Department of Mathematics

Numerical Analysis of Stochastic PDEs

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Dozenten Prof. Christoph Schwab, Dr. Andrea Barth, Dr. Markus Hansen, Dr. Annika Lang
Ort T.b.a.
Zeit T.b.a.
Beginnt am 17.09.2012
Vorbesprechung Mo 17.09.2012, 14:00, HG G19.1
Kontakt Prof. Christoph Schwab

Voraussetzungen Required:
Completed ETH BSc in MATH, RW/CSE.
Completed course in Probability Theory or in Numerical Solution of SPDEs and in Numerical Solution of elliptic & parabolic PDEs or hyperbolic PDEs.

Courses in Functional Analysis and/or Parallel Computing and/or
Computational Methods for Quantitative Finance and/or Numerical Solution of SODEs.

The number of participants is limited to 8. Participation in the first meeting on the 17.09.2012 is mandatory. Registration for the seminar in myStudies will be opened after the first meeting.
Beschreibung In recent years, the mathematical formulation and the development of efficient simulation methods for partial differential equations (PDEs) with random inputs and with noisy data has become increasingly important in engineering and the sciences. We think, among others, of parabolic SPDEs driven by Wiener and Levy noise in term structure models in finance, wave propagation in random media in the geosciences, porous media flow in media with uncertain permeability in subsurface flow models. In the life sciences, PDEs arise on high or even infinite dimensional parameter spaces, such as the master equation or mass action models with hundreds of species in bioengineering.

In the seminar, we discuss the mathematical formulation, regularity, adaptive approximation and numerical analysis of Partial Differential Equations (PDEs) with random input data and on high dimensional parameter spaces.

Mathematical Topics to be discussed include:

multilevel Monte Carlo methods,
multilevel quasi Monte Carlo methods,
polynomial chaos type representation of random fields,
adaptive solvers for PDEs,
Smolyak tensor interpolation algorithms,
Bayesian inverse problems for PDEs,
massively parallel uncertainty quantification algorithms,
SODEs in infinite dimensions.

All topics benefit current research projects in SAM, in particular to the European Research Council project `` Sparse Tensor Approximation of High Dimensional PDE '' and can lead to MSc resp. PhD thesis work in MATH and in
in RW/CSE.

The students will read selected recent research papers on the seminar's mathematical topics, prepare a presentation (and a written summary of their presentation).

Literatur Ch. Schwab and C.J. Gittelson: Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs. Acta Numerica 20 (2011), Cambridge University Press.

G. DaPrato and J. Zabczyk: Stochastic Equations in Infinite Dimensions. Cambridge University Press (1992).

and a (student) selection from recent research papers, e.g.,

Analytic regularity and polynomial approximation of parametric and stochastic elliptic PDEs, A. Cohen, R. DeVore and C. Schwab, SAM-Report 2010-03

Representation of Gaussian fields in series with independent coefficients ,C.J. Gittelson, SAM-Report 2010-15

Multi-Level Monte Carlo Finite Element method for elliptic PDE's with stochastic coefficients, A. Barth, C. Schwab and N. Zollinger, SAM-Report 2010-18

Sparse tensor multi-level Monte Carlo finite volume methods for hyperbolic conservation laws with random intitial data, S. Mishra and Ch. Schwab, SAM-Report 2010-24

Multi-level Monte Carlo finite volume methods for nonlinear systems of conservation laws in multi-dimensions, S. Mishra, Ch. Schwab and J. Šukys, SAM-Report 2011-02

Analytic regularity and polynomial approximation of stochastic, parametric elliptic multiscale PDEs, V.H. Hoang and Ch. Schwab, SAM-Report 2011-07

Stochastic Galerkin approximation of operator equations with infinite dimensional noise, C.J.Gittelson, SAM-Report 2011-10

An adaptive stochastic Galerkin method, C.J.Gittelson, SAM-Report 2011-11

Sparse deterministic approximation of Bayesian inverse problems, Ch. Schwab and A.M. Stuart, SAM-Report 2011-16

Analytic regularity and nonlinear approximation of a class of parametric semilinear elliptic PDEs,M. Hansen and Ch. Schwab, SAM-Report 2011-29

Multievel Monte Carlo Finite Element method for parabolic stochastic partial differential equations, A. Barth, A. Lang and Ch. Schwab,SAM-Report 2011-30

The multi-level Monte Carlo Finite Element Method for a stochastic Brinkman problem, C.J. Gittelson, J. Könnö, Ch. Schwab and R. Stenberg, SAM-Report 2011-31

Static load balancing for multi-level Monte Carlo finite volume solvers, J. Sukys, S. Mishra and Ch. Schwab, SAM-Report 2011-32

Adaptive wavelet methods for elliptic partial differential equations with random operators,C.J. Gittelson, SAM-Report 2011-37

Sparse adaptive Taylor approximation algorithms for parametric and stochastic elliptic PDEs, A. Chkifa, A. Cohen, R. DeVore and Ch. Schwab, SAM-Report 2011-44

First order k-th moment finite element analysis of nonlinear operator equations with stochastic data, A. Chernov and Ch. Schwab, SAM-Report 2011-51

Quasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficients, F.Y. Kuo, Ch. Schwab and I.H. Sloan, SAM-Report 2011-52

Sparse adaptive approximation of high dimensional parametric initial value problems, M. Hansen and Ch. Schwab, SAM-Report 2011-64

Multilevel Monte Carlo method with applications to stochastic partial differential equations, A. Barth and A. Lang, SAM-Report 2011-68

Adaptive Galerkin approximation algorithms for partial differential equations in infinite dimensions, Ch. Schwab and E. Süli, SAM-Report 2011-69

Sparse MCMC gpc Finite Element Methods for Bayesian Inverse Problems, V.H. Hoang, Ch. Schwab and A.M. Stuart, SAM-Report 2012-23


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