In this work we present a new multigrid preconditioner for the linear systems arising in the semismooth Newton method solution process 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 by Draganescu [Optim. Methods Softw., 29 (2004), pp. 786-818] for the aforementioned sub matrices 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.
Andrei Draganescu received his Ph.D. in applied mathematics from the University of Chicago in 2004. After a completing two-year postdoctoral appointment at the Sandia National Labs in Albuquerque, New Mexico he joined the Department of Mathematics and Statistics at UMBC in 2006. His primary current research is focused on developing efficient multigrid preconditioners for optimal control problems constrained by partial differential equations.
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