In this work, a numerical approach based on meshless methods is proposed to obtain eigenmodes of Laplace-Beltrami operator on manifolds, and its performance is compared against existing alternative methods. Radial Basis Function (RBF)-based methods allow one to obtain interpolation and differentiation matrices easily by using scattered data points. We derive expressions for such matrices for the Laplace-Beltrami operator via so-called Reilly’s formulas and use them to solve the respective eigenvalue problem. Numerical studies of proposed methods are performed in order to demonstrate convergence on simple examples of one-dimensional curves and two-dimensional surfaces.
","abstract_html":"<p>In this work, a numerical approach based on meshless methods is proposed to obtain eigenmodes of Laplace-Beltrami operator on manifolds, and its performance is compared against existing alternative methods. Radial Basis Function (RBF)-based methods allow one to obtain interpolation and differentiation matrices easily by using scattered data points. We derive expressions for such matrices for the Laplace-Beltrami operator via so-called Reilly’s formulas and use them to solve the respective eigenvalue problem. Numerical studies of proposed methods are performed in order to demonstrate convergence on simple examples of one-dimensional curves and two-dimensional surfaces.</p>","abstract_has_math":false,"creators":["Delengov, Vladimir"],"institution":null,"degree_name":"Mathematics, PhD","degree_level":"Open Access Dissertation","degree_discipline":"School of Mathematical Sciences","degree_department":null,"school":null,"contributors":["Chiu-Yen Kao","Marina Chugunova","Ali Nadim"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2018,"date_issued":"2018-01-01T08:00:00Z","date_published":"2018-01-01T08:00:00Z","updated_at":"2026-07-24T01:39:44Z","subjects":["radial basis functions","Laplace-Beltrami operator","eigenproblem","meshfree","elliptic operators","manifold","Numerical Analysis and Computation","Partial Differential Equations"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://scholarship.claremont.edu/cgu_etd/113","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Chiu-Yen Kao","Marina Chugunova","Ali Nadim"]},{"key":"dc:creator","label":"Author","values":["Delengov, Vladimir"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2018-10-03T07:00:00Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["School of Mathematical Sciences","Computational Mathematics and Numerical Analysis"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Open Access Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Mathematics, PhD"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["radial basis functions","Laplace-Beltrami operator","eigenproblem","meshfree","elliptic operators","manifold","Numerical Analysis and Computation","Partial Differential Equations"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://scholarship.claremont.edu/cgu_etd/113"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["In this work, a numerical approach based on meshless methods is proposed to obtain eigenmodes of Laplace-Beltrami operator on manifolds, and its performance is compared against existing alternative methods. Radial Basis Function (RBF)-based methods allow one to obtain interpolation and differentiation matrices easily by using scattered data points. We derive expressions for such matrices for the Laplace-Beltrami operator via so-called Reilly’s formulas and use them to solve the respective eigenvalue problem. Numerical studies of proposed methods are performed in order to demonstrate convergence on simple examples of one-dimensional curves and two-dimensional surfaces.
"]},{"key":"dc:title","label":"Title","values":["Computing Eigenmodes of Elliptic Operators on Manifolds Using Radial Basis Functions"]}]}],"canonical_facts":{"dc:contributor":["Chiu-Yen Kao","Marina Chugunova","Ali Nadim"],"dc:creator":["Delengov, Vladimir"],"dc:date.available":["2018-10-03T07:00:00Z"],"dc:description.abstract":["In this work, a numerical approach based on meshless methods is proposed to obtain eigenmodes of Laplace-Beltrami operator on manifolds, and its performance is compared against existing alternative methods. Radial Basis Function (RBF)-based methods allow one to obtain interpolation and differentiation matrices easily by using scattered data points. We derive expressions for such matrices for the Laplace-Beltrami operator via so-called Reilly’s formulas and use them to solve the respective eigenvalue problem. Numerical studies of proposed methods are performed in order to demonstrate convergence on simple examples of one-dimensional curves and two-dimensional surfaces.
"],"dc:identifier":["https://scholarship.claremont.edu/cgu_etd/113"],"dc:subject":["radial basis functions","Laplace-Beltrami operator","eigenproblem","meshfree","elliptic operators","manifold","Numerical Analysis and Computation","Partial Differential Equations"],"dc:title":["Computing Eigenmodes of Elliptic Operators on Manifolds Using Radial Basis Functions"],"thesis:degree_discipline":["School of Mathematical Sciences","Computational Mathematics and Numerical Analysis"],"thesis:degree_level":["Open Access Dissertation"],"thesis:degree_name":["Mathematics, PhD"]},"updated_at":"2026-07-24T01:39:44Z"}