[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83072-en":3,"doc-seo-83072-105":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},83072,1374391975076,"Riley","https://ap-avatar.wpscdn.com/avatar/14000253ca4ec9f6853?x-image-process=image/resize,m_fixed,w_180,h_180&k=1783305029341752051",8,"Research & Report","On Symmetric Kernel Collocation for Nonlinear PDEs","Kernel-based approximation methods for nonlinear partial differential equations are studied through an optimal-recovery generalized interpolation formulation. The PDE problem is posed as an optimization task in a reproducing kernel Hilbert space with nonlinear functional constraints induced by the differential operators and boundary conditions. A convergence analysis is performed directly in the RKHS, relaxing a uniqueness assumption for the PDE solution. For the nonunique regime, the limiting object becomes a minimum-norm solution, and a residual-greedy adaptive collocation strategy is shown to generate convergent generalized interpolants, supported by experiments on nonlinear heat conduction.","arXiv :2607 .06276v1 [math .NA] 7 Jul 2026  \nOn Symmetric Kernel Collocation for Nonlinear  \nPDEs  \nMilan Bacchetta*,1 , Tobias Ehring2 , and Bernard Haasdonk2  \n1Mathematical Optimization and Data Science Group, Saarland University, Germany  \n2Institute of Applied Mathematics and Numerical Simulation, University of Stuttgart, Germany July 8, 2026  \nAbstract  \nThis paper considers kernel-based approximation methods for nonlinear partial differential equations. To this end, the problem is formulated as an optimal-recovery generalized interpolation problem, that is, as an optimization problem in an RKHS with nonlinear functional constraints. This formulation provides the basis for a convergence analysis carried out directly in the RKHS and extends existing results by relaxing the uniqueness assumption on the PDE solution. In the nonunique case, the limiting object is characterized as a minimum-norm solution. Furthermore, a residual-greedy strategy for adaptive collocation point selection is proposed, and convergence of the resulting sequence of generalized interpolants is established. Numerical experiments for a stationary nonlinear heat equation illustrate the method and indicate that residual-greedy point selection can lead to markedly smaller PDE residuals than point sets selected according to fill-distance criteria.  \n1 Introduction  \nSymmetric kernel collocation is a meshless approach for approximating solutions of partial differential equations by functions from the native space of a positive definite kernel. In the linear case, the collocation equations lead to a linear generalized interpolation problem. For nonlinear equations this interpretation is no longer available directly, since imposing the equation at finitely many points leads to nonlinear conditions on the unknown function. To describe the class of nonlinear problems considered here, we separate the linear differential quantities from the nonlinear dependence on them. We consider boundary value problems of the form  \nP (u)(x) = f(x), x ∈ Ω ,  \nB (u)(x) = g(x), x ∈ ∂Ω,  \n*Corresponding author: [e-mail](e-mail milan.bacchetta@math.uni-sb.de)[ milan.bacchetta@math.uni-sb.de](e-mail milan.bacchetta@math.uni-sb.de)  \nwhere  \nP (u)(x) = P 􀀀 LP1u (x),..., LPQu (x)􀀁 , B (u)(x) = B 􀀀 LB1u (x),..., LBRu (x)􀀁 .  \nHere, P and B are continuous, possibly nonlinear functions, while LPi and LBj are linear differential operators. This formulation keeps the differential operations linear, while allowing the equations and boundary conditions to depend nonlinearlyon the resulting quantities. The precise assumptions are stated below.  \nLinear symmetric kernel collocation can be interpreted as an optimal recovery problem in the native space of the kernel. In this setting, the approximation is sought among native-space functions satisfying the differential equation and the boundary conditions at prescribed collocation points. For linear differential and boundary operators, these conditions are linear functional constraints. The nonlinear formulation studied in this work extends this principle by admitting nonlinear pointwise constraints generated by the operators P and B. A natural consistency question is whether the resulting collocation approximants converge to solutions of the boundary value problem as the point sets are refined. We address this question in two regimes: first, for sequences of collocation points with vanishing fill distance, and second, for a novel residual-based, target-dependent greedy strategy for selecting collocation points. In both regimes, convergence is established in the native space of the kernel. Finally, numerical experiments for a stationary nonlinear heat-conduction problem illustrate the proposed greedy strategy.  \nThe analysis builds on reproducing kernel Hilbert spaces, kernel collocation for PDEs, optimal recovery, and greedy point selection for kernel methods. The RKHS perspective goes back to the foundational work of Aronszajn [1], while kernel","cbCaicHIunB44iOs","https://ap.wps.com/l/cbCaicHIunB44iOs","pdf",478067,3,1,20,"English","en",105,"# Abstract\n# Introduction\n# Preliminaries","[{\"question\":\"How is the nonlinear PDE collocation problem formulated in the paper?\",\"answer\":\"It is posed as an optimal-recovery generalized interpolation problem: an optimization task in an RKHS with nonlinear functional constraints generated by the PDE operator and boundary operator evaluated at collocation points.\"},{\"question\":\"What changes in the analysis when the PDE solution is nonunique?\",\"answer\":\"The limiting object is characterized as a minimum-norm solution in the RKHS, rather than relying on uniqueness.\"},{\"question\":\"What is the residual-greedy strategy and what do experiments show?\",\"answer\":\"The method adaptively selects collocation points using a residual-based, target-dependent greedy criterion. Numerical experiments for a stationary nonlinear heat equation indicate it can yield substantially smaller PDE residuals than fill-distance-based point sets.\"}]",1784185013,50,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"on-symmetric-kernel-collocation-for-nonlinear-pdes","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/on-symmetric-kernel-collocation-for-nonlinear-pdes/83072/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-23","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"How is the nonlinear PDE collocation problem formulated in the paper?","Question",{"text":75,"@type":76},"It is posed as an optimal-recovery generalized interpolation problem: an optimization task in an RKHS with nonlinear functional constraints generated by the PDE operator and boundary operator evaluated at collocation points.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What changes in the analysis when the PDE solution is nonunique?",{"text":80,"@type":76},"The limiting object is characterized as a minimum-norm solution in the RKHS, rather than relying on uniqueness.",{"name":82,"@type":73,"acceptedAnswer":83},"What is the residual-greedy strategy and what do experiments show?",{"text":84,"@type":76},"The method adaptively selects collocation points using a residual-based, target-dependent greedy criterion. Numerical experiments for a stationary nonlinear heat equation indicate it can yield substantially smaller PDE residuals than fill-distance-based point sets.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,114,119,122,126,129,133],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":29,"slug":113},6,"Technology","technology",{"id":115,"doc_module":4,"doc_module_name":46,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":22,"slug":125},9,"Religion & Spirituality","religion-spirituality",{"id":22,"doc_module":4,"doc_module_name":46,"category_name":127,"show_sort_weight":22,"slug":128},"World Cup","world-cup",{"id":130,"doc_module":4,"doc_module_name":46,"category_name":131,"show_sort_weight":130,"slug":132},10,"Lifestyle","lifestyle",{"id":134,"doc_module":4,"doc_module_name":46,"category_name":135,"show_sort_weight":106,"slug":136},19,"General","general"]