Presented by:
Christoph Schwab ETH Zürich
Date:
Monday 5th February 2018 - 16:00 to 17:00
Venue:
INI Seminar Room 1
Abstract:
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.