Numerical Methods and Brute Force Optimisation for a General Formulation of the Mean Field Game Equations
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This thesis considers the numerical solution of a general formulation of the mean field game (MFG) equations. MFG are a relatively new field with few general results but with many modelling applications. The MFG equations consist of a Hamilton-Jacobi-Bellman equation (HJB) and a Fokker-Planck equation (FP) which are coupled by an optimal control. In established theory and existing numerical methods for MFG, this optimal control is assumed known as a function of the Hamiltonian in the HJB equation. This reduces the generality of the methods. In this thesis, we instead consider the general formulation for which the optimal control is unknown. We develop brute force optimisation methods to directly compute this based on the discretised Hamiltonian. We also develop robust numerical schemes for the HJB and FP equations that, together with the brute force methods, allow the computation of the solutions of the general formulation of the MFG equations. Of particular note here are methods to evaluate a diffusion tensor in the FP equation. Due to the coupled nature of the MFG equations, the numerical computation of their solutions require a solution procedure. In this procedure the HJB and optimal control are solved at the same time, before the solution of these are used to compute the solution of the FP equation. This solution is again used to solve for the HJB equation and optimal control. It is expected that computing several iterations of this solution procedure is necessary. However, we encountered several cases in which this procedure failed to converge. It is important to remark that this is not a phenomenon unique to our thesis, but is reported for other numerical methods for less general forms of the MFG equations by other authors. After a wide range of numerical tests, we develop an intuition for the causes of this lack of convergence. In addition, we introduce alternative solution procedures with slightly better convergence properties. We are also able to produce solutions that converge with a refined mesh for some cases of the MFG equations for which there to our knowledge does not exist existence theorems for. These results remain speculative. Based on our experiences, we propose topics of future work to deal with the numerical solution of the MFG equations. We also present some ideas for improvements to our solution methods.