Fast Iterative Solution of Elliptic Control Problems in Wavelet Discretization

We investigate wavelet methods for the efficient numerical solution of a class of control problems constrained by a linear elliptic boundary value problem where the cost functional may contain fractional Sobolev norms of the control and the state. Starting point is the formulation of the infinite-dimensional control problem in terms of (boundary-adapted biorthogonal spline-)wavelets, involving only `2 norms of wavelet expansion coefficients (where different norms are realized by a diagonal scaling together with a Riesz map) and constraints in form of an `2 isomorphism. The coupled system of equations resulting from optimization is solved by an inexact conjugate gradient (CG) method for the control, which involves the approximate inversion of the primal and the adjoint operator using again CG iterations. Starting from a coarse discretization level, we use nested iteration to solve the coupled system on successively finer uniform discretizations up to discretization error accuracy on each level. The resulting inexact CG scheme is a ‘fast solver’: it is of asymptotic optimal complexity in the sense that the overall computational effort to compute the solution up to discretization error on the finest grid is proportional to the number of unknowns on that grid, a consequence of grid–independent condition numbers of the linear operators in wavelet coordinates. In the numerical examples we study the choice of different norms and the regularization parameter in the cost functional and their effect on the solution. Moreover, for different situations the performance of the fully iterative inexact CG scheme is investigated, confirming the theoretical results.