The proximal Galerkin finite element method is a high-order, nonlinear numerical method that preserves the geometric and algebraic structure of bound constraints in infinite- dimensional function spaces. In this talk, we will introduce the proximal Galerkin method and apply it to solve free-boundary problems, enforce discrete maximum principles, and develop scalable, mesh-independent algorithms for optimal design. The proximal Galerkin framework is a natural consequence of the latent variable proximal point (LVPP) method, which is an stable and robust alternative to the interior point method that will also be introduced in this talk. In particular, LVPP is a low-iteration complexity, infinite-dimensional optimization al- gorithm that may be viewed as having an adaptive barrier function that is updated with a new informative prior at each (outer loop) optimization iteration. One of the main benefits of this algorithm is witnessed when analyzing the classical obstacle problem. Therein, we find that the original variational inequality can be replaced by a sequence of semilinear partial dif- ferential equations (PDEs) that are readily discretized and solved with, e.g., high-order finite elements. Throughout the talk, we will arrive at several unexpected contributions that may be of independent interest. These include (1) a semilinear PDE we refer to as the entropic Poisson equation; (2) an algebraic/geometric connection between high-order positivity-preserving dis- cretizations and an infinite-dimensional Lie group; and (3) a gradient-based, bound-preserving algorithm for two-field density-based topology optimization. The complete latent variable prox- imal Galerkin methodology combines ideas from nonlinear programming, functional analysis, tropical algebra, and differential geometry and can potentially lead to new synergies among these areas as well as within variational and numerical analysis. This talk is based on [1].

[1] Keith, Brendan, and Thomas M. Surowiec. "Proximal Galerkin: A structure-preserving finite element method for pointwise bound constraints." arXiv preprint arXiv:2307.12444 (2023)
Brown University
Email address: brendan_keith@brown.edu
Simula Research Laboratory
Email address: thomasms@simula.no

Bio: Brendan Keith is an Assistant Professor in the Division of Applied Mathematics at Brown University in Providence, Rhode Island. His research interests mainly relate to modeling, pre- dicting, and optimizing for events arising in natural sciences and engineering, focusing on numerical methods for partial differential equations, scientific machine learning, and PDE- constrained optimization. In 2018, Brendan received his Ph.D. in Computational Science, Engineering, and Mathematics from the Oden Institute for Computational Engineering and Sciences at the University of Texas at Austin. Prior to Brown, he held postdoctoral positions at TU Munich, ICERM, and Lawrence Livermore National Laboratory. Most recently, he was awarded a 2023 DOE Office of Science Early Career Research Award and featured in Popular Science Magazine’s 2023 “Brilliant 10” list of early-career researchers.