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

• Published in: Applied mathematics & optimization Vol. 74; no. 3; pp. 535 - 578
• Main Authors: Achdou, Yves, Laurière, Mathieu
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.12.2016; Springer Nature B.V
• Subjects: Algorithms; Applications of mathematics; BOUNDARY CONDITIONS; Calculus of Variations and Optimal Control; Optimization; CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Congestion; Control; Control theory; Finite difference method; LAGRANGIAN FUNCTION; Linear equations; Mathematical analysis; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematical models; MATHEMATICAL SOLUTIONS; Mathematics; Mathematics and Statistics; MEAN-FIELD THEORY; Numerical analysis; Numerical and Computational Physics; Optimization; PARTIAL DIFFERENTIAL EQUATIONS; PERIODICITY; Probability distribution; Simulation; Studies; Systems Theory; Theoretical; VARIATIONAL METHODS; Finite Difference Schemes; 49M20; 49K20; 65M06; Augmented Lagrangian; 65K10; Mean Field Type Control; Congestion
• ISSN: 0095-4616; 1432-0606
• DOI: 10.1007/s00245-016-9391-z

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.