Generalized optimal transport and mean field control problems for reaction-diffusion systems with high-order finite element computation

TL;DR

This paper develops a numerical framework for solving mean field control problems involving reaction-diffusion systems using high-order finite element discretization and augmented Lagrangian methods, with applications to optimal transport and image density control.

Contribution

It introduces a novel combination of high-order finite element methods and augmented Lagrangian techniques for efficiently solving mean field control problems related to reaction-diffusion systems.

Findings

Effective numerical methods demonstrated on Gaussian and image density examples.

Accurate solutions for generalized optimal transport problems.

Validation of the approach through multiple numerical experiments.

Abstract

We design and compute a class of optimal control problems for reaction-diffusion systems. They form mean field control problems related to multi-density reaction-diffusion systems. To solve proposed optimal control problems numerically, we first apply high-order finite element methods to discretize the space-time domain and then solve the optimal control problem using augmented Lagrangian methods (ALG2). Numerical examples, including generalized optimal transport and mean field control problems between Gaussian distributions and image densities, demonstrate the effectiveness of the proposed modeling and computational methods for mean field control problems involving reaction-diffusion equations/systems.