{"id":{"repo_id":"mit","oai_identifier":"oai:dspace.mit.edu:1721.1/151567"},"canonical_url":"https://search.dev.ndltd.org/etd/mit/oai:dspace.mit.edu:1721.1/151567","repository":{"repo_id":"mit","name":"MIT","base_url":"https://dspace.mit.edu/oai/request"},"display":{"title":"A Framework for Solving Parabolic Partial Differential Equations on Discrete Domains","abstract":"We introduce a framework for solving a class of parabolic partial differential equations on triangle mesh surfaces, including the Hamilton-Jacobi equation and the Fokker- Planck equation. Certain PDE in this class often have nonlinear or stiff terms that cannot be resolved with standard methods on triangle mesh surfaces. To address this challenge, we leverage a splitting integrator combined with a convex optimization step to solve these PDE. Our machinery can be used to compute entropic approximation of optimal transport distances on geometric domains, overcoming the numerical limitations of the state-of-the-art method. In addition, we demonstrate the versatility of our method on a number of linear and nonlinear PDE that appear in diffusion tasks in geometry processing.","abstract_html":"We introduce a framework for solving a class of parabolic partial differential equations on triangle mesh surfaces, including the Hamilton-Jacobi equation and the Fokker- Planck equation. Certain PDE in this class often have nonlinear or stiff terms that cannot be resolved with standard methods on triangle mesh surfaces. To address this challenge, we leverage a splitting integrator combined with a convex optimization step to solve these PDE. Our machinery can be used to compute entropic approximation of optimal transport distances on geometric domains, overcoming the numerical limitations of the state-of-the-art method. In addition, we demonstrate the versatility of our method on a number of linear and nonlinear PDE that appear in diffusion tasks in geometry processing.","abstract_has_math":false,"creators":["Mattos Da Silva, Leticia"],"institution":"Massachusetts Institute of Technology","degree_name":"Master","degree_level":null,"degree_discipline":null,"degree_department":"Massachusetts Institute of Technology. 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Certain PDE in this class often have nonlinear or stiff terms that cannot be resolved with standard methods on triangle mesh surfaces. To address this challenge, we leverage a splitting integrator combined with a convex optimization step to solve these PDE. Our machinery can be used to compute entropic approximation of optimal transport distances on geometric domains, overcoming the numerical limitations of the state-of-the-art method. In addition, we demonstrate the versatility of our method on a number of linear and nonlinear PDE that appear in diffusion tasks in geometry processing."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["S.M."]},{"key":"dc:title","label":"Title","values":["A Framework for Solving Parabolic Partial Differential Equations on Discrete Domains"]}]}],"canonical_facts":{"dc:contributor.advisor":["Solomon, Justin"],"dc:contributor.department":["Massachusetts Institute of Technology. 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In addition, we demonstrate the versatility of our method on a number of linear and nonlinear PDE that appear in diffusion tasks in geometry processing."],"dc:description.degree":["S.M."],"dc:identifier.uri":["https://hdl.handle.net/1721.1/151567"],"dc:publisher":["Massachusetts Institute of Technology"],"dc:rights":["In Copyright - Educational Use Permitted","Copyright retained by author(s)"],"dc:rights.uri":["https://rightsstatements.org/page/InC-EDU/1.0/"],"dc:title":["A Framework for Solving Parabolic Partial Differential Equations on Discrete Domains"],"dc:type":["Thesis"],"thesis:degree_name":["Master","Master of Science in Electrical Engineering and Computer Science"]},"updated_at":"2026-07-22T22:22:03Z"}