[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84414-en":3,"doc-seo-84414-105":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},84414,1099513958607,"Jiven","https://ap-avatar.wpscdn.com/avatar/100002390cf8733938c?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778829742770036399",8,"Research & Report","Walking on Spheres and Talking to Neighbors Variance Reduction for Laplace’s Equation","Walk on Spheres algorithms use Brownian motion properties to build Monte Carlo estimates for solutions of elliptic partial differential equations. A new caching strategy is introduced by exploiting the continuity of Brownian paths so estimates incorporate relationships between solutions at neighboring points. For Laplace’s equation with Dirichlet boundary conditions, the method improves asymptotic runtime over prior approaches. Fixed-size information reuse enables performance bounds and demonstrations on problems of increasing complexity.","Walking on Spheres and Talking to Neighbors: Variance Reduction for Laplace’s Equation  \nMichael T. Czekanski  \n[mc2589@cornell.edu](mc2589@cornell.edu)[ ](mc2589@cornell.edu)Cornell University Ithaca, NY, USA  \nBenjamin J. Faber  \nUniversity of Wisconsin-Madison Madison, WI, USA  \nMargaret E. Fairborn Columbia University New York, NY, USA  \narXiv :2404 . 17692v3 [physics .comp-ph] 2 Jul 2026  \nAdelle M. Wright  \nUniversity of Wisconsin-Madison Madison, WI, USA  \nDavid S. Bindel  \nCornell University Ithaca, NY, USA  \n(a) Standard Walk on Spheres (b) Equal Weighted Info. Reuse (ours) (c) Variance Weighted Info. Reuse (ours)  \nFigure 1: Standard Walk on Spheres output (left) vs. our variance equal-weighted (middle) and inverse variance-weighted (right) reuse strategies with 10 walks per pixel. Time for information reuse is negligible (\u003C 1% of time to perform walks)  \nAbstract  \nWalk on Spheres algorithms leverage properties of Brownian Motion to create Monte Carlo estimates of solutions to elliptic partial differential equations. We propose a new caching strategy that  \nPermission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission [and/or a fee. Request permissions from permissions@acm.org](and/or a fee. Request permissions from permissions@acm.org).  \nConference acronym ’XX, Woodstock, NY  \n© 2026 Copyright held by the owner/author(s) . Publication rights licensed to ACM. ACM ISBN 978-x-xxxx-xxxx-x/YYYY/MM [https://doi.org/10.1145/nnnnnnn.nnnnnnn](https://doi.org/10.1145/nnnnnnn.nnnnnnn)  \nleverages the continuity of paths of Brownian Motion. Until recently, estimates were constructed pointwise and did not use the relationship between solutions at nearby points within a domain. In the case of Laplace’s equation with Dirichlet boundary conditions, our algorithm has improved asymptotic runtime compared to previous approaches. Our results are achieved by information reuse from a cache of fixed size. We also provide bounds on the performance of our algorithm and demonstrate our approach on example problems of increasing complexity.  \nCCS Concepts  \n• Mathematics of computing → Partial differential equations; Stochastic processes.  \nACM Reference Format:  \nMichael T. Czekanski, Benjamin J. Faber, Margaret E. Fairborn, Adelle M. Wright, and David S. Bindel. 2026. Walking on Spheres and Talking to Neighbors: Variance Reduction for Laplace’s Equation. In Proceedings of Make sure to enter the correct conference title from your rights confirmation email (Conference acronym ’XX). ACM, New York, NY, USA, 9 pages. [https:](https:)//[doi.org/10.1145/nnnnnnn.nnnnnnn](doi.org/10.1145/nnnnnnn.nnnnnnn)  \n1 Introduction  \nPhysics-based modeling plays an important role in enhancing the realism of virtual environments. We do this by solving ordinary or partial differential equations (PDEs) . A particularly important class of PDEs, often arising in the description of steady-state systems, are elliptic PDEs. The most fundamental of these, Laplace’s equation, describes situations arising naturally in heat conduction, fluid dynamics, and electrostatics.  \nOften we need to obtain solutions to Laplace’s equation on geometrically complex domain representing real-world objects and environments. Generally, this must be done numerically. There exist a variety of mesh-based and mesh-free methods each with distinct advantages. We present improvements to the mesh-free Walk on Spheres algorithm which has two advantages: it is not necessary to discretize the domain, and solutions can be computed on subsets o","cbCaistJUrZFhkUr","https://ap.wps.com/l/cbCaistJUrZFhkUr","pdf",10982010,1,9,"English","en",105,"# Introduction\n## Laplace’s Equation\n## Contribution","[{\"question\":\"What problem does the paper address?\",\"answer\":\"The paper addresses numerical solution of Laplace’s equation on complex domains using a mesh-free Monte Carlo method (Walk on Spheres).\"},{\"question\":\"How does the proposed method reduce variance?\",\"answer\":\"It introduces a caching and information-reuse strategy that leverages continuity of Brownian motion paths to reuse information across nearby points, improving estimation efficiency.\"},{\"question\":\"What are the reported benefits for Laplace’s equation with Dirichlet boundary conditions?\",\"answer\":\"The approach improves asymptotic runtime compared with previous methods, achieved through fixed-size cache-based information reuse, along with bounds and experiments on increasingly complex example 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