Eigenvalues and s-numbers
Finite element methods are not always optimal
Advances in Applied Mathematics
Information-based complexity
s-numbers in information-based complexity
Journal of Complexity
The computational complexity of differential and integral equations: an information-based approach
The computational complexity of differential and integral equations: an information-based approach
n-Widths and Singularly Perturbed Boundary Value Problems
SIAM Journal on Numerical Analysis
Complexity of linear probelms with a fixed output basis
Journal of Complexity
Approximation characteristics for diagonal operators in different computational settings
Journal of Approximation Theory
Optimal approximation of elliptic problems by linear and nonlinear mappings II
Journal of Complexity
The quantum query complexity of elliptic PDE
Journal of Complexity - Special issue: Information-based complexity workshops FoCM conference Santander, Spain, July 2005
Optimal approximation of elliptic problems by linear and nonlinear mappings III: Frames
Journal of Complexity
Sampling numbers and function spaces
Journal of Complexity
Linear information versus function evaluations for L2-approximation
Journal of Approximation Theory
Randomized approximation of Sobolev embeddings, III
Journal of Complexity
Approximation characteristics for diagonal operators in different computational settings
Journal of Approximation Theory
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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: H0s(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.