A multilevel correction method for optimal controls of elliptic equation
This method addresses the computational bottleneck of solving large-scale PDE-constrained optimization problems for researchers in scientific computing and engineering.
The paper proposes a multilevel correction method for solving optimal control problems constrained by elliptic equations, transforming the large-scale optimization problem into solving linear boundary value problems on multilevel meshes and optimization problems on a coarse space, thereby improving computational efficiency.
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.