{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T18:57:23Z","timestamp":1774637843799,"version":"3.50.1"},"reference-count":50,"publisher":"Association for Computing Machinery (ACM)","issue":"4","funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["GU 945\/3-1"],"award-info":[{"award-number":["GU 945\/3-1"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Graph."],"published-print":{"date-parts":[[2025,8,1]]},"abstract":"<jats:p>\n                    In recent years, grid-free Monte Carlo methods have gained increasing popularity for solving fundamental partial differential equations. For a given point in the domain, the\n                    <jats:italic toggle=\"yes\">Walk-on-Spheres<\/jats:italic>\n                    method solves a boundary integral equation by integrating recursively over the largest possible sphere. When the walks approach boundaries with Dirichlet conditions, the number of path vertices increases considerably, since the step size becomes smaller with decreasing distance to the boundary. In practice, the walks are terminated once they reach an epsilon-shell around the boundary. This, however, introduces bias, leading to a trade-off between accuracy and performance. Instead of using spheres, we propose to utilize geometric primitives that share more than one point with the boundary to increase the likelihood of immediately terminating. Along the boundary of those new geometric primitives a sampling probability is needed, which corresponds to the exit probability of a Brownian motion. This is known as a first passage problem. Utilizing that Laplace equations are invariant under conformal maps, we transform exit points from unit circles to the exit points of our geometric primitives, for which we describe a suitable placement strategy. With this, we obtain a novel approach to solve the Laplace equation in two dimensions, which does not require an epsilon-shell, significantly reduces the number of path vertices, and reduces inaccuracies near Dirichlet boundaries.\n                  <\/jats:p>","DOI":"10.1145\/3730942","type":"journal-article","created":{"date-parts":[[2025,7,27]],"date-time":"2025-07-27T04:02:41Z","timestamp":1753588961000},"page":"1-11","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":["Conformal First Passage for Epsilon-free Walk-on-Spheres"],"prefix":"10.1145","volume":"44","author":[{"ORCID":"https:\/\/orcid.org\/0009-0000-3825-045X","authenticated-orcid":false,"given":"Paul","family":"Himmler","sequence":"first","affiliation":[{"name":"Department of Computer Science, Friedrich-Alexander-Universit\u00e4t Erlangen-N\u00fcrnberg, Erlangen, Germany"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3020-0930","authenticated-orcid":false,"given":"Tobias","family":"G\u00fcnther","sequence":"additional","affiliation":[{"name":"Department of Computer Science, Friedrich-Alexander-Universit\u00e4t Erlangen-N\u00fcrnberg, Erlangen, Germany"}]}],"member":"320","published-online":{"date-parts":[[2025,7,27]]},"reference":[{"key":"e_1_2_2_1_1","doi-asserted-by":"publisher","DOI":"10.1145\/3618374"},{"key":"e_1_2_2_2_1","doi-asserted-by":"publisher","DOI":"10.1002\/9780470611418"},{"key":"e_1_2_2_3_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00039-012-0161-z"},{"key":"e_1_2_2_4_1","doi-asserted-by":"publisher","DOI":"10.1201\/9781315266008"},{"key":"e_1_2_2_5_1","doi-asserted-by":"publisher","DOI":"10.1145\/3658199"},{"key":"e_1_2_2_6_1","doi-asserted-by":"publisher","DOI":"10.1137\/19M125337X"},{"key":"e_1_2_2_7_1","doi-asserted-by":"publisher","DOI":"10.1016\/0167-7977(87)90014-1"},{"key":"e_1_2_2_8_1","unstructured":"Michael Czekanski Benjamin Faber Margaret Fairborn Adelle Wright and David Bindel. 2024. Walking on Spheres and Talking to Neighbors: Variance Reduction for Laplace's Equation. arXiv:2404.17692 [physics.comp-ph]"},{"key":"e_1_2_2_9_1","doi-asserted-by":"publisher","DOI":"10.1145\/3623264.3624457"},{"key":"e_1_2_2_10_1","doi-asserted-by":"publisher","unstructured":"David H Douglas and Thomas K Peucker. 1973. Algorithms for the reduction of the number of points required to represent a digitized line or its caricature. Cartographica: the international journal for geographic information and geovisualization 10 2 (1973) 112\u2013122. 10.3138\/FM57-6770-U75U-7727","DOI":"10.3138\/FM57-6770-U75U-7727"},{"key":"e_1_2_2_11_1","doi-asserted-by":"publisher","DOI":"10.1109\/PacificVis.2013.6596151"},{"key":"e_1_2_2_12_1","doi-asserted-by":"publisher","DOI":"10.1063\/1.473428"},{"key":"e_1_2_2_13_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-21617-1"},{"key":"e_1_2_2_14_1","doi-asserted-by":"publisher","DOI":"10.1109\/TVCG.2018.2867478"},{"key":"e_1_2_2_15_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2015.10.002"},{"key":"e_1_2_2_16_1","volume-title":"Classical electrodynamics (3 ed.)","author":"Jackson John David","unstructured":"John David Jackson. 1998. Classical electrodynamics (3 ed.). John Wiley & Sons, Nashville, TN. ISBN:978-0-471-30932-1."},{"key":"e_1_2_2_17_1","doi-asserted-by":"publisher","DOI":"10.1145\/1618452.1618462"},{"key":"e_1_2_2_18_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00033-004-3114-z"},{"key":"e_1_2_2_19_1","volume-title":"The theory of electromagnetism","author":"Jones Douglas Samuel","unstructured":"Douglas Samuel Jones. 1964. The theory of electromagnetism. Elsevier, Oxford, UK."},{"key":"e_1_2_2_20_1","doi-asserted-by":"publisher","DOI":"10.1145\/2766997"},{"key":"e_1_2_2_21_1","doi-asserted-by":"publisher","DOI":"10.1201\/9781315180236"},{"key":"e_1_2_2_22_1","doi-asserted-by":"publisher","DOI":"10.1145\/3610548.3618141"},{"key":"e_1_2_2_23_1","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827503422221"},{"key":"e_1_2_2_24_1","doi-asserted-by":"publisher","DOI":"10.1145\/3592400"},{"key":"e_1_2_2_25_1","doi-asserted-by":"publisher","DOI":"10.1145\/3687913"},{"key":"e_1_2_2_26_1","doi-asserted-by":"publisher","DOI":"10.1145\/3658153"},{"key":"e_1_2_2_27_1","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177728169"},{"key":"e_1_2_2_28_1","doi-asserted-by":"publisher","DOI":"10.1145\/3450626.3459809"},{"key":"e_1_2_2_29_1","volume-title":"Complex analysis and conformal mapping. Lecture Notes","author":"Olver Peter J","unstructured":"Peter J Olver. 2017. Complex analysis and conformal mapping. Lecture Notes, University of Minnesota, available at https:\/\/www-users.cse.umn.edu\/olver\/ln_\/cml.pdf, accessed on April 17, 2025."},{"key":"e_1_2_2_30_1","doi-asserted-by":"publisher","DOI":"10.1145\/1360612.1360691"},{"key":"e_1_2_2_31_1","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781139507486"},{"key":"e_1_2_2_32_1","doi-asserted-by":"publisher","DOI":"10.1111\/cgf.14586"},{"key":"e_1_2_2_33_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0146-664X(72)80017-0"},{"key":"e_1_2_2_34_1","doi-asserted-by":"publisher","DOI":"10.1145\/3550454.3555450"},{"key":"e_1_2_2_35_1","first-page":"1076","article-title":"Vector algorithms in the Monte-Carlo method for solving systems of second-order elliptic equations and Lam\u00e9s equation","volume":"262","author":"Sabelfeld Karl Karlovich","year":"1982","unstructured":"Karl Karlovich Sabelfeld. 1982. Vector algorithms in the Monte-Carlo method for solving systems of second-order elliptic equations and Lam\u00e9s equation. Doklady Akademii Nauk 262, 5 (1982), 1076\u20131080. https:\/\/www.mathnet.ru\/eng\/dan45069 (in Russian).","journal-title":"Doklady Akademii Nauk"},{"key":"e_1_2_2_36_1","doi-asserted-by":"publisher","DOI":"10.1515\/9783110942026"},{"key":"e_1_2_2_37_1","doi-asserted-by":"publisher","DOI":"10.1145\/3132705"},{"key":"e_1_2_2_38_1","doi-asserted-by":"publisher","DOI":"10.1145\/3386569.3392374"},{"key":"e_1_2_2_39_1","doi-asserted-by":"publisher","DOI":"10.1145\/3592398"},{"key":"e_1_2_2_40_1","doi-asserted-by":"publisher","DOI":"10.1016\/0010-4485(90)90039-F"},{"key":"e_1_2_2_41_1","doi-asserted-by":"publisher","DOI":"10.1007\/s41478-016-0019-0"},{"key":"e_1_2_2_42_1","doi-asserted-by":"publisher","DOI":"10.1145\/3641519.3657405"},{"key":"e_1_2_2_43_1","doi-asserted-by":"publisher","DOI":"10.1145\/3592109"},{"key":"e_1_2_2_44_1","doi-asserted-by":"publisher","DOI":"10.1145\/3680528.3687599"},{"key":"e_1_2_2_45_1","doi-asserted-by":"publisher","DOI":"10.1109\/TVCG.2025.3558263"},{"key":"e_1_2_2_46_1","volume-title":"A High Quality Solver for Diffusion Curves. Master's thesis","author":"de Gronde Jasper Van","unstructured":"Jasper Van de Gronde. 2011. A High Quality Solver for Diffusion Curves. Master's thesis. University of Groningen."},{"key":"e_1_2_2_47_1","doi-asserted-by":"publisher","DOI":"10.1007\/s41095-023-0368-y"},{"key":"e_1_2_2_48_1","doi-asserted-by":"publisher","DOI":"10.1145\/1833349.1778815"},{"key":"e_1_2_2_49_1","doi-asserted-by":"publisher","DOI":"10.1145\/3687990"},{"key":"e_1_2_2_50_1","doi-asserted-by":"publisher","DOI":"10.1145\/3641519.3657460"}],"container-title":["ACM Transactions on Graphics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/3730942","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T17:59:54Z","timestamp":1774634394000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3730942"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,27]]},"references-count":50,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2025,8,1]]}},"alternative-id":["10.1145\/3730942"],"URL":"https:\/\/doi.org\/10.1145\/3730942","relation":{},"ISSN":["0730-0301","1557-7368"],"issn-type":[{"value":"0730-0301","type":"print"},{"value":"1557-7368","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,7,27]]},"assertion":[{"value":"2025-01-23","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2025-03-29","order":2,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2025-07-27","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}