A multilevel correction method for optimal controls of elliptic equation

TL;DR

This paper introduces a multilevel correction method that enhances the efficiency of solving elliptic PDE-constrained optimal control problems by combining multigrid techniques with low-dimensional optimization.

Contribution

The paper presents a novel multilevel correction scheme that reduces computational complexity by transforming large-scale problems into simpler subproblems using multigrid and low-dimensional spaces.

Findings

Significant reduction in computational cost.

Effective in solving large-scale elliptic optimal control problems.

Improved convergence compared to traditional methods.

Abstract

We propose in this paper a multilevel correction method to solve optimal control problems constrained by elliptic equations with the finite element method. In this scheme, solving optimization problem on the finest finite element space is transformed to a series of solutions of linear boundary value problems by the multigrid method on multilevel meshes and a series of solutions of optimization problems on the coarsest finite element space. Our proposed scheme, instead of solving a large scale optimization problem in the finest finite element space, solves only a series of linear boundary value problems and the optimization problems in a very low dimensional finite element space, and thus can improve the overall efficiency for the solution of optimal control problems governed by PDEs.