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A Face-Upwinded Spectral Element Method and Machine Learning for Optimal Meshes

Per-Olof Persson
UC Berkeley


The talk will cover two new developments in our work on efficient high-order methods for conservation laws: (1) A simple stabilization technique for spectral element methods, which uses continuous solution spaces and is provably convergent for linear problems at arbitrary orders of accuracy. The main motivation for the new scheme is its lower cost, which comes from having fewer degrees of freedom, no Riemann solvers, and a line-based sparsity pattern. However, it also has other attractive properties such as an improved CFL condition and allowing for other solvers including static condensation. (2) A deep reinforcement learning approach for generation of meshes with optimal connectivities. Starting from a Delaunay mesh, we formulate the mesh optimization process as a "game" where the moves are standard topological element operations, and the goal is to maximize the number of regular nodes. The agent is trained in a self-play framework using the proximal policy optimization (PPO) algorithm running on GPUs. Our approach works for 2D triangular and quadrilateral meshes with minimal modification, and it routinely produces close-to-perfect meshes.

Tuesday, May 14, 2024
11:00AM AP&M 2402 and Zoom ID 982 8500 1195