Optimal-order preconditioners for linear systems arising in the semismooth Newton solution of a class of control-constrained problems

TL;DR

This paper introduces a new multigrid preconditioner with non-conforming coarse spaces for linear systems in semismooth Newton methods solving control-constrained optimal control problems, achieving optimal approximation quality.

Contribution

It develops a novel multigrid preconditioner using non-conforming coarse spaces that outperforms previous methods for certain control-constrained problems.

Findings

Preconditioner approximates inverse of submatrix to optimal order.

Numerical tests show improved convergence on elliptic and image-deblurring problems.

Preconditioner is effective under geometric assumptions on inactive constraints.

Abstract

In this article we present a new multigrid preconditioner for the linear systems arising in the semismooth Newton method solution of certain control-constrained, quadratic distributed optimal control problems. Using a piecewise constant discretization of the control space, each semismooth Newton iteration essentially requires inverting a principal submatrix of the matrix entering the normal equations of the associated unconstrained optimal control problem, the rows (and columns) of the submatrix representing the constraints deemed inactive at the current iteration. Previously developed multigrid preconditioners for the aforementioned submatrices were based on constructing a sequence of conforming coarser spaces, and proved to be of suboptimal quality for the class of problems considered. Instead, the multigrid preconditioner introduced in this work uses non-conforming coarse spaces, and it is shown that, under reasonable geometric assumptions on the constraints that are deemed inactive, the preconditioner approximates the inverse of the desired submatrix to optimal order. The preconditioner is tested numerically on a classical elliptic-constrained optimal control problem and further on a constrained image-deblurring problem.