Discontinuous information in the worst case and randomized settings

We believe that discontinuous linear information is never more powerful than continuous linear information for approximating continuous operators. We prove such a result in the worst case setting. In the randomized setting we consider compact linear operators defined between Hilbert spaces. In this case, the use of discontinuous linear information in the randomized setting cannot be much more powerful than continuous linear information in the worst case setting. These results can be applied when function evaluations are used even if function values are defined only almost everywhere

Linear average and stochastic n-widths of Besov embeddings on Lipschitz domains

AbstractIn this paper, we determine the asymptotic degree of the linear average and stochastic n-widths of the compact embeddings Bq0s+t(Lp0(Ω))↪Bq1s(Lp1(Ω)),t>max{d(1/p0−1/p1),0},1≤p0,p1,q0,q1≤∞, where Bq0s+t(Lp0(Ω)) is a Besov space defined on the bounded Lipschitz domain Ω⊂Rd

Optimal Algorithms for Numerical Integration: Recent Results and Open Problems

We present recent results on optimal algorithms for numerical integration and several open problems. The paper has six parts: 1. Introduction 2. Lower Bounds 3. Universality 4. General Domains 5. iid Information 6. Concluding RemarksComment: Survey written for the MCQMC conference in Linz, 26 pages. arXiv admin note: text overlap with arXiv:2108.0205

On the power of iid information for linear approximation

This survey is concerned with the power of random information for approximation in the (deterministic) worst-case setting, with special emphasis on information that is obtained independently and identically distributed (iid) from a given distribution on a class of admissible information. We present a general result based on a weighted least squares method and derive consequences for special cases. Improvements are available if the information is "Gaussian" or if we consider iid function values for Sobolev spaces. We include open questions to guide future research on the power of random information in the context of information-based complexity.Comment: 61 page