Multigrid preconditioning of linear systems for interior point methods applied to a class of box-constrained optimal control problems

In this article we construct and analyze multigrid preconditioners for discretizations of operators of the form D+K* K, where D is the multiplication with a relatively smooth positive function and K is a compact linear operator. These systems arise when applying interior point methods to the minimization problem min_u (||K u-f||^2 +b||u||^2) with box-constraints on the controls u. The presented preconditioning technique is closely related to the one developed by Draganescu and Dupont in [11] for the associated unconstrained problem, and is intended for large-scale problems. As in [11], the quality of the resulting preconditioners is shown to increase with increasing resolution but decreases as the diagonal of D becomes less smooth. We test this algorithm first on a Tikhonov-regularized backward parabolic equation with box-constraints on the control, and then on a standard elliptic-constrained optimization problem. In both cases it is shown that the number of linear iterations per optimization step, as well as the total number of fine-scale matrix-vector multiplications is decreasing with increasing resolution, thus showing the method to be potentially very efficient for truly large-scale problems.

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