Sparse Tensor Product Discretization of Stochastic PDE

Description

Parameters in boundary value problems are often incompletely known. Modeling them as random variables or random fields, the solution to a boundary value problem also becomes a random field. In general, it can be interpreted as a parametric element of a function space, with an infinite dimensional parameter domain. Stochastic moments of the solution can be computed directly as functions on higher dimensional domains.

Various sparse tensorization techniques exist to deal with the high dimensionality either of the parameter domain or in the moment equations. We consider multilevel Monte Carlo methods, sparse tensor collocation, and Galerkin methods using generalized polynomial chaos bases. In particular, we develop adaptive techniques for selecting active polynomial chaos modes.

Funding

• SNF Grant No. 200021-120290/1

References

• C.J. Gittelson, Representation of Gaussian fields in series with independent coefficients, SAM Report 2010-15
  http://www.sam.math.ethz.ch/reports/2010/15
• C.J. Gittelson, Stochastic Galerkin discretization of the log-normal isotropic diffusion problem, M3AS 2010
  http://www.worldscinet.com/m3as/20/2002/S0218202510004210.html
• B. Pentenrieder and Ch. Schwab, hp-FEM for second moments of elliptic PDEs with stochastic data. Part 2: Exponential convergence, SAM Report 2010-9
  http://www.sam.math.ethz.ch/reports/2010/09
• B. Pentenrieder and Ch. Schwab, hp-FEM for second moments of elliptic PDEs with stochastic data. Part 1: Analytic regularity, SAM Report 2010-8
  http://www.sam.math.ethz.ch/reports/2010/08
• B. Pentenrieder, Exponential convergence of hp-FEM for second moments of elliptic PDEs with stochastic data, PhD Thesis 2009
  http://e-collection.ethbib.ethz.ch/view/eth:574
• M. Bieri, Sparse tensor discretizations of elliptic PDEs with random input data, PhD Thesis 2009
  http://e-collection.ethbib.ethz.ch/view/eth:339
• M. Bieri, A sparse composite collocation finite element method for elliptic sPDEs, SAM Report 2009-8
  http://www.sam.math.ethz.ch/reports/2009/08
• M. Bieri, R. Andreev, Ch. Schwab, Sparse Tensor Discretization of Elliptic sPDEs, SIAM J. Sci. Comput. 2009
  http://scitation.aip.org/
• M. Bieri, Ch. Schwab, Sparse high order FEM for elliptic sPDEs, CMAME 2008
  http://www.sciencedirect.com/
• R.A. Todor, Ch. Schwab, Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients, IMAJNA 2007
  http://imajna.oxfordjournals.org/cgi/content/abstract/27/2/232
• Ch. Schwab, R.A. Todor, Karhunen–Loève approximation of random fields by generalized fast multipole methods, JCP 2006
  http://www.sciencedirect.com/

Contacts

Prof. Christoph Schwab
Claude Gittelson
Dr. Andrea Barth