Quasi-Monte Carlo integration in uncertainty quantification of elliptic PDEs with log-Gaussian coefficients

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Description:	Herrmann, L Tuesday 18th June 2019 - 15:40 to 16:30


Created:	2019-06-19 09:02
Collection:	Approximation, sampling, and compression in high dimensional problems
Publisher:	Isaac Newton Institute
Copyright:	Herrmann, L
Language:	eng (English)


Abstract:	Quasi-Monte Carlo (QMC) rules are suitable to overcome the curse of dimension in the numerical integration of high-dimensional integrands. Also the convergence rate of essentially first order is superior to Monte Carlo sampling. We study a class of integrands that arise as solutions of elliptic PDEs with log-Gaussian coefficients. In particular, we focus on the overall computational cost of the algorithm. We prove that certain multilevel QMC rules have a consistent accuracy and computational cost that is essentially of optimal order in terms of the degrees of freedom of the spatial Finite Element discretization for a range of infinite-dimensional priors. This is joint work with Christoph Schwab. References: [L. Herrmann, Ch. Schwab: QMC integration for lognormal-parametric, elliptic PDEs: local supports and product weights, Numer. Math. 141(1) pp. 63--102, 2019], [L. Herrmann, Ch. Schwab: Multilevel quasi-Monte Carlo integration with product weights for elliptic PDEs with lognormal coefficients, to appear in ESAIM:M2AN], [L. Herrmann: Strong convergence analysis of iterative solvers for random operator equations, SAM report, 2017-35, in review]

Abstract:

Quasi-Monte Carlo (QMC) rules are suitable to overcome the curse of dimension in the numerical integration of high-dimensional integrands. Also the convergence rate of essentially first order is superior to Monte Carlo sampling. We study a class of integrands that arise as solutions of elliptic PDEs with log-Gaussian coefficients. In particular, we focus on the overall computational cost of the algorithm. We prove that certain multilevel QMC rules have a consistent accuracy and computational cost that is essentially of optimal order in terms of the degrees of freedom of the spatial Finite Element discretization for a range of infinite-dimensional priors. This is joint work with Christoph Schwab. References: [L. Herrmann, Ch. Schwab: QMC integration for lognormal-parametric, elliptic PDEs: local supports and product weights, Numer. Math. 141(1) pp. 63--102, 2019], [L. Herrmann, Ch. Schwab: Multilevel quasi-Monte Carlo integration with product weights for elliptic PDEs with lognormal coefficients, to appear in ESAIM:M2AN], [L. Herrmann: Strong convergence analysis of iterative solvers for random operator equations, SAM report, 2017-35, in review]

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