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Deterministic Multilevel Methods for Forward and Inverse UQ in PDEs

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UNQW04 - UQ for inverse problems in complex systems

We present the numerical analysis of Quasi Monte-Carlo methods for high-dimensional integration applied to forward and inverse uncertainty quantification for elliptic and parabolic PDEs. Emphasis will be placed on the role of parametric holomorphy of data-to-solution maps. We present corresponding results on deterministic quadratures in Bayesian Inversion of parametric PDEs, and the related bound on posterior sparsity and (dimension-independent) QMC convergence rates. Particular attention will be placed on Higher-Order QMC , and on the interplay between the structure of the representation system of the distributed uncertain input data (KL, splines, wavelets,...) and the structure of QMC weights. We also review stable and efficient generation of interlaced polynomial lattice rules, and the numerical analysis of multilevel QMC Finite Element PDE discretizations with applications to forward and inverse computational uncertainty quantification. QMC convergence rates will be compared with those afforded by Smolyak quadrature. Joint work with Robert Gantner and Lukas Herrmann and Jakob Zech (SAM, ETH ) and Josef Dick, Thong LeGia and Frances Kuo (Sydney). References: [1] R. N. Gantner and L. Herrmann and Ch. Schwab Quasi-Monte Carlo integration for affine-parametric, elliptic PDEs: local supports and product weights, SIAM J . Numer. Analysis, 56/1 (2018), pp. 111-135. [2] J. Dick and R. N. Gantner and Q. T. Le Gia and Ch. Schwab Multilevel higher-order quasi-Monte Carlo Bayesian estimation, Math. Mod. Meth. Appl. Sci., 27/5 (2017), pp. 953-995. [3] R. N. Gantner and M. D. Peters Higher Order Quasi-Monte Carlo for Bayesian Shape Inversion, accepted (2018) SINUM , SAM Report 2016-42. [4] J. Dick and Q. T. Le Gia and Ch. Schwab Higher order Quasi Monte Carlo integration for holomorphic, parametric operator equations, SIAM Journ. Uncertainty Quantification, 4/1 (2016), pp. 48-79 [5] J. Dick and F.Y. Kuo and Q.T. LeGia and Ch. Schwab Multi-level higher order QMC Galerkin discretization for affine parametric operator equations, SIAM J . Numer. Anal., 54/4 (2016), pp. 2541-2568

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