Domain Uncertainty Quantification

We address the numerical analysis of domain uncertainty in UQ for partial differential and integral equations. For small amplitude shape variation, a first order, kth moment perturbation analysis and sparse tensor discretization produces approximate k-point correlations at near optimal order: work and memory scale log-linearly w.r. to N, the number of degrees of freedom for approximating one instance of the nominal (mean-field) problem [1,3]. For large domain variations, the notion of shape holomorphy of the solution is introduced. It implies (the `usual') sparsity and dimension-independent convergence rates of gpc approximations (e.g., anisotropic stochastic collocation, least squares, CS, ...) of parametric domain-to-solution maps in forward UQ. This property holds for a broad class of smooth elliptic and parabolic boundary value problems. Shape holomorphy also implies sparsity of gpc expansions of certain posteriors in Bayesian inverse UQ [7], [->WS4]. We discuss consequences of gpc sparsity on some surrogate forward models, to be used e.g. in optimization under domain uncertainty [8,9]. We also report on dimension independent convergence rates of Smolyak and higher order Quasi-Monte Carlo integration [5,6,7]. Examples include the usual (anisotropic) diffusion problems, Navier-Stokes [2] and time harmonic Maxwell PDEs [4], and forward UQ for fractional PDEs. Joint work with Jakob Zech (ETH), Albert Cohen (Univ. P. et M. Curie), Carlos Jerez-Hanckes (PUC, Santiago, Chile). Work supported in part by the Swiss National Science Foundation. References: [1] A. Chernov and Ch. Schwab: First order k-th moment finite element analysis of nonlinear operator equations with stochastic data, Mathematics of Computation, 82 (2013), pp. 1859-1888. [2] A. Cohen and Ch. Schwab and J. Zech: Shape Holomorphy of the stationary Navier-Stokes Equations, accepted (2018), SIAM J. Math. Analysis, SAM Report 2016-45. [3] H. Harbrecht and R. Schneider and Ch. Schwab: Sparse Second Moment Analysis for Elliptic Problems in Stochastic Domains, Numerische Mathematik, 109/3 (2008), pp. 385-414. [4] C. Jerez-Hanckes and Ch. Schwab and J. Zech: Electromagnetic Wave Scattering by Random Surfaces: Shape Holomorphy, Math. Mod. Meth. Appl. Sci., 27/12 (2017), pp. 2229-2259. [5] 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. [6] J. Zech and Ch. Schwab: Convergence rates of high dimensional Smolyak quadrature. In review, SAM Report 2017-27. [7] 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. [8] P. Chen and Ch. Schwab: Sparse-grid, reduced-basis Bayesian inversion: Nonaffine-parametric nonlinear equations. Journal of Computational Physics, 316 (2016), pp. 470-503. [9] Ch. Schwab and J. Zech: Deep Learning in High Dimension. In review, SAM Report 2017-57.