Upcoming Event: CSEM Student Forum

Proximal Galerkin for Inequality-Constrained Variational Problems

Matthew Meeker, Oden Institute

Abstract

Many variational problems in science and engineering involve inequality constraints, such as non-penetration in contact, irreversibility in fracture, and positivity of densities in multiphase flows and optimal design. In contrast to equality constraints, these problems define feasible sets with boundaries, so standard numerical methods do not automatically preserve admissibility. As a result, practitioners rely on piecemeal, ad hoc approaches tailored to specific applications.

In this talk, we introduce the Latent Variable Proximal Point (LVPP) framework and its finite element realization, Proximal Galerkin (PG), as a systematic approach to these difficulties. The central idea is to replace Euclidean updates with proximal steps adapted to the geometry of the feasible set, using Legendre functions and the associated Bregman divergences to encode the constraints. To make these abstract ingredients concrete, we will work through a standard topology optimization problem for heat diffusion, which involves pointwise bounds and a volume constraint. We conclude by briefly discussing error analysis and surveying other applications, which include obstacle problems, Signorini/contact, and eikonal and Monge-Ampère equations.

 

Biography

Matthew started the CSEM PhD program in Fall 2025. He was born and raised in the metro Atlanta area and then obtained his bachelor's degree from Brown University, where he studied applied mathematics and computer science. His interests sit between analysis, machine learning, and scientific computing.