into the plasma before ablation. Experiments at JET determined a minimum threshold pellet size which was necessary to reach the top of the pedestal in order to trigger ELMs~\\cite{Lang2011}, where the pellet frequency exceeded the natural ELM frequency. For example, Lang\\cite{Lang2015} discusses the use of pellets of $1.5-3.7\\times10^{20}$D, introduced into a plasma with particle inventory of $6\\times10^{20}$D, i.e. $25-60\\%$ of the total plasma inventory. It has also been observed that in a 2016 series of discharges in JET, the highest fusion performance was observed using a particle fuelling scheme consisting of pellet injection combined with low gas puffing~\\cite{Kim2018}. Lang \\cite{Lang2015} discussed pellets added at lower frequencies (higher $\\Delta t_P$) with pellet timing aligned to ELM onset. These pellets triggered ELMs. Lang\\cite{Lang2015} observes that as pellets increase the plasma density, this in turn increases the L-H threshold. At DIII-D, pellet injection has been observed to trigger synchronous ELMs with a frequency of $12$ times the natural $f_{ELM}$\\cite{Huijsmans2015,Baylor2013}. It is proposed that a dual pellet injection system will be used in ITER with large pellets to provide fuelling, and smaller pellets to trigger ELMs\\cite{Baylor2015}, and it has been suggested that a pellet frequency of $\\sim45$ times the natural $f_{ELM}$ will be required to provide the necessary reduction in heat flux.\n\nOne way of understanding the impact of pellet injection on both confinement and ELM behaviour is to seek to identify a physical system whose relaxation processes have characteristics similar to those of the ELMing process under consideration. Of particular interest is the sandpile~\\cite{Bak1987}, whose relevance to fusion plasmas is well known~\\cite{Chapman1999,Dendy1997}. \n\nSandpile models generate avalanches, which may be internal or result in loss of particles from the system. These avalanches are the response to steady fuelling of a system which relaxes through coupled near-neighbour transport events that occur whenever a critical gradient is locally exceeded. The possibility that, in some circumstances, ELMing may resemble avalanching was raised~\\cite{Chapman2001A} in studies of the specific sandpile model of Chapman~\\cite{Chapman2000}. This simple one-dimensional N-cell sandpile model~\\cite{Chapman2000,Chapman2001A} incorporates other established models~\\cite{Bak1987,Dendy1998A} as limiting cases. It is centrally fuelled at cell $n = 1\/500$, and its distinctive feature is the rule for local redistribution of sand near a cell (say at $n = k$) at which the critical gradient $Z_{c}$ is exceeded. The sandpile is conservatively flattened around the unstable cell over a fast redistribution lengthscale $L_{f}$, which spans the cells $n = k - (L_{f}  - 1), k - (L_{f} - 2), ... , k+1$, so that the total amount of sand in the fluidization region before and after the flattening is unchanged. Because the value at cell $n = k+1$ prior to the redistribution is lower than the value of the cells behind it (at $n<k+1$), the redistribution results in the relocation of sand from the fluidization region, to the cell at $n = k + 1$. If redistributions are sequentially triggered outwards across neighbouring cells, leading to sand ultimately being output at the edge of the sandpile, an avalanche is said to have occurred. The sandpile is then fuelled again, after the sandpile has iterated to stability so that sand ceases to escape from the system.  \n\nThe lengthscale $L_{f}$, normalized to the system scale $N$, is typically ~\\cite{Chapman1999,Chapman2001A,Chapman2001B,Chapman2003,Chapman2004} treated as the model's primary control parameter $L_{f}\/N$, which governs different regimes of avalanche statistics and system dynamics. Here, we employ a modification to the classic model in which the lengthscale is variable over a distance from the edge, which itself depends upon the energy of the system~\\cite{Bowie2018}. As $L_f$ reduces near the edge, the gradient increases at the edge, resulting in a pedestal which is subject to feedback due to the dependence of the distance on the energy. The resulting pedestal was introduced as a proxy for the pedestal of a fusion plasma in a H-mode plasma~\\cite{Bowie2018}. The feedback loops were seen to be analagous to the feedback effects intrinsic to the H-mode pedestal in a fusion plasma~\\cite{Bowie2018}. It was suggested that reduction of feedback in the pedestal could result in ELM suppression within a H-mode plasma~\\cite{Bowie2018}.\n\nTypically, the model is centrally fuelled only. Here, we introduce a new feature, being dual fuelling, in which the sandpile is constantly fuelled concurrently at two locations, in order to observe the effect on energy confinement and mass loss event (MLE) size. We observe that by adding $\\sim$2.5\\% of the sand at a location near the top of the pedestal (near the edge of the plasma), the maximum amount of sand lost in an MLE ($\\Delta S_{max}$) is significantly reduced.\n\n\n\\section{Dual-fuelled sandpile}\n\nWe begin with a feedback model in which sand is added only at the core (as is typical for other implementations of the model). We add sand at a constant rate ($dx_{fc}=1.2$) until the sandpile builds up and enters a `steady state' in which the time averaged amount of sand lost in MLEs equals the amount of sand added. The median waiting time, $\\Delta t_n$, between MLEs is  $\\sim$$135000$, and $\\Delta S_{max}$ is  $\\sim$$630000$. The energy of the system ($E_p$), measured by the sum of the squares of the values of the cells, is $\\sim$$2.7\\times10^9$. The parameters chosen are based on Bowie and Hole~\\cite{Bowie2018}.\n\nFor the sandpile chosen, the width of the pedestal, $P_w$, is $\\sim$$15\/500$ cells, meaning that the top of the pedestal is located at $n=485$. Due to the feedback effects built into the model, the pedestal edge moves with time, approximately synchronously with $E_p$. The resulting shape of the sandpile is shown in Figure \\ref{fig:Sandpile, Ep, and Max MLE}(a), with the values of $E_p$ and $P_w$ over 2 million iterations shown in Figure \\ref{fig:Sandpile, Ep, and Max MLE}(b) and (c). \n\n\\begin{figure\n\\centering\n\\begin{minipage}{1.2in\n\\includegraphics[clip,trim=0cm 0 0cm 0,width=\\linewidth]{.\/\"Sandpile_-_Pellet_-_Model_3_-Cell_480-492__RLstart300_LFstart_100_Time_1_NO_BONUS_SAND\"}\n\\end{minipage\n\\begin{minipage}{1.2in\n\\includegraphics[clip,trim=0cm 0 0cm 0,width=\\linewidth]{.\/\"Potential_Energy_-_Pellet_-_Model_3_-Cell_490__RLstart300_LFstart_100_Time_NO_BONUS_SAND\"}\n\\end{minipage\n\\begin{minipage}{1.2in\n\\includegraphics[clip,trim=0cm 0 0cm 0,width=\\linewidth]{.\/\"Larmor_Radius_-Pellet_-_Model_3_-Cell_480-492__RLstart300_LFstart_100_Time_1_NO_ADDED_SAND\"}\n\\end{minipage}\n\n\\begin{minipage}{1.2in\n\\includegraphics[width=\\linewidth]{.\/\"Sandpile_-_Pellet_-_Model_3_-Cell_487__RLstart300_LFstart_100_Time_1_Pellet_0_03\"}\n\\end{minipage\n\\begin{minipage}{1.2in\n\\includegraphics[width=\\linewidth]{.\/\"Potential_Energy_-_Pellet_-_Model_3_-Cell_487__RLstart300_LFstart_100_Time_1_Pellet_0_03\"}\n\\end{minipage\n\\begin{minipage}{1.2in\n\\includegraphics[width=\\linewidth]{.\/\"Larmor_radius_-_Pellet_-_Model_3_-Cell_487__RLstart300_LFstart_100_Time_1_Pellet_0_03\"}\n\\end{minipage\n\\caption{(L to R) Sandpile, $E_p$, and $P_w$ plots for base case ($dx_{fe} = 0$) (top); $dx_{fe} = 0.03$, added at $n=487$ (bottom)\n\\label{fig:Sandpile, Ep, and Max MLE\n\\end{figure}\n\nWe then modify the model, by adding most of the sand ($dx_{fc}=1.2$) at the core ($n=1$), and some of the sand ($dx_{fe}$)at another location within the sandpile, $n_{fe}$. Sand is added continuously at both the core and $n_{fe}$, representing dual fuelling rather than time separated pellets. We record the average value of $E_p$, and $\\Delta S_{max}$. We then repeat the process for a number of values in the range from $n_{fe}=2$ to $n_{fe}=500$. The sandpile, and values of $E_p$ and $P_w$, using this dual fuelling model, are shown in Figure \\ref{fig:Sandpile, Ep, and Max MLE}(d-f). We observe that, consistent with the reduction in $\\Delta S_{max}$, the $E_p$ and $P_w$ traces are much smoother where dual fuelling is employed. Figure \\ref{fig:Sandpile, Ep, and Max MLE}(f) shows us that for $dx_{fe}=0.03$, $P_w$ is about $13$ when $n_{fe}=487$, i.e. the sand is added at about the top of the pedestal.\n\n\n\n\\begin{figure\n\\centering\n\\includegraphics[clip,trim={0cm 0cm 0cm 0cm},width=\\linewidth]{.\/\"dual_fuelling_MLE02\"}\n\\caption[Pellet size 0.03]{$E_p$ and $\\Delta S_{max}$ for $dx_{fe}= 0.03$, added at $n_{fe}=480$ to $500$ (bottom) and $n_{fe}=1$ to $500$ (top). Figure \\ref{fig:pellet-size-0_03}(a) shows the full range from $n=1-500$, while Figure \\ref{fig:pellet-size-0_03}(b) shows detail within the pedestal. The average amount of sand in the sandpile is $\\sim$$10^6$ units, meaning that $\\Delta S_{max}$ is up to $50\\%$ for the base case, and is reduced to $\\sim$$0.5\\%$ when $n_{fe}=487, dx_{fe}=0.03$.}\n\\label{fig:pellet-size-0_03}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[clip,trim={0cm 0cm 0cm 0cm},width=\\linewidth]{\"dual_fuelling_490_1\"}\n\\caption[Variable pellet sizes]{$E_p$ and $\\Delta S_{max}$ for $dx_{fe}= 0.01$ to $0.2$. In each case $n_{fe}=490$. $\\Delta S_{max}$ is significantly reduced where $dx_{fe}>0.03$, while $E_p$ declines more slowly, suggesting that $dx_{fe}=0.03$ is optimal for $\\Delta S_{max}$, while maintaining $E_p$.}\n\\label{fig:variable-pellet-sizes---max-mle-and-pe}\n\\end{figure}\n\nFigure \\ref{fig:pellet-size-0_03} shows how $E_p$, and $\\Delta S_{max}$ vary as we change $n_{fe}$, for $dx_{fe}=0.03$. Both $E_p$, and $\\Delta S_{max}$ are minimised when $n_{fe}$ is located within the pedestal. MLEs are maximally suppressed when $n_{fe}$ is in the range from $487 - 497$, with the maximum $E_p$ in that range at $n_{fe}=487$ (i.e. the top of the pedestal). When $n_{fe}$ is located at the top of the pedestal, $E_p$ declines by about 30\\%, with a concurrent $\\sim$93\\% reduction in $\\Delta S_{max}$. If $n_{fe}$ is located just outside the pedestal, a reduction in $\\Delta S_{max}$ of $\\sim$50\\% can be achieved with little effect on $E_p$. By contrast, dual fuelling significantly outside the pedestal has little effect on either $E_p$ or $\\Delta S_{max}$, as shown in Figure \\ref{fig:pellet-size-0_03}(a).\n\nEssentially, what is observed is that $n_{fe}$, when located at the top of, or within, the pedestal, sets a maximum value for $P_w$, by suppressing further growth of $P_w$. This in turn prevents the sandpile from becoming sufficiently large that it collapses.\n\nThe trade-off between reduction in $\\Delta S_{max}$ and $E_p$ can also be seen if $dx_{fe}$ is varied. In Figure \\ref{fig:variable-pellet-sizes---max-mle-and-pe}, we show $\\Delta S_{max}$ and $E_p$ for a range of pellet sizes, added at $n_{fe}=490$, which is near the top of the pedestal. We see that as we increase $dx_{fe}$, $\\Delta S_{max}$ and $E_p$ both decline. $\\Delta S_{max}$ has been reduced by an order of magnitude at $dx_{fe}=0.03$ and remains relatively steady after that, while $E_p$ continues to decrease as we increase $dx_{fe}$. \n\nIn addition, generally speaking, for values of $dx_{fe}$ below $0.03$, the `dip' in $E_p$ and $\\Delta S_{max}$ is smaller, and occurs over a smaller range of values of $n_{fe}$. For higher values, the dip is larger over a $\\sim17$ cell range, representing an approximate radial width of $17\/500=0.034$ of the plasma. The `sweet spot' appears where the dip is over a wide enough range such that extreme precision in adding $dx_{fe}$ is not required, without resulting in a large decrease in $E_p$.\n\nTaking these factors into account, we suggest that the optimal value for $dx_{fe}$ is about $0.03$, or $2.5\\%$ of $dx_{fc}$. As noted above, for $dx_{fe}=0.03$, maximal suppression of MLEs, coupled with minimal reduction in $E_p$, occurs at about $n_{fe}=487$, being the top of the pedestal.\n\n\\section{Discussion}\n\nTo date, pellet fuelling in fusion plasmas has been aimed at the triggering of an ELM immediately following the introduction of a pellet, so as to increase $f_{ELM}$, and consequently decrease $W_{ELM}$, on the basis that $f_{ELM}\\times W_{ELM}=constant$.~\\cite{Hong1999,Baylor2005,Baylor2007,Baylor2013,Baylor2015,Lang2004,Lang2014,Lang2015,Pegourie2007,Rhee2012}. Here we suggest a potentially different path to ELM reduction, as the dual fuelling proposed here is constant, rather than pelletized, and therefore does not produce MLEs synchronised with the introduction of additional fuelling. Instead, the constant injection of fuel at or about the top of the pedestal in a feedback modified sandpile, when coupled with the feedback mechanism, triggers MLEs more regularly, but still with a waiting time of at least several thousand time steps.\n\nWe observe that MLE suppression does not occur when $n_{fe}$ is significantly outside the pedestal in which feedback occurs. MLE suppression also does not occur for dual fuelling in the classic sandpile model, in which no feedback occurs. This suggests that MLE suppression by dual fuelling is directly related to modification of feedback in the pedestal.\n\nThe feedback model, including a pedestal, has been suggested to be analogous to a fusion plasma, including a H-mode pedestal in which feedback effects occur\\cite{Bowie2018}, perhaps because a common underlying dynamical behaviour occurs in both the model and the fusion plasma. As a result, we suggest that dual fuelling in a fusion plasma may similarly lead to ELM suppression. Specifically, it may be advantageous to operate a fusion plasma in a mode in which most of the fuelling occurs at the core, while 2.5\\% of the fuelling occurs at the top of the pedestal. If our conjecture is correct, and the fuelling properties\/insights of the MLE model are portable to a tokamak, such an operating mode will result in the suppression of ELMs at a low energy density and temperature cost.\n\nNotwithstanding that existing pellet fuelling schemes have been aimed at the triggering of an ELM immediately following the introduction of a pellet, there may nonetheless be a relationship between the proposal here and pellet fuelling schemes employed to date. Minimum pellet sizes have been suggested for production of ELMs in experiments, as a consequence of the practical requirement that pellets be large enough to reach the top of the pedestal. The minimum size is also a function of pellet velocity, as the pellet size necessary to reach the top of the pedestal decreases as pellet velocity increases. These minimum sizes are coupled with the maximum practically achievable injection frequency in each experiment. If our analogy is correct, the minimum necessary size to reach the top of the pedestal will couple with the injection frequency to produce an optimal injection frequency, which may be less than the maximum achievable injection frequency.\n\nIn order to make a comparison with the proposed ITER scheme, we have 'pelletized' $dx_{fe}$ by adding sand at every $4,000$ time steps (being approximately the natural waiting time in the model, divided by $45$, based on the assumption that the pellet frequency in ITER will be $45$Hz\\cite{Baylor2015}), with $f_{ELM}=1$Hz\\cite{Baylor2015}. The amount of sand added in total is equal to the amount added continuously, i.e. $4000\\times0.03=120$. On the assumption that pellets take effect over their ablation time, rather than instanteously, we have delivered the pellet over $400$ time steps, adopting an observed ablation time for a MAST pellet of $13\\times200 \\micro \\second = 2.6 \\milli \\second$ \\cite{Garzotti2010}, which equates to $\\sim 400$ time steps in our model. The result is that at each time step during pellet injection, $dx=1.2$ and $dx_{fe}=0.3$, while for all other time steps $dx=1.2$ and $dx_{fe}=0$. We also observe that the amount of sand in the pedestal in the model is about $11,000$ units, so that a pellet size of $120$ units is $\\sim 1\\%$ of the particles in the pedestal, which is consistent with modelling by Hayashi\\cite{Hayashi2013}, suggesting that the pellet size should be 1\\% of the pedestal particle content. With these parameter settings, $E_p\\sim1.9\\times10^9$ (a reduction of $\\sim30\\%$ from the base case), and $\\Delta S_{max}\\sim13000$ (a reduction of $\\sim98\\%$). \n\nBy contrast, if pellets are injected at a rate equal to the natural MLE frequency, consistent with pellet pacing experiments at JET, then while $E_p\\sim1.9\\times10^9$ (the same as for the reduction from the base case of $\\sim30\\%$), $\\Delta S_{max}\\sim99000$ (a reduction of only $\\sim75\\%$). The continuing occurrence of significant MLEs is consistent with the result observed at JET in which ELMs still occurred during pellet pacing, rather than being fully suppressed.\n\nThis suggests that a series of pellets, such as those to be used in ITER, represent a good approximation to the continuous edge fuelling proposed here, particularly with regard to the practical limitations of implementing such a scheme. Our model also suggests that the relevant consideration for pellet pacing is whether the total amount of particles delivered reaches the ELMing threshold, whether delivered continuously, or over several pellets or gas puffs. This result contrasts with pellet pacing schemes in which pellet timing is aligned to ELM onset \\cite{Lang2015} - our result suggests that it is not synchronisation of the pellets which is relevant in this regime, but instead the total amount of fuelling delivered (at least quasi-continuously) at the top of the pedestal.\n\nThe scheme may alternatively be implemented by gas puffing, to the extent that gas puffs can be controllably injected at the top of the pedestal as part of a dual fuelling scheme in the proportions suggested here.\n\n\\section{Conclusion}\n\nWe have implemented a feedback modified sandpile model, to which we have added dual fuelling. The sandpile model incorporates feedback effects within an edge pedestal. We have observed that when additional fuelling is added at the top of the pedestal, MLEs are almost entirely suppressed while $E_p$ is reduced to a lesser extent.\n\nWe observe that optimal MLE suppression, with minimal $E_p$ reduction, occurs when edge fuelling represents approximately 2.5\\% of core fuelling, and the edge fuelling is added at the top of the pedestal. We conjecture that this MLE suppression results from suppression of feedback in the pedestal of the model. We suggest that a similar scheme employed in a fusion plasma may result in the suppression of ELMs at a low particle density and temperature cost.\n\nWe have shown that this scheme is related to a scheme of pellet injection at frequencies up to 45 times the natural $f_{ELM}$ proposed for use in ITER\\cite{Baylor2015}, and tested in DIII-D\\cite{Baylor2013}, and to a scheme modelled by Hayashi\\cite{Hayashi2013},who suggests that small pellets of the order of 1\\% of the pedestal particle content, which are fully ablated at the top of the pedestal, may be sufficient to trigger ELMs, and thereby reduce their size. However, significant ELM suppression may not occur unless the pellet rate significantly exceeds $f_{ELM}$. Our result suggests that it is not the synchronisation of pellets to ELMs which is relevant for ELM suppression in this regime, but rather the total amount of fuel delivered (at least quasi-continuously) at the top of the pedestal.\n\nGas puffing which provides relatively constant edge fuelling may also suppress ELMs at the same ratio of core to edge fuelling.\n\nWe suggest that others may wish to implement the scheme proposed here in a fusion plasma, to determine whether edge fuelling can suppress ELMs at a particle density and temperature cost which is considered acceptable for the experiment in question, consistent with the results of our model.\n\n\n\\section*{Acknowledgments}\nThis work was jointly funded by the Australian Research Council through grant FT0991899 and the Australian National University. One of the authors, C. A. Bowie, is supported through an ANU PhD scholarship, an Australian Government Research Training Program (RTP) Scholarship, and an Australian Institute of Nuclear Science and Engineering Postgraduate Research Award.\n\n\\input{Dual_fuelling_article_Arxiv.bbl}\n\n\\renewcommand\\refname{}\n\n\\end{document}\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec1}\nThe era of Artificial Intelligence and Big Data Analytics have been coming into play, marking a new revolution, widely referred as the Fourth Industrial Revolution (IR 4.0) \\cite{ir4}. On one hand, the physical world have been progressively evolving towards digitization thanks to the convergence of several technological advances including notably sensor technologies and ubiquitous access to the Internet. Such digital transformation therefore heralds for an always-connected world in which the digital shadow of the physical world has been practically created. On the other hand, the introduction of novel and bandwidth-hungry services such as tele-presence holography and tactile control of robotics have been accelerating in an unprecedented manner. Such two driving forces have been push-pulling the scale of challenges posed by global consumption of data and the explosive growth of Internet traffic \\cite{crunch}. According to the recently released report from Cisco \\cite{Cisco20}, annual global IP traffic will reach 4.8 ZB per year by 2022, exhibiting a three-fold increase over a span of 5-year and such multiplicative growth has shown no signs of stopping. In this context, optical core networks forming the backbone of Internet infrastructure have been under the critical pressure for a radical re-consideration across different phases ranging from designing, and planning to operation and management to achieve greater capital and operational efficiency. Indeed, the ultimate goal is to transfer more information at a lower cost on less spectrum resources and doing so helps to produce low-latency and high-throughput backbone networks, enabling the so-called global connectivity at scale. \\\\\n\nIn facing with the unprecedented traffic growth, optical transport networks have been advancing accordingly. On one hand, from the architectural perspective, core optical networks have been evolving from the opaque mode to translucent and eventually fully transparent operation. Thanks to the enormous advancements in optical components, transmission technologies and photonic switching, the vision of all-optical\/transparent core networks have been experimentally and practically realized, bringing in significant savings of cost, footprint, and power by eliminating unnecessary O-E-O regenerations \\cite{all-optical, efficient, 20years}. On the other hand, driven by the fact that the spectrum is limited and therefore, the capacity limit of conventional fiber might soon be reached (i.e., fiber capacity crunch), elastic optical networks technologies have been proposed and developed with the objective of using fiber capacity more efficiently. In particular, thanks to significant progress in optical transmission technologies, rate-adaptive optics has been emerging as promising solution to meet ever growing bandwidth demands and simultaneously reduce network cost. In EONs, the spectrum is divided into slices, breaking the traditional rigid frequency grid in WDM networks and hence, paving the way for adaptive spectrum provisioning tailoring to specific demand including its bandwidth requirement and transmission quality \\cite{EON, EON3, EON4}. On another front, as optical transport networks have been the key enabler for information society, the issue of resilience has become of greater importance \\cite{Amazon}. In practice, securing the core networks against single link failures have been an integral part of network planning process and among survivability schemes, dedicated path protection has been widely deployed thanks to its rapid recovery speed and ease of operations \\cite{hai_csndsp, dedicated2}. \\\\\n\nPlanning EONs with survivability requirements has been the focus of extensive research works, covering a wide range of protection schemes \\cite{hai_ps2, hai_wiley, 1+1rsa}. It is evident that as far as protection services are provided, a certain amount of redundant capacity must be required and to optimize such redundancy, network coding has been used recently for protection purposes \\cite{eon-new1, hai_comletter, hai_systems}. The majorities of these work have nevertheless relied on the conventional concept of full protection service for all demands. However, in practice, such over-provisioning appears to miss the heterogeneous nature of services. Indeed, it is well-known that roughly $25\\%$ traffic traveling in optical fiber belongs to premium class while $75\\%$ remaining are best-effort type and hence can be temporarily discarded on occasion of failures \\cite{sla1, sla2, sla3}. Moreover, it appears that link failures are typically fixed quickly and the customers may be willing to tolerate a decreased rate if it is accompanied by reduced cost. This means that instead of providing full protection, just some part of the traffic could be delivered to receivers upon failure events and by reducing the amount of protection requirement, less spare resources might be needed and this brings to the reduction of power consumption and cost as well \\cite{sla1}. This background sets the stage for the arrival of a concept, entitled, partial protection (PP) in which the differentiation of protection level for demands are taken into account to further improve network performances. Partial protection had been raised in the context of traditional WDM networks with pioneering works from \\cite{pp1, pp2}. In \\cite{pp1}, the authors examined the throughput gain when applying partial protection in optical WDM Networks. In \\cite{pp2}, partial protection was examined to guarantees a minimum grade of service upon a link failure by utilizing backup capacity sharing between demands and such application resulted in a remarkable reductions (e.g., $83\\%$) in protection resources. The scenario of traffic grooming with sub-wavelength capacity connection was studied by leveraging partial protection and it was shown in \\cite{pp9} that applying PP brought out lower blocking probability and lower network costs. Multi-path routing was suitable with PP strategies and the works in \\cite{pp12} proposed an effective multi-path algorithm attempting to provision bandwidth requests and simultaneously guaranteeing the maximum partial-protection possible. A related scheme, called partial path protection were covered in \\cite{pp10, pp11} where a collection of backup path were used to protect an active path. Partial protection was furthermore studied in the context of disaster-survivable networks with a pioneering work from \\cite{pp3}. The use of partial protection for mixed line rates networks were investigated in \\cite{pp4} and the authors took advantage of multipath routing and volume discount of higher-line-rate transponders for maximizing the network performances.  However, due to the fixed nature of WDM networks, there has been little room for further improvement with partial protection and consequently the concept of partial protection had been somehow faded. The development of elastic optical networks (EONs) with adaptive resource allocation according to specific traffic requirement and\/or transmission qualities opens up therefore new opportunities for a re-consideration of partial protection. In fact, a closely related strategy is called, quality-of-service provisioning, in EONs have been studied in \\cite{pp5, pp6, pp7}. In \\cite{pp8}, an effective machine learning-based algorithm for identifying the most probable cause of failure impacting a given service and a re-optimization algorithm to re-route affected demands with the objective of minimizing Service Level Agreement (SLA) were proposed. Multi-path allocation has been proposed as a simple way to achieve traffic protection in different networking domains and in EONs, it was shown in \\cite{pp13} that dual-path allocation assuming dynamic offered traffic outperformed traditional techniques. The works in \\cite{sla3} exploited the observation that traffic traveling in fiber could be classified into the premium and best-effort type and currently premium traffic made up approximately $25\\%$ total traffic, better spectrum allocation could be offered. In leveraging this idea, in our recent works \\cite{qos1, qos2}, the framework for adaptive provisioning satisfying the quality-of-service of individual demands have been addressed. Our works developed an optimal network design formulation taking into account the protection level of each demand as an input to the design process and it was shown that by varying the quality-of-service provisioning, greater spectrum efficiency could be achieved. Existing works in the literature nevertheless have largely focused on the direction that the level of protection traffic is specified in advance for each demand and hence provided non-optimal solution for the network design with partial protection. Different from this perspective, we are interested in a more realistic scenario as the protection level for each demand should be considered as variables of network design process and the task is to determine the optimum protection level for each demand so as to the overall service level agreement is met while minimizing the spectrum occupancy. To the best of our knowledge, this is the first time such issue has been raised and investigated.  \\\\ \n\nThis paper marks a departure from conventional full protection provisioning in elastic optical networks with a concept of partial protection scheme in order to achieve greater spectrum efficiency. Leveraging partial protection offers a new perspective to re-design EONs taking into account the diversity of protection services that a traffic demand can request instead of treating all demands with same service. To this end, we present a new research problem entitled, routing, modulation level, spectrum and protection service assignment which is an extension of the well-established one, that is, routing, modulation level, and spectrum assignment as the (partial) protection service for each demand is taken into account and optimized. Our proposal of partial protection strategies in EONs is illustrated and discussed comprehensively in Section 2. In order to optimally determine the protection service for each demand, we develop the network planning framework based on mixed integer linear programming formulation in Section 3. The important point of this formulation is the consideration of protection level for each demand as a design variable to be optimized so as to meet the overall SLA requirement and to minimize the spectrum resources. To assess the efficacy of our partial protection proposal, extensive numerical simulations have been applied on the realistic topology COST239 covering diverse scenarios and the result is reported in Section 4. Finally, the conclusion of the paper and perspectives for extension is addressed in Section \\ref{sec: conclusion}.\n\n\n\n\n\\section{The Three Shades of Partial Protection versus Full Protection}\n\\label{sec: QoS}\nIn this part, the concept of partial protection is introduced in the context of EONs. We also highlight various shades when it comes to adapting partial protection to EONs context and compare them to the conventional full protection. In backing up our proposal for partial protection in EONs, we propose a simple cost-efficient optical node architecture for practical realization of such partial protection mechanism. \\\\\n\nLet us consider a conventional network planning case with full dedicated protection as illustrated clearly in Fig. 1. In this illustrative example, supposing that there are only two traffic demands in the network, that is, the first demand is from node $A$ to node $Z$ with 100 Gbps while the second demand is from node $B$ to node $Z$ with 175 Gbps requirement. Provisioning such two demands in the context of elastic optical networks means that we must select the routing, modulation format and spectrum allocation tailored to the transmission condition of each demand.  Assuming that for the first demand, both its working path and protection path have almost the same transmission quality and therefore, make use of the same modulation format (PM) 16-QAM. For a spectrum slot of $6.25$ GHz, it hence requires two spectrum slots for both working and protection signal to support $100$ Gbps traffic (i.e., each spectrum slot with PM-16-QAM modulation is capable of carrying 50 Gbps). With respect to the second demand, supposing that due to the considerable transmission length difference between its working and protection path, its working signal and protection signal operate at different modulation formats. Specifically, its working route is shorter and hence, its working signal can be modulated on higher-order modulation format (e.g., 64-QAM) than its protection signal (e.g., 8-QAM). In order to support $175$ Gbps of the second demand, three spectrum slots are needed for the working path while the protection path have to utilize five spectrum slots. A comprehensive routing, modulation format selection and spectrum assignment for the full protection is shown in Fig. 1. The spectral cost for such full protection provisioning is $7$ spectrum slots for the entire network and $31$ units of spectrum link usage. \\\\\n\n\\begin{figure*}[!ht]\n\t\\centering\n\t\\includegraphics[width=14cm, height=8cm]{fullprotection}\n\t\\caption{The traditional approach: full protection}\n\t\\label{fig: fullprotection}\n\\end{figure*}\n\nIn an attempt to reduce the spectrum occupancy, we consider a different approach for offering protection service, that is, partial protection. Instead of providing full protection for all demands, let us here assume that different demands require different amount of protection traffic and that specific amount is negotiated a priori and thus becomes the inputs to the network planning process. Figure 2 illustrates the first case of partial protection where each demand is specified in advance the amount of protected traffic. Specifically, in this case, the first demand requires $50\\%$ traffic upon protection mode while demand 2 needs $75\\%$ amount of traffic for protection. It has been shown in Fig. 2 that doing so clearly brings about spectral savings. As far as the spectral cost is evaluated, only $5$ spectrum slots, in contrast to $7$ slots for full protection, are required for the whole network and the spectrum link usage have been reduced accordingly from $31$ units in full protection to $25$ units thanks to the reduction of protection traffic for both two demands. \\\\\n\n\\begin{figure*}[!ht]\n\t\\centering\n\t\\includegraphics[width=10cm, height=8cm]{partial_c}\n\t\\caption{Partial Protection: Conventional approach}\n\t\\label{fig: partial_c}\n\\end{figure*}\n\n\nLet us turn the attention to a different variant of partial protection strategy where instead of specifying the protection level for each demand a priori, network operators receive the service level agreement (SLA) for the entire traffic. For example, one such SLA could be that (at least ) $75\\%$ of the total traffic must be protected in case of failures. In satisfying such requirement, network operators could rely basically on two approaches. The first one is naive when all demands are treated with the same figure of protection level required by SLA. For example, given the SLA of $75\\%$, all demands are offered uniformly $75\\%$ amount of protection traffic. Figure 3 reveals comprehensively the application of such naive approach and it can be seen that $one$ spectrum slot is saved for the entire network compared to the full protection case. It is important to note that while shrinking from 175 Gbps to 131.25 Gbps results in $one$ spectrum slice saving for the second demand, a reduction from 100 Gbps to 75 Gbps is not significant enough to produce spectral saving for the first demand. Indeed, for demand 1, providing full protection and $75\\%$ protection does not make any difference from spectral occupancy perspectives. It is observed that although the one-size-fits-all approach is intuitive, its spectral benefit is limited and hence it leads to a question of whether it exists a more spectrally efficient way to meet SLA requirement. \\\\\n\n\\begin{figure*}[!ht]\n\t\\centering\n\t\\includegraphics[width=10cm, height=8cm]{partial_a}\n\t\\caption{Partial Protection: One-size-fits-all approach}\n\t\\label{fig: partial_a}\n\\end{figure*}\n\nConsidering the case that SLA of $75\\%$, let us propose an alternative approach shown in Fig. 4. Instead of treating uniformly all demands with the same amount of protection level, we now vary the protection level specifically for each demand so as to the overall SLA is met. One such provisioning is illustrated in Fig. 4 as protection traffic of demand 1 is reduced by half from 100 Gbps to 50 Gbps and protection traffic of demand 2 is shrinked from 175 Gbps to 106.25 Gbps to meet SLA of $75\\%$ for the entire network traffic. Overall, an amount of $75\\%$ of entire traffic is secured as protection traffic. For this demand-wise approach, it is easy to verify from Fig. 4 that, $3$ spectrum slots saving could be achieved compared to the full protection scenario in Fig. 1. Moreover, it is highlighted that the demand-wise approach could generate greater spectral efficiency than the one-size-fits-all one. \\\\\n\n\\begin{figure*}[!ht]\n\t\\centering\n\t\\includegraphics[width=10cm, height=8cm]{partial_b}\n\t\\caption{Partial Protection: Demand-wise approach}\n\t\\label{fig: partial_b}\n\\end{figure*}\n\nIn supporting our proposed partial protection scheme, the optical node architecture has to be taken into account. Here we propose a cost-effective architecture leveraging the usage of a shared elastic transponder for both working and protection mode. The elastic transponder is capable of providing multiple modulation formats and\/or baud-rate in order to support a wide range of traffic demands. The transponder is directed to the working path in the normal condition and upon the detection of failure event, the optical-layer switch is reconfigured to re-direct the transponder to the protection path. It is important to note that during this transition, the transponder might be reconfigured itself to operate at a different baud-rate and\/or different modulation format to support partial protection operation. Such transition from working mode to protection mode with this node architecture has been experimentally demonstrated and it has been revealed that the overall reconfiguration could be rapidly taken place, being kept within a few tens of milliseconds satisfying the requirements of many operators \\cite{review7, review9, transponder}. \\\\\n\n\\begin{figure*}[!ht]\n\t\\centering\n\t\\includegraphics[width=10cm, height=8cm]{node_architecture}\n\t\\caption{The optical node architecture supporting partial protection schemes}\n\t\\label{fig: node}\n\\end{figure*}\n\nThe illustration provided in this part demonstrates a new venue for improving spectrum efficiency of optical networks by relying on the idea of reducing protection traffic with partial protection strategies. In general, network operators are interested in the question that given the SLA requirement for the entire network traffic, what would be the optimal assignment of protection level for each demand so that the spectral occupancy is minimized while satisfying SLA. We aim to tackle this issue in this paper and the next part provides a framework for optimal network planning with partial protection \\\\ \n\n\n\\section{Mathematical Formulation for Routing, Modulation Level, Spectrum and Protection Service Assignment Problem}\n\\label{sec: math}\nIn order to realize the spectral benefits of introducing partial protection in elastic optical networks, we present a mathematical formulation for optimal network dimensioning. The formulation takes the fiber network topology, set of traffic demands and service level agreement for protection traffic as the key inputs and returns the optimum routing, modulation level, spectrum assignment for each demand together with the partial protection coefficient so as to minimize the spectrum occupancy while satisfying SLA requirement. In our formulation, the spectrum assignment with continuity and contiguity constraints are processed with the concept of pre-calculated channels \\cite{modelRSA} as it was proven highly computational efficient. \\\\  \n\n\\noindent{Raw Input Data:}\n\n\\begin{itemize}\n\t\\item  $G(V,E)$: The given physical topology with $|V|$ nodes and $|E|$ fiber links\n\t\\item $S$: Set of available spectrum slices in each fiber link, indexed by $s$\n\t\\item  $D$: Set of all traffic demands and each demand $d \\in D$ requires $t^d$ Gbps\n\t\\item $M$: Set of supportable modulation formats, indexed by $m$\n\t\\item A function relates optical reach to modulation format and bit-rate operation $L = 145.741 + (n-14.0344) \\times (60.2196 \\times ln(BR) - 465.82)$: L $[km]$, n  $[bit\/s\/Hz]$ and bit rate (BR) $[Gbps]$ \\cite{optical_reach}\n\t\\item $K$: Set of allowable protection services, indexed by $k$. For example, $k=1$ may refer to the amount of protection of $25\\%$ while $k=2$ represents to an agreement of $50\\%$ protection traffic\t\n\t\\item $SLA$: Service Level Agreement specifies the overall minimum protection traffic for entire network \n\\end{itemize}\n\n\\noindent{Additional Input Data obtained from Pre-processing Process:}\n\n\\begin{itemize}\t \n\t\\item $P^{d}$: Set of $k$ pairs of link-disjoint routes for a demand $d$, each route pair $p^d$ of a demand $d$ consists of a working route and a protection route for that demand\n\t\n\t\\item $P=\\cup_{{d \\in D}} {P^{d}}$: Set of all pairs of link-disjoint routes for all demands in the network\n\t\t\n\t\\item $\\beta_{p, k}^{d} $: a coefficient equals to 1 if a pair of routes $p \\in P$ is feasible for demand $d \\in D$ at protection index $k$, 0 otherwise. The pair of routes $p \\in P^{d}$ for demand $d$ is considered to be feasible at protection service $k$ if there is at least one feasible modulation format carrying $t^d$ $Gbps$ traffic on the working path and there is at least one feasible modulation format carrying a partial amount of $t_d$ $Gbps$ corresponding to the protection service $k$ on protection route. \n\t\n\t\\item $h_{p, w}^{d}$: the most spectrum-efficient (highest-order) modulation format for working path belonging to a route pair $p \\in P^{d}$ of demand $d \\in D$\n\t\n\t\\item $h_{p, b}^{d, k}$: the most spectrum-efficient (highest-order) modulation format for backup path belonging to a route pair $p \\in P^{d}$ of demand $d \\in D$ with respect to protection service $k$\n\t\n\t\\item $C_{p, w}^d$: Set of admissible spectrum channels for working path of a route pair $p \\in P^{d}$\n\t\n\t\\item $C_{p, b}^{d, k}$: Set of admissible spectrum channels for backup path of of a route pair $p \\in P^{d}$ with respect to protection service $k$\n\t\n\t\\item $C=\\cup_{{p \\in P^{d}, d \\in D}} {(C_{p, w}^d \\cup C_{p, b}^{d, k})}$: Set of all possible spectrum channels\n\t\n\t\\item $\\alpha_{c, s} $: a coefficient equals to 1 if the slice $s \\in S$ belongs to channel $c \\in C$, 0 otherwise\n\t\n\t\\item $\\gamma_{p, e} $: a coefficient equals to 1 if link $e \\in E$ belongs to the working path of a route pair $p \\in P$, 0 otherwise\n\t\n\t\\item $\\delta_{p, e} $: a coefficient equals to 1 if link $e \\in E$ belongs to the backup path of a route pair $p \\in P$, 0 otherwise\n\t\n\n\n\t\n\\end{itemize}\nVariables:\n\\begin{itemize}\n\t\\item $x_{p, c}^{d} \\in \\{0,1\\}$: equals to 1 if demand $d$ is provisioned by finding a pair of routes $p \\in P^{d}$ and a spectrum channel $c \\in C_{p, w}^d$ for working path \n\t\n\t\\item $y_{p, c} ^{d, k} \\in \\{0,1\\}$: equals to 1 if demand $d$ is provisioned by finding a pair of routes $p \\in P^{d}$ and a spectrum channel $c \\in C_{p, b}^{d, k}$ for protection path with respect to protection service $k$\n\t \n\t\\item $z_k^{d} \\in \\{0,1\\}$: equals to 1 if demand $d$ if demand $d$ is provided the protection service $k$, 0 otherwise\n\t \n\t\\item $r_{e, s} \\in \\{0,1\\}$: equals to 1 if the spectrum slice $s \\in S$ is occupied in link $e \\in E$ \\\\\n\t\n\t\\item $\\theta_{s} \\in \\{0,1\\}$: equals to 1 if the spectrum slice $s \\in S$ is utilized in the network, 0 otherwise \\\\\n\t\n\n\t\n\\end{itemize}\t\n\n\\noindent{Objective:} \n\\begin{equation}\n\\label{eq:obj}\n\\textit{Minimize} \\; \\phi= \\sum_{s \\in S} {\\theta_s}\n\\end{equation}\n\n\\noindent{Subject to the following constraints:}\n\n\n\n\\begin{equation}\n\\label{eq:protectionselection}\n\\sum_{k \\in K} z^{d}_k = 1 \\qquad \\forall d \\in D \n\\end{equation}\n\n\\begin{equation}\n\\label{eq:partial}\n\\sum_{p \\in {P^d}: \\beta_{p}^{d, k} = 1} \\sum_{c \\in C_{p, b}^{d, k}} y_{p, c}^{d, k} = z^{d}_k \\qquad \\forall k \\in K \n\\end{equation}\n\n\\begin{equation}\n\\label{eq:dcdf}\n\\sum_{c \\in C_{p, w}^d}x_{p, c}^d= \\sum_{k \\in K}\\sum_{c \\in C_{p, b}^{d, k}}y_{p, c}^{d, k} \\qquad \\forall d \\in D, p \\in P^d: \\beta_{p}^{d, k} = 1\n\\end{equation}\n\n\\begin{equation}\n\\label{eq:sla}\n\\sum_{d \\in D} \\sum_{k \\in K} z^{d}_k \\times t^d \\geq SLA \\times \\sum_{d \\in D} t^d \\qquad \\forall d \\in D \n\\end{equation}\n\n\\begin{equation}\n\\label{eq:sliceunique}\n\\sum_{d \\in D} \\sum_{k \\in K} \\sum_{p \\in {P^d}} \\beta_{p}^{d, k} (\\sum_{c \\in C_{p, w}^d} x_{p, c}^d \\times  \\gamma_{p, e} \\times \\alpha_{c, s} + \\sum_{c \\in C_{p, b}^{d, k}} y_{p, c}^{d, k} \\times \\delta_{p, e} \\times  \\alpha_{c, s}) = r_{e, s} \\qquad \\forall e \\in E, s \\in S\n\\end{equation}\n\n\\begin{equation}\n\\label{eq:useslice}\n\\sum_{e \\in E}{r_{e, s}}  \\leq |E| \\theta_s \\qquad\\forall s \\in S \n\\end{equation}\n\n\nThe objective in Eq. \\ref{eq:obj} aims at minimizing the spectrum usage measured in terms of number of spectrum slices to support all traffic demands so as to meet the protection service level agreement for the entire traffic. Constraints in Eq. \\ref{eq:protectionselection} guarantees that each demand must select one and only one protection service. The constraints in Eq. \\ref{eq:partial} say that for the selected protection service of a demand $d$, a protection route and a protection channel must be found to support that demand. The coherence between the working route and protection of a demand is expressed by the constraints in Eq. \\ref{eq:dcdf}. The constraints in Eq. \\ref{eq:sla} are to satisfy the service level agreement for protection. The traditional constraints involving the unique usage for a spectrum slice is ensured by Eq. \\ref{eq:sliceunique}. Finally, the definition of using a spectrum slice in the network, that is, a spectrum slice is considered to be used if it is occupied in any link of the network, is provided by Eq. \\ref{eq:useslice}. \\\\   \n\n\\section{Numerical Results}\t\n\\label{sec: results}\nThis section is dedicated to evaluate our proposal of partial protection on the realistic network topology COST239 as shown in Fig. \\ref{fig:cost239}. This network is composed of 11 nodes and 52 fiber links whose length are ranged from $\\approx$ 200 km to $\\approx$ 1000 km. Such diversity in link length is a good match for the differentiation of transmission properties between the working and protection path where working signal and protection signal could be modulated at different modulation format rather than having the same one based on the worse case. For traffic generation, we assume that $50\\%$ of node-pairs having the traffic and the traffic between each node pair is randomly generated between 100 Gbps and 200 Gbps with a step of 25 Gbps. Ten traffic instances are generated and six modulation formats are considered as shown in Table 1. The number of pairs of route per demand $k$ is 4 and we make use of a variant of Suurballe algorithm to find such 4 shortest pairs for a demand. The mixed integer linear programming formulation in Section 3 is applied and CPLEX solver with an academic version \\cite{IBM} is used to obtain optimal results. \\\\\n\n\\begin{figure*}[!ht]\n\t\\begin{minipage}[t]{.45\\linewidth}\n\t\t\\vspace{0pt}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth, height=6cm]{cost239_distance.eps}\n\t\\end{minipage}%\n\t\\begin{minipage}[t]{.6\\linewidth}\n\t\t\\vspace{2cm}\n\t\t\\centering\n\t\t\\begin{tabular}{|c|c|}\n\t\t\t\\hline\n\t\t\t\\multicolumn{2}{|c|}{\\small{Descriptive Features}} \\\\\n\t\t\t\\hline\n\t\t\tNodes & 11  \\\\\n\t\t\tLinks & 52  \\\\\n\t\t\tAverage nodal degree & 4.7 \\\\\n\t\t\tShortest link distance & $\\approx 200 km$ \\\\\n\t\t\tLongest link distance & $\\approx 1000 km$\\\\ \n\t\t\t\\hline\n\t\t\\end{tabular}\n\t\\end{minipage}\n\t\\caption{COST239 network with link distance in $km$}\n\t\\label{fig:cost239}\n\\end{figure*}\n\n\\begin{table*}[!ht]\n\t\\caption{Bit-rate Operation per $6.25$ $GHz$ Spectrum Slot for different Modulation Formats}\n\t\\label{tab:result0}\n\t\\centering\n\t\\begin{tabular}{cc}\n\t\t\\hline\n\t\tModulation Formats & Bit-rate per Spectrum Slot \\\\\n\t\t\\hline\n\t\tPM-BPSK & 12.5 Gbps \\\\\n\t\tPM-QPSK & 25 Gbps \\\\\n\t\tPM-8QAM & 37.5 Gbps \\\\\n\t\tPM-16QAM & 50 Gbps \\\\\n\t\tPM-32QAM & 62.5 Gbps \\\\\n\t\tPM-64QAM & 75 Gbps \\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table*}\n\nFor benchmarking, in the first scenario, we compare the traditional full protection design with the partial protection design where the relative amount of protection traffic for each demand is randomly selected among $25\\%$, $50\\%$ and $75\\%$ and provided as input to the design process. Figure 7 shows the comparison of such two designs in terms of used spectrum slices. As expected, by reducing the amount of protection traffic, less spectrum slots are required to accommodate the traffic. In this simulation scenario, the result reveals that instead of over-provisioning with full protection traffic for all demands, remarkable spectral saving could be achieved, up to $32\\%$ for partial protection approach \\\\\n\n \n\\begin{figure*}[!ht]\n\t\\centering\n\t\\includegraphics[width=14cm, height=6.5cm]{pp1}\n\t\\caption{Full protection versus Partial Protection in terms of used spectrum slices}\n\t\\label{fig:result1}\n\\end{figure*}\n\n \nLet us move to the second scenario of applying partial protection. For this case, all demands are enforced to have the same protection services specified by the SLA for the entire network. Three kind of SLA protection services are considered, that is, $25\\%$, $50\\%$ and $75\\%$. Figure 8 reports the impact of spectral occupancy as the amount of protection traffic is varied from $75\\%$ to $25\\%$ in comparison with the full protection. It is clearly shown that shrinking the protection trafffic generates spectral savings and specifically such saving could be up to $26\\%$ for $75\\%$ partial protection, $46\\%$ for $50\\%$ partial protection and $55\\%$ for $25\\%$ partial protection. It can also be observed that the spectral saving does not move proportinally to the reduction of protection traffic. In our studied case, shrinking protection traffic up to $50\\%$ tends to achieve the best revenue while going beyond $50\\%$ brings about relatively small impact. \\\\\n\n\\begin{figure*}[!ht]\n\t\\centering\n\t\\includegraphics[width=14cm, height=6.5cm]{pp2}\n\t\\caption{Full protection versus Partial Protection $25\\%$, $50\\%$ and $75\\%$}\n\t\\label{fig:result2}\n\\end{figure*}\n\n\nWe turn the attention to third scenario where the relative amount of protection traffic for each demand becomes a design variable and it has to be optimized to meet the SLA of entire network. We reconsider the second scenario but now instead of enforcing the same amount of protection traffic for all demands, each demand can select the protection services among three types $25\\%$, $50\\%$ and $75\\%$ according to the optimization model in Section 3. Figure 9 compares the one-size-fits-all approach and the demand-wise one when the SLA is $50\\%$. It can be clearly seen that by optimizing the selection of protection service for each demand, better spectrum usage can be realized. In our simulation of ten traffic instances, 8 out 10 cases results in greater spectrum efficiency with the demand-wise approach compared to the one-size-fits-all one and in the most favorable case, the demand-wise approach could be $25\\%$ more efficient than its counterpart.  \n\n\\begin{figure*}[!ht]\n\t\\centering\n\t\\includegraphics[width=14cm, height=6.5cm]{pp3}\n\t\\caption{One-size-fits-all partial protection versus Demand-wise partial protection}\n\t\\label{fig:result3}\n\\end{figure*}\n\n.  \n\\section{Conclusions}\n\\label{sec: conclusion}\nIn an attempt to achieve greater capacity efficiency for optical core networks, this paper proposed the concept of partial protection to be applied in elastic optical networks, that is, instead of providing full protection for all traffic demands, the assignment of (partial) protection traffic for each demand is taken into account and optimized. We examined various scenarios that applying partial protection could potentially generate spectral savings and introduced the network design framework incorporating partial protection. A new problem, entitled, routing, modulation level, spectrum and protection service assignment was developed for optimal network designs based on mixed integer linear programming formulation. In verifying the efficacy of our proposal, extensive simulation on the realistic network topology COST239 was carried out and our findings indicated that significant savings of spectrum resources could be achieved by reducing the protection traffic. Moreover, our results demonstrated that the demand-wise approach far outperformed the one-size-fits-all and the difference could be up to $25\\%$ in our studied cases. \\\\\n\n\n\\section*{Conflict of interest}\nThe authors declare that they have no conflict of interest.\n\n\\bibliographystyle{elsarticle-num}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nAnderson localization \\cite {And}, which is one of the most fundamental physical phenomena in condensed matter physics and optics, occurs due to the interference of wave components multiply scattered by randomly placed scattering centers \\cite{Lee,She}. It has been predicted theoretically and observed experimentally in many different types of waves including microwaves \\cite{Dal,Cha}, optical waves \\cite{Sch,Lah} and matter waves \\cite{Bil,Roa}. Until now, localization has been studied mainly in conservative systems, as it is usually believed that the onset of localization requires multiple scattering by the real part of the potential.\n\nUsing a non-degenerate perturbation theory, Thouless has found that, for the case of weak and uncorrelated disorder, the Lyapunov exponent for the one-dimensional (1D) Anderson model obeys $\\gamma(E)=W^{2}\/(96V^{2}-24E^{2})$, where $W$ represents the strength of disorder and $V$ is the nearest-neighbor hopping parameter \\cite{Thouless1}. However, the numerical calculations performed by Czycholl {\\it et al.} have indicated that Thouless' perturbation expansion breaks down at zero energy \\cite{Czycholl}. This discrepancy between analytical and numerical results, termed the band center anomaly, was resolved by Kappus and Wegner, when they developed a degenerate perturbation theory near the band center and\n obtained the Lyapunov exponent of the form $\\gamma(E)=W^{2}V^{-2}f^{-1}(6EV\/W^{2})$ with $f(0)=105.045$ and $f(\\pm \\infty)=96$ \\cite{Kappus}. Within this context, a full analytical solution for the problem of the band center anomaly in the 1D Anderson model has been obtained as well \\cite{Kravtsov1,Kravtsov2,Tessieri1}.\n\n Apart from the band center anomaly, the anomalies have also been observed in the vicinities of the band edges, $E=\\pm 2$ \\cite{Deych1,Deych2,Hern}, as well as at other spectral points \\cite{Derrida,Alloatti,Sepehrinia1,Nguyen,Sepehrinia2}. Remarkably, based on a weak-disorder expansion of the Lyapunov exponent, Derrida and Gardner systematically treated the anomalies at the energies $E=2V\\cos(\\pi\\alpha)$ with $\\alpha$ a rational number \\cite{Derrida}. In addition, other related anomalous behaviors have been studied. For example, in the vicinity of the band center, the conductance distribution \\cite{Schomerus} deviates from that predicted by the single-parameter scaling hypothesis \\cite{Abrahams}. A sharp increase in the localization length has been observed in the weak-disorder limit \\cite{Krimer}. A comprehensive study of the anomalous localization in low-dimensional systems with correlated disorder has recently been done \\cite{Izrailev}. We emphasize that all the studies of anomalous localization mentioned so far share the common feature that the disorder is introduced by real random potentials. One of the main findings has been that the anomaly leads to unusual enhancement of the localization length and suppression of the Anderson localization.\n\n\nIn the past years, considerable attention has also been paid to localization phenomena in non-Hermitian systems \\cite{Hat,Asa,Sen,Sil,Kal,Bas,Vaz,Mej,Kar}.  A representative example of this kind of systems is the Hatano-Nelson model \\cite{Hat}. In this model, the presence of a constant imaginary vector potential in the Anderson Hamiltonian can give rise to a transition from a real to a complex spectrum, which is associated with the mobility edge in 1D problems. Recently, Basiri {\\it et al.} have studied a different kind\nof non-Hermitian lattice model with randomness only in the imaginary part of the on-site potential and reported that the eigenstates of such a model are localizaed, but the nature of the localization mechanism is qualitatively different from that of the usual Anderson localization \\cite{Bas}.\nIn this context, a fundamental question we ask is whether the phenomenon of anomalous localization occurs in disordered systems characterized by a non-Hermitian Hamiltonian with an imaginary random potential. This problem, to the best of our knowledge,  has so far not been considered in the literature. This provides us with the motivation for pursuing this research direction.\n\nThe rest of this paper is organized as follows. In Sec.~\\ref{sec:2}, we describe the 1D disordered lattice model characterized by a random non-Hermitian Hamiltonian within the nearest-neighbor tight-binding approximation. We also describe the numerical calculation method and the physical quantities of interest. In Sec.~\\ref{sec:3}, we present our numerical results and discussions. Finally, in Sec.~\\ref{sec:4}, we conclude the paper.\n\n\n\\section{Theoretical model}\n\\label{sec:2}\n\\subsection{Model}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=10cm]{Fig1.eps}\n\\caption{Real and imaginary parts of the energy eigenvalues for the system under consideration, when $\\epsilon_{n}^{\\rm R}$ is zero and $\\epsilon_{n}^{\\rm I}$ is selected randomly from a uniform distribution in the interval $[-W\/2, W\/2]$ with $W=0.5$.\nThe system size is $N=1000$.}\n\\end{figure}\n\nLet us consider an array of $N$ weakly coupled optical waveguides with a constant tunneling amplitude $V$ with which light is transferred from waveguide to waveguide. In the approximation where only the coupling between nearest-neighbor waveguides is considered, the wave propagation process in such a system can be described by a set of coupled discrete Schr\\\"{o}dinger equations for the field amplitudes $C_{n}$ given by\n\\begin{eqnarray}\ni \\frac{dC_{n}}{dz}=-V\\left(C_{n-1}+C_{n+1}\\right) + \\epsilon_{n}C_{n},\n\\label{equation1}\n\\end{eqnarray}\nwhere $z$ is the paraxial propagation distance, $n$ ($=1,2,\\cdots, N$) is the waveguide number, and $\\epsilon_{n}$ is the on-site potential.\nThe stationary solutions of Eq.~(\\ref{equation1}) can be represented in the conventional form, $C_{n}(z)=\\psi_{n}e^{-iEz}$, with $E$ as the energy of an eigenstate $n$. Then we obtain the stationary discrete Schr\\\"{o}dinger equation of the form\n\\begin{eqnarray}\n E{\\psi_{n}}=-V\\left(\\psi_{n-1}+\\psi_{n+1}\\right)+\\epsilon_{n}\\psi_{n}.\n\\label{equation2}\n\\end{eqnarray}\nTo obtain the eigenstates and the corresponding eigenvalues, we solve Eq.~(\\ref{equation2}) numerically with the supplementary conditions $\\psi_{0}=\\psi_{N+1}=0$, which are the usual fixed boundary conditions (or hard-wall boundary conditions).\n\nIn the present work, the on-site potential $\\epsilon_{n}$ is considered to be complex. Specifically, it takes the form,\n\\begin{eqnarray}\n\\epsilon_{n}=\\epsilon_{n}^{\\rm R}+i\\epsilon_{n}^{\\rm I},\n\\end{eqnarray}\nwhere the real part $\\epsilon_{n}^{\\rm R}$ is chosen to be the same for all lattice sites (below we put $\\epsilon_{n}^{\\rm R}=0$ for simplicity), whereas the imaginary part $\\epsilon_{n}^{\\rm I}$ is selected randomly from a uniform distribution in the interval $[-W\/2,W\/2]$, where $W$ denotes the disorder strength. When the random on-site potential is complex, the energy eigenvalues are no longer real, but become complex. An example of the distribution of the eigenvalues in the complex plane is shown in Fig.~1.\nIt is worth noting that an approximate analytical expression describing the envelope of the points shown in Fig.~1 was derived in Refs.~37 and 38.\n\n\n\\subsection{Participation number and information entropy}\n\nIn infinite disordered systems, the localization properties can be characterized in terms of the localization length, which is commonly defined from the decay of the amplitude of eigenstates in the limit $N\\to\\infty$. In finite-size systems, however, one needs to use other physical quantities that are valid for both finite and infinite systems such as the participation number. For the $k$-th eigenstate $(\\psi_{1}^{(k)},\\psi_{2}^{(k)},\\cdots,\\psi_{N}^{(k)})^{T}$ with the corresponding eigenvalue $E_{k}$, the participation number $P(E_{k})$ is defined by \\cite{Thouless}\n\\begin{eqnarray}\nP(E_{k})=\\frac{\\left(\\sum_{n=1}^{N}\\big\\vert\\psi_{n}^{(k)}\\big\\vert^2\\right)^2}{\\sum_{n=1}^{N}\\big\\vert\\psi_{n}^{(k)}\\big\\vert^4},\n\\label{equation3}\n\\end{eqnarray}\nwhich gives approximately the number of lattice sites to which the $k$-th eigenstate extends.\n\nThe participation number estimates the degree of spatial extension or localization of eigenstates. For a finite-size system, $P$ increases with increasing the system size $N$ in the extended regime. The completely extended state which spreads over the entire system uniformly corresponds to $P=N$. On the other hand, localized states exhibit much smaller values (in comparison with $N$) of $P$, which converge to constant values as $N\\to \\infty$. The most strongly localized state corresponds to $P=1$. Therefore, the participation number is bounded within the range $1\\le P \\le N$. The finite-size scaling analysis of $P$ gives a very useful information about the localized or extended nature of the eigenstates.\n\nIn addition to $P$, we will also calculate the Shannon entropy, which is another basic quantity for the description of localized single-mode states in disordered systems. The Shannon entropy is defined by \\cite{Oliveira, Bogomolny}\n\\begin{eqnarray}\nS(E_{k})=-\\sum_{n=1}^{N}\\vert\\phi_{n}^{(k)}\\vert^2\\ln\\left(\\vert\\phi_{n}^{(k)}\\vert^2\\right),\n\\label{equation4}\n\\end{eqnarray}\nwhere $\\phi_{n}^{(k)}$ denotes the normalized amplitude of the $k$-th eigenstate wave function at site $n$,  $\\phi_{n}^{(k)}=\\psi_{n}^{(k)}\/(\\sum_{n=1}^{N}\\vert\\psi_{n}^{(k)}\\vert^2)$. It is well-known that the Shannon entropy measures the deviation of the actual wave function from the uniform distribution. As the system size increases to infinity, $S$ is expected to behave as $\\exp(S)\\propto N$ for extended states and  $\\exp(S)\\to$ constant in the case of localized states. Therefore, it is bounded as $1\\le \\exp(S)\\le N$ or $0\\le S\\le \\ln N$.\n\nIt is worth mentioning that the Schr\\\"odinger equation, Eq.~(\\ref{equation1}), which describes the propagation of light in the optical systems, is effectively identical to the equation which describes the propagation of noninteracting electrons in the electronic systems. The key difference is that the evolution coordinate in the optical systems is the paraxial propagation distance, while it is replaced by the time in the electronic systems. Therefore, all the results obtained in this work are valid for electronic as well as optical systems.\n\n\\section{Numerical results}\n\\label{sec:3}\n\nIn our calculations, all energy quantities are measured in the unit of $V$, which we set equal to 1 without loss of generality. The participation number $P$ and the Shannon entropy $S$ were obtained by averaging over the eigenstates with eigenvalues in a small interval around a fixed ${\\rm Re}(E)$ and ensemble averaging over 10000 distinct disorder configurations.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=9.5cm]{Fig2.eps}\n\\caption{(a) Participation number $P$ versus system size $N$ for different disorder strengths $W=2$, 2.2 and 2.4. (b) $P$ versus disorder strength $W$, when the system size is fixed to $N=1000$. The dash-dotted line indicates a power-law fit, $P(W)\\propto W^{-2}$, to the date\n in the region $W< 4$. $P$ is obtained by averaging over the eigenstates with eigenvalues in the interval $[-0.1,0.1]$ around ${\\rm Re}(E)=0$ and ensemble averaging over 10000 distinct disorder realizations.}\n\\end{figure}\n\nIn Fig.~2(a), we plot the participation number $P$, which was obtained by averaging over a small window around the band center, ${\\rm Re}(E)\\in [-0.1,0.1]$, as a function of the system size $N$ for several typical values of the disorder strength $W$. For a given $W$, we find that $P$ initially increases as $N$ increases and rapidly approaches a saturation value $P_{s}$ at $N>N_{c}$). The value of $P_{s}$ is much smaller than the corresponding $N_{c}$. This indicates that the eigenstates near the band center are localized states. We will see later that all states in the whole energy band are localized as well. Similarly to in the standard Anderson model where the real part of the on-site potential is random, in the present model, the participation number is reduced, hence the localization is enhanced, as the disorder strength increases, as shown in Fig.~2(b). In the weak-disorder limit, we find that the participation number is proportional to the disorder strength as a power law, $P\\propto W^{-\\alpha}$, with $\\alpha=2$ (see the dash-dotted line in Fig.~2(b)) \\cite{Bas}. In the strong-disorder limit, the participation number tends to $P=1$, indicating the occurrence of complete localization.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=10cm]{Fig3.eps}\n\\caption{Structural entropy $S_{\\rm str}$ plotted versus spatial filling factor $q$ of eigenstates. The shaded area represents the allowed domain of the localization diagram, which is bounded by $0\\le S_{\\rm str}\\le -\\ln (q)$ with $0<q\\le 1$. The symbols represent our numerical results for the eigenstates with ${\\rm Re}(E)=0$, 0.3 and 1.4. Calculations were performed for the system size $N=1000$. The disorder strength is fixed to $W=2$.}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=10cm]{Fig4a.eps}\n\\includegraphics[width=10cm]{Fig4b.eps}\n\\caption{(a) Participation number $P$ and (b) Shannon entropy $S$ plotted versus real part of the energy eigenvalue ${\\rm Re}(E)$ for disorder strengths $W=1.5$, 1.75 and 2. $\\exp(S)$ instead of $S$ is plotted for an easier comparison with $P$. For a given value of ${\\rm Re}(E)$,\n$P$ and $S$ are obtained by averaging over the states in a small energy interval $\\left[{\\rm Re}(E)-0.05, {\\rm Re}(E)+0.05\\right]$. In the inset of Fig.~4(a), we also show the results obtained for the standard Anderson model with the corresponding real random potential when $W=1$, 1.25 and 1.5.}\n\\end{figure}\n\nThe participation number is commonly used as a basic quantity for measuring the degree of localization. However, many localized states with completely different internal structures such as Gaussian, exponential and power-law decay forms can give rise to the same values of $P$. Therefore, we need a method to determine the shape of the localized eigenstate wave function explicitly. To this end, we use the quantity called structural entropy, which characterizes the shape of the localized wave function and is defined by \\cite{Casati2,Pipek}\n\\begin{eqnarray}\nS_{\\rm str}(E_{k})=S(E_{k})-\\ln P(E_{k}).\n\\label{equation5}\n\\end{eqnarray}\nIn Ref.~43, it has been shown that there exists a basic inequality between the structural entropy $S_{\\rm str}$ and the spatial filling factor $q$ $(=P\/N)$ given by $0\\le S_{\\rm str}\\le -\\ln (q)$ with $0<q\\le 1$. This bound defines an allowed domain of the localization diagram, which has been proved to be universal. Therefore, the pair of the parameters $(q,S_{\\rm str})$ should lie in this allowed domain for any generalized localized state. In Fig.~3, we plot $S_{\\rm str}$ as a function of $q$ for several representative eigenstates with ${\\rm Re}(E)=0$, 0.3 and 1.4. We see clearly that all points indicated by the pairs of the localization quantities $(q,S_{\\rm str})$ are well included in the allowed domain of the localization diagram. More specifically, we observe that apart from in the vicinity of ${\\rm Re}(E)=0$ where $S_{\\rm str}\\sim 0.39$, we get $S_{\\rm str}\\sim 0.34$ for all other cases, as $q\\to 0$ (or $N\\to \\infty$). These results are consistent with the theoretically predicted value for exponentially localized eigenstates, $S_{\\rm str}=1-\\ln 2\\sim 0.31$, which was determined based on the continuous lattice model approach \\cite{Pipek}. We believe that a substantial difference between numerical result and theoretical prediction in the vicinity of the band center can be ascribed to the existence of  anomalous localization of eigenstates in that region.\n\n\\begin{table*}\n\t\\caption{\\label{tab:table3} Numerical values of the wave function amplitude $\\psi_{m}$ and the corresponding eigenvalue $E$ for several typical eigenstates in the strong-disorder region are listed. All eigenstates have ${\\rm Re}(E)=0$. The used parameters are $N=50$ and $W=10$. The approximate values of ${\\rm Im}(E)$ are obtained by Eq.~(\\ref{equation6}).}\n\t\\begin{ruledtabular}\n\t\t\\begin{tabular}{ccccccc}\n\t\t\t&eigenstate label&$m$ (center position)&$\\epsilon_{m}^{\\rm I}$&$\\vert\\psi_{m}\\vert$&${\\rm Im}(E_{\\rm exact})$&${\\rm Im}(E_{\\rm approx}$)\\\\\n\t\t\t\\hline\n\t\t\t&1&7&4.87183992&0.93916916&4.56421869&4.50660618 \\\\\n\t\t\t&2&1&4.66220070&0.98818589&4.52704191&4.52045924 \\\\\n\t\t\t&3&50&-4.57092533&0.99348626&-4.46465683&-4.46121117\\\\\n\t\t\t&4&29&-4.67401248&0.98511495&-4.43615195&-4.43805708\\\\\n\t\t\t&5&11&-4.84019334&0.94008911&-4.35074321&-4.39101455 \\\\\n\t\t\t&6&6&-4.55733493&0.98565603&-4.32524531&-4.32636897 \\\\\n\t\t\t&7&49&4.54366806&0.98127570&4.28462152&4.28389259\\\\\n\t\t\t&8&44&-4.48680400&0.94718793&-4.01643280&-4.04678787\\\\\n\t\t\\end{tabular}\n\t\\end{ruledtabular}\n\\end{table*}\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=9.5cm]{Fig5.eps}\n\t\\caption{Real part of the energy eigenvalue versus eigenstate label arranged in order of increasing value of ${\\rm Re}(E)$,\nwhen the disorder strength $W$ is 1.5, 1.75 and 2. All the cases show that there exist a large number of eigenstates with pure imaginary\neigenvalues. The number of such eigenstates increases with $W$.}\n\\end{figure}\n\nIn Fig.~4, we show the participation number $P$ and the Shannon entropy $S$ as a function of the real part of the energy eigenvalue, ${\\rm Re}(E)$, for disorder strengths $W=1.5$, 1.75 and 2, when the system size is fixed to $N=1000$. For the considered values of $W$, we see that the eigenstates near the band edges are more strongly localized than those near the band center as in the usual Anderson model. As one moves from the band edge towards the band center, $P$ and $S$ increase gradually until they decrease abruptly and exhibit {\\it deep concave} regions around ${\\rm Re}(E)=0$. This non-analytical dependence of $P$ and $S$ on ${\\rm Re}(E)$ indicates an anomalous localization behavior of eigenstates. Remarkably, in contrast to the standard Anderson model in which one observes a sharp increase of the participation number (as illustrated in the inset of Fig.~4(a)), the present anomaly gives rise to a strong decrease of the participation number for eigenstates around ${\\rm Re}(E)=0$. Moreover, in the former case, the band center anomaly exists only when the disorder strength gets sufficiently small ($W\\le 1$) \\cite{Krimer}, whereas, in the present non-Hermitian case, a change in the disorder strength does not alter the height of the deep concave region, but gives rise to a change in the number of the eigenstates with ${\\rm Re}(E)=0$ (see Fig.~5 below). We have also verified that the ratio $\\exp(S)\/P$ is always larger than unity for all values of ${\\rm Re}(E)$. This is in good agreement with Eq.~(\\ref{equation5}).\n\nIn order to provide a quantitative explanation of the effect of anomalous localization enhancement which appears in 1D random non-Hermitian lattices, in Fig.~5, we show the real part of the energy eigenvalue versus eigenstate label arranged in order of increasing value of ${\\rm Re}(E)$,\nfor the same parameters as in Fig.~4. From the numerical results, we find that the anomalous localization behavior at the band center is closely associated with the appearance of a large number of eigenstates with pure imaginary eigenvalues. If we consider only the real part of the eigenvalue\nto distinguish different eigenvalues, this is equivalent to having a large number of {\\it degenerate} states with ${\\rm Re}(E)=0$. From numerical calculations and approximate analytical treatments given below, we have verified that these states are strongly localized. Therefore, any superposition of these states results in strongly localized states. We have found that the number of such degenerate states increases with increasing the disorder strength $W$. As a consequence of this, in the limit of strong disorder, the distribution function of ${\\rm Re}(E)$ will exhibit a huge $\\delta$ peak at ${\\rm Re}(E)=0$. Within the second-order approximation in $V$ $(V\/W\\ll 1)$ in the perturbation theory proposed in Ref.~31, the eigenstates with pure imaginary eigenvalues have the eigenstate wave function and the energy eigenvalue given by\n\\begin{eqnarray}\n\\psi_{n}\\approx\\delta_{n,m} \\nonumber\n\\end{eqnarray}\nand\n\\begin{eqnarray}\nE\\approx i\\epsilon_{m}^{\\rm I}-i\\left(\\frac{1}{\\epsilon_{m}^{\\rm I}-\\epsilon_{m-1}^{\\rm I}}+\\frac{1}{\\epsilon_{m}^{\\rm I}-\\epsilon_{m+1}^{\\rm I}}\\right)V^2,\n\\label{equation6}\n\\end{eqnarray}\nwhere $m$ is the position at which the wave function is localized and $V$ is set equal to unity in our study. In Table~1, we show our numerical results for the wave function amplitude and the corresponding eigenvalue for several representative eigenstates with ${\\rm Re}(E)=0$ obtained by solving Eq.~(\\ref{equation2}) when $N=50$ and $W=10$ directly and compare the results with those obtained from the approximate analytical result, Eq.~(\\ref{equation6}).\nWe find that the two results agree very well in this regime of strong disorder. We note that the eigenstates with pure imaginary eigenvalues seem to be favored to localize at lattice sites, where the corresponding values of $\\epsilon_{m}^{\\rm I}$ are large ($\\sim \\pm W\/2$).\n\nFrom Fig.~4, we can easily notice that the localization of the eigenstates near the band center is more strongly enhanced than that of those near the band edges when the disorder strength becomes large. From this observation, we predict that, as the disorder strength gets sufficiently large, the eigenstates with ${\\rm Re}(E)=0$ will become more strongly localized than all other eigenstates. This prediction is supported by the numerical results presented in Fig.~6, where several typical wave functions for the eigenstates with zero and non-zero real parts of the energy are shown. It is clearly seen that the eigenstates with pure imaginary eigenvalues (5th, 6th and 7th eigenstates in Table 1) are more strongly localized than the remaining ones.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=10cm]{Fig6.eps}\n\t\\caption{Spatial distributions of several typical wave functions for the eigenstates with zero and non-zero real parts of the energy are shown in the case of $N=50$ and $W=10$. The dashed curve indicates the imaginary part $\\epsilon_{n}^{\\rm I}$ of the on-site potential. It is clear that, as the disorder strength gets sufficiently large, the eigenstates with ${\\rm Re}(E)= 0$ become more strongly localized than those with ${\\rm Re}(E)\\neq 0$.}\n\\end{figure}\n\nIt is worth mentioning that although it is not presented in this paper, we have checked that all the results reported above are also valid in the situations where the imaginary part of the on-site potential $\\epsilon_{n}^{\\rm I}$ takes random values which are only positive or negative.\n\n\n\\section{Conclusion}\n\\label{sec:4}\n\nIn this paper, we have presented a numerical study of the localization properties of eigenstates in a 1D disordered lattice characterized by a random non-Hermitian Hamiltonian, where both the randomness and the non-Hermiticity are introduced simultaneously in the on-site potential. After obtaining the eigenstates and the corresponding eigenvalues by solving Eq.~(\\ref{equation2}) numerically, we have calculated the averaged participation number, Shannon entropy and structural entropy as a function of other parameters. We have found that, in the presence of an imaginary random potential, all eigenstates are exponentially localized in the thermodynamic limit. Most importantly, we have found that anomalous Anderson\nlocalization occurs at the band center in the system under consideration. In contrast to the usual anomalies where a weaker localization is observed, our numerical results have clearly shown that the localization of the states at the band center is strongly enhanced. This phenomenon is associated with the occurrence of a large number of strongly-localized states with ${\\rm Re}(E)=0$. We speculate that, in the strong-disorder regime, eigenstates with pure imaginary eigenvalues will become more strongly localized than all other eigenstates with non-zero real parts of the energy eigenvalue.\n\nWe hope that the results presented here will be a useful contribution to the topic of anomalous Anderson localization in disordered systems. Moreover, the numerical method presented in this paper can be used to study more complicated cases such as quasi-1D random non-Hermitian Hamiltonian systems. This will be considered in future publications.\n\n\\acknowledgments B.P.N is thankful to F. M. Izrailev and D. Leykam for reading the manuscript and giving useful comments. This research was funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 103.01-2014.10. It is also supported by a National Research Foundation of Korea Grant (NRF-2015R1A2A2A01003494) funded by the Korean Government.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}