Eigenvalues and s-numbers
Information-based complexity
On the degree of nonlinear spline approximation in Besov-Sobolev spaces
Journal of Approximation Theory
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
The real number model in numerical analysis
Journal of Complexity
A posteriori error estimation and adaptive mesh-refinement techniques
ICCAM'92 Proceedings of the fifth international conference on Computational and applied mathematics
A convergent adaptive algorithm for Poisson's equation
SIAM Journal on Numerical Analysis
A posteriori error estimates for elliptic problems in two and three space dimensions
SIAM Journal on Numerical Analysis
Journal of Complexity
Stable multiscale bases and local error estimation for elliptic problems
Applied Numerical Mathematics - Special issue on multilevel methods
On the cost of uniform and nonuniform algorithms
Theoretical Computer Science - Special issue on computability and complexity in analysis
Composite wavelet bases for operator equations
Mathematics of Computation
Continuous algorithms in n-term approximation and non-linear widths
Journal of Approximation Theory
Complexity of linear probelms with a fixed output basis
Journal of Complexity
Adaptive wavelet methods for elliptic operator equations: convergence rates
Mathematics of Computation
Adaptive Wavelet Methods for Saddle Point Problems---Optimal Convergence Rates
SIAM Journal on Numerical Analysis
Adaptive Wavelet Schemes for Nonlinear Variational Problems
SIAM Journal on Numerical Analysis
The randomized information complexity of elliptic PDE
Journal of Complexity
Optimal approximation of elliptic problems by linear and nonlinear mappings I
Journal of Complexity - Special issue: Algorithms and complexity for continuous problems Schloss Dagstuhl, Germany, September 2004
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
Randomized approximation of Sobolev embeddings, III
Journal of Complexity
Linear average and stochastic n-widths of Besov embeddings on Lipschitz domains
Journal of Approximation Theory
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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 Bqr(Lp(Ω)) and Ω ⊂ 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 since the optimal Galerkin space usually depends on the operator and on the shape of the domain Ω. For p f is again optimal. Our main theoretical tool is the best n-term approximation with respect to an optimal Riesz basis and related nonlinear widths. These general results are used to study the Poisson equation in a polygonal domain. It turns out that best n-term wavelet approximation is (almost) optimal. The main results of this paper are about approximation, not about computation. However, we also discuss consequences of the results for the numerical complexity of operator equations.