Optimal Approximation of Elliptic Problems by Linear and Nonlinear Mappings

We study the optimal approximation of the solution of an operator equation A(u) = f by four types of mappings: a) linear mappings of rank n; b) n-term approximation with respect to a Riesz basis; c) approximation based on linear information about the right hand side f; d) continuous mappings. We consider worst case errors, where f is an element of the unit ball of a Sobolev or Besov space Br q (Lp()) and Rd is a bounded Lipschitz domain; the error is always measured in the Hs-norm. The respective widths are the linear widths (or approximation numbers), the nonlinear widths, the Gelfand widths, and the manifold widths. As a technical tool, we also study the Bernstein numbers. Our main results are the following. If p 2 then the order of convergence is the same for all four classes of approximations. In particular, the best linear approximations are of the same order as the best nonlinear ones. The best linear approximation can be quite difficult to realize as a numerical algorithm si...

• Type: Conference
• Year: 2004
• Where: DAGSTUHL
• Authors: Erich Novak, Stephan Dahlke, Winfried Sickel
• Keywords: Approximation, DAGSTUHL 2004, DAGSTUHL 2007, Linear, Nonlinear Widths