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Adaptive wavelet methods for elliptic partial differential equations with random operators

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Abstract

We apply adaptive wavelet methods to boundary value problems with random coefficients, discretized by wavelets in the spatial domain and tensorized polynomials in the parameter domain. Greedy algorithms control the approximate application of the fully discretized random operator, and the construction of sparse approximations to this operator. We suggest a power iteration for estimating errors induced by sparse approximations of linear operators.

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Notes

  1. As above, \(\log ^+x :=\log (\max (x,1))\).

  2. Proposition 4.3 initially only implies that the first term in (9.1) is bounded by the third. However, if (9.1) does not hold, we can replace \(\bar{e}_{0,j_0(r)}\) by \(\tilde{d}_{0,s} r^{-s}\) in (8.1).

  3. This heuristic is actually used to distribute tolerances for a subproblem in [24, 27]; it is not clear whether the resulting error in the approximation of \(u\) is distributed evenly among all active coefficients.

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Acknowledgments

This research was supported in part by the Swiss National Science Foundation grant No. 200021-120290/1. The author expresses his gratitude to Ch. Schwab for his many insightful remarks and suggestions.

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  1. Claude Jeffrey Gittelson

    Present address: Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA

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  1. Seminar for Applied Mathematics, ETH Zurich, Rämistrasse 101, 8092 , Zurich, Switzerland

    Claude Jeffrey Gittelson

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Gittelson, C.J. Adaptive wavelet methods for elliptic partial differential equations with random operators. Numer. Math. 126, 471–513 (2014). https://doi.org/10.1007/s00211-013-0572-2

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