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Sci. Comput."],"published-print":{"date-parts":[[2023,4,30]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>We develop a new meshfree geometric multilevel (MGM) method for solving linear systems that arise from discretizing elliptic PDEs on surfaces represented by point clouds. The method uses a Poisson disk sampling-type technique for coarsening the point clouds and new meshfree restriction\/interpolation operators based on polyharmonic splines for transferring information between the coarsened point clouds. These are then combined with standard smoothing and operator coarsening methods in a V-cycle iteration. MGM is applicable to discretizations of elliptic PDEs based on various localized meshfree methods, including RBF finite differences (RBF-FD) and generalized finite differences (GFD). We test MGM both as a standalone solver and preconditioner for Krylov subspace methods on several test problems using RBF-FD and GFD and numerically analyze convergence rates, efficiency, and scaling with increasing point cloud sizes. We also perform a side-by-side comparison to algebraic multigrid methods for solving the same systems. Finally, we further demonstrate the effectiveness of MGM by applying it to three challenging applications on complicated surfaces: pattern formation, surface harmonics, and geodesic distance.<\/jats:p>","DOI":"10.1137\/22m1490338","type":"journal-article","created":{"date-parts":[[2023,3,27]],"date-time":"2023-03-27T11:55:52Z","timestamp":1679918152000},"page":"A312-A337","source":"Crossref","is-referenced-by-count":15,"title":["MGM: A Meshfree Geometric Multilevel Method for Systems Arising from Elliptic Equations on Point Cloud Surfaces"],"prefix":"10.1137","volume":"45","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7441-1791","authenticated-orcid":false,"given":"Grady B.","family":"Wright","sequence":"first","affiliation":[{"name":"Department of Mathematics, Boise State University, Boise, ID 83725 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Andrew","family":"Jones","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Boise State University, Boise, ID 83725 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4987-2972","authenticated-orcid":false,"given":"Varun","family":"Shankar","sequence":"additional","affiliation":[{"name":"School of Computing, University of Utah, Salt Lake City, UT 84112 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2023,3,28]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1002\/nla.374"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1007\/s10915-020-01399-3"},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.1137\/0732037"},{"key":"ref4","doi-asserted-by":"publisher","DOI":"10.1137\/15M1028807"},{"key":"ref5","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2016.12.008"},{"key":"ref6","doi-asserted-by":"publisher","DOI":"10.1137\/110838844"},{"key":"ref7","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492904000212"},{"key":"ref8","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.2001.6937"},{"key":"ref9","doi-asserted-by":"publisher","DOI":"10.1145\/1278780.1278807"},{"key":"ref10","doi-asserted-by":"publisher","DOI":"10.1137\/130929497"},{"key":"ref11","doi-asserted-by":"publisher","DOI":"10.1145\/2516971.2516977"},{"key":"ref12","first-page":"398","volume":"39","author":"Davydov O.","year":"2019","journal-title":"IMA J. 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