224 IGPM224.pdf        December 2002
TITLE Adaptive Wavelet Methods for Linear-Quadratic Elliptic Control Problems: Convergence Rates
AUTHORS Wolfgang Dahmen, Angela Kunoth
ABSTRACT We propose an adaptive algorithm based on wavelets for the fast numerical solution of control problems governed by elliptic boundary value problems with distributed or Neumann boundary control. A quadratic cost functional that may involve fractional Sobolev norms of the state and the control is to be minimized subject to linear constraints in weak form. Placing the problem into the framework of (biorthogonal) wavelets allows to formulate the functional and the constraints equivalently in terms of l2 of wavelet expansion coefficients and constraints in form of an l2 automorphism. The resulting first order necessary conditions are then derived as a (still infinite) system in l2. Applying the machinery developed in [CDD1, CDD2], we propose an adaptive method which can be interpreted as an inexact gradient method, where in each iteration step the primal and the adjoint system needs to be solved up to a prescribed accuracy. In particular, we show that the adaptive algorithm is asymptotically optimal, that is, the convergence rate achieved for computing the solution up to a desired target tolerance is asymptotically the same as the wavelet best N term approximation of the solution, and the total computational work is proportional to the number of computational unknowns.
KEYWORDS Optimal control, elliptic boundary value problem, wavelets, infinite l2-system, preconditioning, adaptive refinements, inexact iterations, convergence, convergence rates, optimal complexity
DOI 10.1137/S0363012902419199
PUBLICATION SIAM journal on control and optimization
43(5), 1640-1675 (2005)