Space-Time Adaptive Wavelet Methods for Optimal Control Problems Constrained by Parabolic Evolution Equations

Abstract

An adaptive algorithm based on wavelets is proposed for the efficient numerical solution of a control problem governed by a linear parabolic evolution equation. First, the constraints are represented by means of a full weak space-time formulation as a linear system in $\ell_2$ in wavelet coordinates, following a recent approach by Schwab and Stevenson. Second, a quadratic cost functional involving a tracking-type term for the state and a regularization term for the distributed control is also formulated in terms of $\ell_2$ sequence norms of wavelet coordinates. This functional serves as a representer for a functional involving different Sobolev norms with possibly nonintegral smoothness parameter. Standard techniques from optimization are then used to derive the first order necessary conditions as a coupled system in $\ell_2$-coordinates. For this purpose, an adaptive method is proposed, which can be interpreted as an inexact gradient method for the control. In each iteration step, the primal and adjoint systems are solved up to a prescribed accuracy by the adaptive algorithm. It is shown that the adaptive algorithm converges. Moreover, the algorithm is proved to be asymptotically optimal: the convergence rate achieved for computing each of the components of the solution (state, adjoint state, and control) up to a desired target tolerance is asymptotically the same as the wavelet best N-term approximation of each solution component, and the total computational work is proportional to the number of computational unknowns.