Information-based 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
Lower bounds for the complexity of Monte Carlo function approximation
Journal of Complexity
The complexity of definite elliptic problems with noisy data
Journal of Complexity - Special issue for the Foundations of Computational Mathematics conference, Rio de Janeiro, Brazil, Jan. 1997
A new algorithm and worst case complexity for Feynman-Kac path integration
Journal of Computational Physics
Quantum integration in Sobolev classes
Journal of Complexity
Worst case complexity of multivariate Feynman-Kac path integration
Journal of Complexity
Quantum approximation I. Embeddings of finite-dimensional Lp spaces
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
Monte Carlo approximation of weakly singular integral operators
Journal of Complexity
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
The randomized complexity of initial value problems
Journal of Complexity
Randomized approximation of Sobolev embeddings, II
Journal of Complexity
Randomized approximation of Sobolev embeddings, III
Journal of Complexity
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We study the information complexity in the randomized setting of solving a general elliptic PDE of order 2m in a smooth, bounded domain Q ⊂ Rd with smooth coefficients and homogeneous boundary conditions. The solution is sought on a smooth submanifold M ⊆ Q of dimension O≤d1≤d, the right-hand side is supposed to be in Cr (Q), the error is measured in the L∞(M) norm. We show that the nth minimal error is (up to logarithmic factors) of order n-min((r+2m)/d1,r/d+1/2).For comparison, in the deterministic setting the nth minimal error is of order n-r/d, for all d1.