27,202 research outputs found

    Discontinuous information in the worst case and randomized settings

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    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

    Tractability of multivariate analytic problems

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    In the theory of tractability of multivariate problems one usually studies problems with finite smoothness. Then we want to know which ss-variate problems can be approximated to within ε\varepsilon by using, say, polynomially many in ss and ε1\varepsilon^{-1} function values or arbitrary linear functionals. There is a recent stream of work for multivariate analytic problems for which we want to answer the usual tractability questions with ε1\varepsilon^{-1} replaced by 1+logε11+\log \varepsilon^{-1}. In this vein of research, multivariate integration and approximation have been studied over Korobov spaces with exponentially fast decaying Fourier coefficients. This is work of J. Dick, G. Larcher, and the authors. There is a natural need to analyze more general analytic problems defined over more general spaces and obtain tractability results in terms of ss and 1+logε11+\log \varepsilon^{-1}. The goal of this paper is to survey the existing results, present some new results, and propose further questions for the study of tractability of multivariate analytic questions

    Some Results on the Complexity of Numerical Integration

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    This is a survey (21 pages, 124 references) written for the MCQMC 2014 conference in Leuven, April 2014. We start with the seminal paper of Bakhvalov (1959) and end with new results on the curse of dimension and on the complexity of oscillatory integrals. Some small errors of earlier versions are corrected
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