Mean Field Type Control with Congestion (II): An Augmented Lagrangian Method

TL;DR

This paper introduces a numerical approach using an augmented Lagrangian method to solve mean-field control problems with congestion, building on prior theoretical work and demonstrating effectiveness through various tests.

Contribution

It develops a finite difference scheme with a variational interpretation and applies an ADMM algorithm for efficient numerical solutions of mean-field control with congestion.

Findings

Finite difference scheme is monotone and variational.

ADMM algorithm effectively solves the variational problem.

Numerical tests validate the method's robustness and applicability.

Abstract

This work deals with a numerical method for solving a mean-field type control problem with congestion. It is the continuation of an article by the same authors, in which suitably defined weak solutions of the system of partial differential equations arising from the model were discussed and existence and uniqueness were proved. Here, the focus is put on numerical methods: a monotone finite difference scheme is proposed and shown to have a variational interpretation. Then an Alternating Direction Method of Multipliers for solving the variational problem is addressed. It is based on an augmented Lagrangian. Two kinds of boundary conditions are considered: periodic conditions and more realistic boundary conditions associated to state constrained problems. Various test cases and numerical results are presented.