BEGIN:VCALENDAR
VERSION:2.0
PRODID:-//Talks.cam//talks.cam.ac.uk//
X-WR-CALNAME:Talks.cam
BEGIN:VEVENT
SUMMARY:Domain Uncertainty Quantification - Christoph Schwab (ETH Zürich)
DTSTART:20180205T160000Z
DTEND:20180205T170000Z
UID:TALK99871@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION: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 t
 ensor discretization produces approximate k-point correlations at near opt
 imal order: work and memory scale log-linearly w.r. to N\, the number of d
 egrees of freedom for approximating one instance of the nominal (mean-fiel
 d) problem [1\,3]. For large domain variations\, the notion of shape holom
 orphy of the solution is introduced. It implies (the `usual&#39\;) sparsit
 y and dimension-independent convergence rates of gpc approximations (e.g.\
 , anisotropic stochastic collocation\, least squares\, CS\, ...) of parame
 tric domain-to-solution maps in forward UQ. This property holds for a broa
 d class of smooth elliptic and parabolic boundary value problems. Shape ho
 lomorphy 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 conve
 rgence rates of Smolyak and higher order Quasi-Monte Carlo integration [5\
 ,6\,7]. Examples include the usual (anisotropic) diffusion problems\, Navi
 er-Stokes [2] and time harmonic Maxwell PDEs [4]\, and forward UQ for frac
 tional 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. Che
 rnov and Ch. Schwab: First order k-th moment finite element analysis of no
 nlinear operator equations with stochastic data\, Mathematics of Computati
 on\, 82 (2013)\, pp. 1859-1888. [2] A. Cohen and Ch. Schwab and J. Zech: S
 hape Holomorphy of the stationary Navier-Stokes Equations\, accepted (2018
 )\, SIAM J. Math. Analysis\, SAM Report 2016-45. [3] H. Harbrecht and R. S
 chneider and Ch. Schwab: Sparse Second Moment Analysis for Elliptic Proble
 ms 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\, param
 etric operator equations. SIAM Journ. Uncertainty Quantification\, 4/1 (20
 16)\, pp. 48-79. [6] J. Zech and Ch. Schwab: Convergence rates of high dim
 ensional Smolyak quadrature. In review\, SAM Report 2017-27. [7] J. Dick a
 nd 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-ba
 sis 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.
LOCATION:Seminar Room 1\, Newton Institute
END:VEVENT
END:VCALENDAR