Multigrid method for optimal control problem

Abstract

In this talk, we will present Multigrid Methods applied to optimal control problems arising from parabolic partial differential equations (PDEs). For a given state y governed by a heat equation, we wish to drive the solution of this PDE to a desired state $\hat y$ through a control $u$. The goal is to find the optimal control $u^*$ which minimizes the discrepancy between these states (i.e. original state $y$ and desired state $\hat y$). We first use the Lagrange multiplier approach to derive a forward-backward system. By eliminating the adjoint state $p$ and the control $u$, we find a simplified PDE on the state y which is second order in time and fourth order in space. This simplified problem can then be solved by multigrid methods. Traditional multigrid methods does not work well for dealing with the parabolic problem, and a space-time method is needed to improve this situation. To better understand the smoothing behaviour, we provide a convergence analysis for the damped Jacobi method and block Jacobi method. We conclude with some numerical results.

卢 柳 䃅

Postdoctoral Fellow in Applied Mathematics

My research interests include Numerical Analysis, Mathematical Biology, Scientific Computing, Optimization and Control.