[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81522-en":3,"doc-seo-81522-105":30,"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":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},81522,1099513958762,"Logic","https://ap-avatar.wpscdn.com/avatar/1000023916a998db790?x-image-process=image/resize,m_fixed,w_180,h_180&k=1784791008015729253",8,"Research & Report","Characterizing and Computing Solutions to Regularized Semi-Discrete Optimal Transport via an Ordinary Differential Equation","Investigates the semi-discrete optimal transport problem with entropic regularization. The work characterizes the optimal solution by formulating a well-posed ordinary differential equation whose curve governs how the dual potentials vary with the regularization parameter. This construction yields a numerical algorithm with strong convexity properties, including a global strong convexity result linked to inverting the dual Hessian in the scheme. Extensive experiments compare the ODE method against Newton’s method, showing strong performance for squared and higher-power Euclidean costs and special effectiveness when target points lie outside the source support.","arXiv :2504 .03030v2 [math .NA] 10 Jul 2026  \nCharacterizing and computing solutions to regularized semi-discrete optimal transport via an ordinary differential equation  \nLuca Nenna∗,† Daniyar Omarov ‡and Brendan Pass §  \nMonday 13th July, 2026  \nKeywords. Semi-discrete optimal transport, entropic regularization, ODE, convex analysis. 2020 Mathematics Subject Classification. Primary: 49Q22; Secondary: 49N15, 94A17, 49K40 .  \nThis paper investigates the semi-discrete optimal transport (OT) problem with entropic regularization. We characterize the solution using a governing, well-posed ordinary differential equation (ODE) . This naturally yields an algorithm to solve the problem numerically, which we prove has desirable properties, notably including global strong convexity of a value function whose Hessian must be inverted in the numerical scheme. Extensive numerical experiments are conducted to validate our approach. We compare the solutions obtained using the ODE method with those derived from Newton’s method. Our results demonstrate that the proposed algorithm is competitive for problems involving the squared Euclidean distance and exhibits superior performance when applied to various powers of the Euclidean distance. In addition, it proves particularly effective in scenarios where the target points lie outside the support of the source measure. Finally, we note that the ODE approach yields an estimate on the rate of convergence of the solution as the regularization parameter vanishes, for a generic cost function.  \n∗ Universit´e Paris-Saclay, CNRS, Laboratoire de math´ematiques d’Orsay, ParMA, Inria Saclay, 91405, Orsay, [France. email:](France. email:) [luca.nenna@universite-paris-saclay.fr](luca.nenna@universite-paris-saclay.fr)  \n†Institut Universitaire de France, I.U.F.  \n‡Department of Mathematical and Statistical Sciences, 632 CAB, University of Alberta, Edmonton, Alberta, Canada, [T6G 2G1. email:](T6G 2G1. email:) [daniyar@ualberta.ca](daniyar@ualberta.ca)  \n§ Department of Mathematical and Statistical Sciences, 632 CAB, University of Alberta, Edmonton, Alberta, Canada, [T6G 2G1. email:](T6G 2G1. email:) [pass@ualberta.ca](pass@ualberta.ca)  \n1. Introduction  \nThe optimal transport (OT) problem, which was first proposed by Monge in 1781 [31] and subsequently relaxed by Kantorovich during the 1940-s [26, 27], involves identifying the most efficient way to transfer mass from one probability measure to another, all while minimizing a specified cost function. This profound problem is intricately connected to many areas of mathematics, including partial differential equations [25, 34] and statistics [37, 7, 39], and its applications are vast, extending into fields such as economics [20], fluid mechanics [3, 9], image processing [35], and machine learning [38] . In recent decades, advancements in computational techniques, notably entropic regularization [12, 4, 21, 11], have facilitated the practical implementation of OT in large-scale scenarios as well asthe numerical resolution of many variational problems involving optimal transport terms [36, 13, 11, 5, 8, 15, 2] .  \nIn this paper, we focus specifically on the semi-discrete optimal transport problem, that is  \nγ a(xρ,µ) ZX×Y b (x, y)dγ , (1)  \nwhere b (x, y) is a cost function, the source measure ρ (x) is absolutely continuous with respect to the Lebesgue measure, the target measure µ is supported on a finite set Y and Π(ρ,µ) is the set of couplings having ρ and µ as marginals. This particular variant has garnered increased interest recently due to its relevance in applications such as geometric optics [10, 14, 18] and mesh generation [19, 40] .  \nNumerous studies have discussed various numerical algorithms relevant to this problem [28, 17]; we refer the reader to [30] and the references within for an overview on this topic. Motivated by some recent works in the discrete case [32, 24], our focus here is on the entropic regularized version of the semi-discrete OT proble","cbCaisA28KXLwhO0","https://ap.wps.com/l/cbCaisA28KXLwhO0","pdf",3024852,3,1,27,"English","en",105,"# Introduction\n## Problem setting and entropic regularization\n## Dual formulation and ODE characterization\n## Computational implications and numerical validation","[{\"question\":\"What is the main contribution of the paper for regularized semi-discrete optimal transport?\",\"answer\":\"The paper characterizes the entropic regularized semi-discrete OT solution by deriving a well-posed ordinary differential equation governing the relevant dual potential curve as the regularization parameter varies.\"},{\"question\":\"How does the proposed ODE approach help compute solutions numerically?\",\"answer\":\"The ODE characterization provides an algorithm whose numerical properties are studied, including strong convexity of a value function and a boundedness property of the dual Hessian eigenvalues that helps avoid instabilities.\"},{\"question\":\"How does the ODE-based algorithm compare with Newton’s method?\",\"answer\":\"Numerical experiments validate the approach and demonstrate competitiveness with Newton’s method, with particularly strong performance for squared Euclidean distance and improved results for several powers of the Euclidean distance, including cases where target points fall outside the source 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is the main contribution of the paper for regularized semi-discrete optimal transport?","Question",{"text":75,"@type":76},"The paper characterizes the entropic regularized semi-discrete OT solution by deriving a well-posed ordinary differential equation governing the relevant dual potential curve as the regularization parameter varies.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the proposed ODE approach help compute solutions numerically?",{"text":80,"@type":76},"The ODE characterization provides an algorithm whose numerical properties are studied, including strong convexity of a value function and a boundedness property of the dual Hessian eigenvalues that helps avoid instabilities.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the ODE-based algorithm compare with Newton’s method?",{"text":84,"@type":76},"Numerical experiments validate the approach and demonstrate competitiveness with Newton’s method, with particularly strong performance for squared 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