Fourier Galerkin approximation of a mean field control problem
The goal of this talk is to introduce a finite dimensional approximation of the solution to a mean field optimal control problem set on the d-dimensional torus, without relying on particle-based methods. Our approximation is obtained by means of a Fourier-Galerkin method, the main principle of which is to truncate the Fourier expansion of probability measures on the torus. However, this operation has the main feature not to leave the space of probability measures invariant, which drawback is know as Gibbs' phenomenon. First, we manage to prove that, for initial conditions in the "interior" of the space of probability measures and for sufficiently large levels of truncation, the Fourier-Galerkin method actually induces a new finite dimensional control problem whose trajectories take values in the space of probability measures with a finite number of Fourier coefficients. Subsequently, our main result asserts that, whenever the cost functionals are smooth and convex, the optimal control, trajectory, and value function from the approximating problem converge to their counterparts in the original mean field control problem. Noticeably, we show that our method yields a polynomial convergence rate directly proportional to the data's regularity. This convergence rate is faster than the one achieved by the usual particles approach, offering a more efficient alternative. Furthermore, our technique also provide an explicit method for constructing an approximate optimal control along with its corresponding trajectory. This talk is based on a joint work with François Delarue.
Area: IS7 - Stochastic optimal control of McKean-Vlasov equations (Elena Bandini)
Keywords: Mean Field Control; Convergence; Fourier analysis
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