2024 FFT

Efficient multigrid methods for the numerical solution of optimization problems constrained by partial differential equations

Andrei Draganescu (UMBC)

Abstract: In this talk, we highlight our recent results on designing and analyzing multigrid preconditioners for the numerical solution of large-scale optimization problems constrained by partial differential equations (PDECO). Solving these large-scale PDECO problems involves iterative methods that repeatedly solve the partial differential equation (PDE) representing the constraints, as well as the linearized adjoint equation, which is a PDE of similar character to the constraint. As a result, solving PDECO problems can be very costly, necessitating preconditioning techniques to accelerate convergence.

Multigrid methods were originally introduced as scalable solvers for partial differential equations (PDEs), primarily for elliptic equations. The core idea is that a well-coordinated combination of multiple resolutions of the PDE can significantly speed up the solution process. However, the multigrid approach for PDE-constrained optimization (PDECO) differs from the original method. This difference arises because, in PDECO, we need to invert integral operators instead of differential operators. The multigrid strategy we present was originally developed for PDECO of elliptic equations, where it was shown to be optimal with respect to the order of approximation of the numerical method. In this talk, we will extend this method to PDECO of parabolic equations, discussing associated results and challenges. Specifically, we will describe an optimal-order multigrid preconditioner that incorporates coarsening in both space and time.