Explore a comprehensive lecture on convergent semi-Lagrangian methods for the Monge-Ampère equation on unstructured grids. Delve into the challenges of numerically solving fully nonlinear second-order partial differential equations, focusing on Monge-Ampère type equations. Discover a new approach that establishes an equivalent Bellman formulation and designs monotone numerical methods for general triangular grids. Learn about the application of Howard's algorithm for robust computation of numerical approximations on fine meshes. Examine the rigorous convergence analysis, comparison principle for the Bellman operator, and treatment of boundary conditions. Gain insights into the connection between Monge-Ampère and Hamilton-Jacobi-Bellman equations, and understand their applications in optimal transport and inverse reflector problems. Follow the presentation's structure, covering motivation, viscosity solutions, equivalence, comparison principles, convergence, boundary conditions, and numerical experiments.
Convergent Semi-Lagrangian Methods for the Monge-Ampère Equation on Unstructured Grids
Hausdorff Center for Mathematics via YouTube
Overview
Syllabus
Intro
The outline
Motivation: Optimal transport
Motivation: Inverse reflector problem
Simple Monge Ampere equation
Ellipticity and convexity
Without convexity: Loss of uniqueness
Viscosity solution of the Monge Ampere equation
Summary of Part 1
Classical equivalence
Viscosity solutions of HJB
Equivalence in viscosity sense
Comparison principle
Summary of Part 2
Summary of Part 3
Towards convergence
How should we pose boundary conditions?
Summary of Part 4
Two numerical experiments
Summary of the presentation
Taught by
Hausdorff Center for Mathematics
