Department of Mathematics

Numerical Analysis of Stochastic PDEs

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Dozenten Prof. Christoph Schwab, Dr. Andrea Barth
Ort T.b.a.
Zeit T.b.a.
Beginnt am 20.09.2011
Vorbesprechung 20.09.2011, 10:00-12:00, HG G57.1
Kontakt Prof. Christoph Schwab
Voraussetzungen required:
Completed ETH BSc in MATH, RW/CSE.
Completed course in Probability Theory or in Numerical Solution of SPDEs (FS11) or 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.
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 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 will 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:

Multi-Level Monte Carlo Methods,
Multi-Level 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.

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.

Format of the seminar:
MATH students will read selected recent research papers on the seminar's mathematical topics, prepare
+an oral presentation and
+a written summary of their presentation.
Student presentations will take place either during November or during the last week of HS2011.

RW/CSE students can replace the written mathematical summary by an implementation.

The number of participants is limited to 10.

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).

To get an impression of the topics covered, attend (some of) the lectures at the
2011 FIM workshop Sept 12-16:  
Numerical Analysis of Infinite Dimensional Problems

and the recent research papers

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

Tensor-structured Galerkin approximation of parametric and stochastic elliptic PDEs, B.N. Khoromskij and C. Schwab, SAM-Report 2010-04

Sparse tensor Galerkin discretizations for parametric and random parabolic PDEs. I: Analytic regularity and gpc-approximation, V.H. Hoang and Ch. Schwab, SAM-Report 2010-11

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

Regularity and generalized polynomial chaos approximation of parametric and random 2nd order hyperbolic partial differential equations, V.H. Hoang and Ch. Schwab, SAM-Report 2010-19

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

Multi-level 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

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


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