[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83029-en":3,"doc-seo-83029-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},83029,7971461740909,"Levi","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","A Sub-linear Low-Rank Solver for Poisson’s Equation using Machine Learning Frameworks for GPU Acceleration","A fast Poisson solver is developed for problems whose solutions exhibit low-rank structure, aiming to reduce computational cost and memory growth for elliptic PDEs. The method uses an adaptive, warm-started cross approximation (Cross-DEIM) alternating index selection with cross approximation to construct a low-rank representation. Implementation leverages PyTorch as a GPU-oriented array language and studies statistical leverage scores to reduce overhead from pivoted DEIM/QDEIM-style index selection. The approach is paired with a DST-based Poisson solver and evaluated on A100 GPUs and AMD EPYC CPUs.","arXiv :2607 .0602 1v 1 [ cs .PF] 7 Jul 2026  \nA Sub-linear Low-Rank Solver for Poisson’s Equation using Machine Learning Frameworks for  \nGPU Acceleration  \nMåns I. Andersson 1 ,2[0000−0002−6384−2630] and Daniel  \nAppelö 1[0000−0002−0378−4563]  \n1 Virginia Polytechnic Institute and State University (Virginia Tech),  \nBlacksburg, Virginia, United States of America  \n[2](2 mansande@vt.edu)[ mansande@vt.edu](2 mansande@vt.edu)  \nAbstract. In this paper we explore a fast Poisson solver for problems with a solution that is known to be low-rank. We use an adaptive and warm started cross approximation called Cross-DEIM that iterates between index selection and and cross approximation to generate a low-rank solution. This paper focuses on leveraging a modern machine learning framework, PyTorch, as a general purpose array language to implement low-rank solvers based on Cross-DEIM. PyTorch enables native access to GPUs and accelerators but with a user-friendly high-level interface.  \nWe investigate statistical leverage scores for the index selection for the cross approximation due to the cost associated with the pivoted algorithms used with the discrete empirical interpolation methods (DEIMand QDEIM) which are historically preferred. The cross approximation is naturally paired with a Discrete Sine Transform (DST) Poisson solver.  \nThis allows the Fast Fourier Transform (FFT) to be evaluated in batches along dimensions independently without any global transpose even in higher dimensions. We present performance results running on a A100 GPU and AMD EPYC CPU demonstrating the usefulness of the approach that enables problems sizes that previously were not feasible.  \nKeywords: Cross approximation · Poisson’s equation · GPU · PyTorch.  \n1 Introduction  \nSolving Partial Differential Equations (PDE) is important in many fields of science and engineering with applications such as computational fluid dynamics [17] and computational electromagnetics [18] . Assuming a PDE has been discretized with NDOF number of degrees of freedom the computational cost and memory usage is traditionally expected to scale, at best, as O (NDOF ) . For certain problems (e.g. elliptic PDE) the solution to the PDE may exhibit so called lowrank structure [5] . When present, such low-rank structures makes it possible to design numerical approximation methods with exponentially better complexity, O (rq (NDOF ) ~~ 1~~D ) in D dimensions. Here r is a measure of the rank of the solution  \n2 M. I. Andersson and D. Appelö  \n(in this paper r will be the matrix rank) and the parameter q depends on the precise low-rank compression format used but is typically 2-4 .  \nIn two dimensions a natural low-rank format is the singular value decomposition. In higher dimensions there are multiple competing formats for example Tucker tensors and Tensor Trains (TT) (see [6]) . In this study we focus on the two dimensional case and low-rank techniques that can accelerate PDE discretization methods based on logically Cartesian meshes (e.g. finite difference methods and finite / spectral elements on quad elements) .  \nThe methods we consider here are based on access to elements of a nonlinear function G (i1 , i2 ,..., iD ) defined on a D-dimensional tensor. For example, G may come from a spatial discretization of a PDE. As mentioned previously we specialize to the case D = 2 and thus assume that G is a matrix. Note that we allow that G depend on another matrix X , i.e. an element of G (X) : G (X)(i,j) can be a nonlinear function of elements X (i′, j ′) . An example would be that G isan approximation to Xxx + Xyy + X3 using a local finite difference stencil, then (i′, j ′) would take values in, e.g. , (i ± 1, i ± 1) . When X is stored as a singular value decomposition with rank r the elements in X (i,j) = Prk=1 σkuk (i)vTk(j)  \ncan be accessed at O (r) cost.  \nOne of the main challenges for efficient low-rank methods arises from thenonlinearity of G. For simplicity let the indices i, j ∈ [1, n] s","cbCaigpqN8kFfW2B","https://ap.wps.com/l/cbCaigpqN8kFfW2B","pdf",781115,1,16,"English","en",105,"# Introduction\n## Low-rank structure and complexity reduction\n## Cross approximation and Cross-DEIM\n## GPU implementation with PyTorch","[{\"question\":\"What problem does the paper address?\",\"answer\":\"It addresses fast numerical solving of Poisson’s equation when the solution is known to have low-rank structure, focusing on reducing complexity beyond conventional scaling.\"},{\"question\":\"How does Cross-DEIM construct a low-rank solution?\",\"answer\":\"Cross-DEIM iterates between index selection and cross approximation, using an adaptive warm-start strategy to generate a low-rank representation efficiently.\"},{\"question\":\"Why is PyTorch used in the proposed GPU-accelerated solver?\",\"answer\":\"PyTorch provides GPU-native access with a user-friendly high-level interface, allowing the low-rank solver components based on Cross-DEIM to be implemented effectively and paired with a DST/FFT-style Poisson 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problem does the paper address?","Question",{"text":75,"@type":76},"It addresses fast numerical solving of Poisson’s equation when the solution is known to have low-rank structure, focusing on reducing complexity beyond conventional scaling.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does Cross-DEIM construct a low-rank solution?",{"text":80,"@type":76},"Cross-DEIM iterates between index selection and cross approximation, using an adaptive warm-start strategy to generate a low-rank representation efficiently.",{"name":82,"@type":73,"acceptedAnswer":83},"Why is PyTorch used in the proposed GPU-accelerated solver?",{"text":84,"@type":76},"PyTorch provides GPU-native access with a user-friendly high-level interface, allowing the low-rank solver components based on Cross-DEIM to be implemented effectively and paired with a DST/FFT-style Poisson 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