Optimal approximation of elliptic problems by linear and nonlinear mappings I

We study the optimal approximation of the solution of an operator equation A(u) = f by linear mappings of rank n and compare this with the best n-term approximation with respect to an optimal Riesz basis. We consider worst case errors, where f is an element of the unit ball of a Hilbert space. We apply our results to boundary value problems for elliptic PDEs that are given by an isomorphism A : Hs 0(Ω) → H−s(Ω), where s > 0 and Ω is an arbitrary bounded Lipschitz domain in Rd. We prove that approximation by linear mappings is as good as the best n-term approximation with respect to an optimal Riesz basis. We discuss why nonlinear approximation still is important for the approximation of elliptic problems. AMS subject classification: 41A25, 41A46, 41A65, 42C40, 65C99