Numerical Analysis of Stochastic PDEs

MSc Seminar - DMATH ETHZ HS2011

Room and date: T.b.a.

Link to D-MATH Lectures

1. Goal

Prepare
ETH MSc Students in Applied Math and
ETH MSc Students in RW/CSE
for Master Thesis work.

2. Instructor

Prof. Ch. Schwab,
Seminar for Applied Mathematics,
ETHZ, HG G57.1,
schwab@math.ethz.ch

3. First meeting

Vorbesprechung

Tuesday, September 20, 2011, 10:00-12:00, HG G57.1.

4. General Outline of the seminar

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.

5. 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 impinge on 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.

6. 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 algorithm implementation.

The number of participants is limited to 10.

7. Hours/ Credits

Hours/ Credits: S2

8. Prerequisites 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 ell.&para. PDEs or hyp. PDEs.

9. Recommended

recommended: Courses in
Functional Analysis and/or
parallel computing and/or
Computational Methods for Quantitative Finance and / or
Numerical Solution of Stochastic ODEs.

10. Representative Reading List

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).

11. To get an impression of the topics covered

A. Lectures
Attend (some of) the lectures at the
2011 FIM workshop Sept 12-16:
Numerical Analysis of Infinite Dimensional Problems

B. Recent research papers

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

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

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

Report Nr. 2011-41
Ph. Grohs and Ch. Schwab
Sparse twisted tensor frame discretization of parametric transport operators

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

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

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

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

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

Report Nr. 2011-19
C.J. Gittelson
Uniformly convergent adaptive methods for parametric operator equations

Report Nr. 2011-17
D. Marazzina, O. Reichmann and Ch. Schwab
hp-DGFEM for Kolmogorov-Fokker-Planck equations of multivariate Lévy processes

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

Report Nr. 2011-12
C.J. Gittelson
Adaptive stochastic Galerkin methods: beyond the elliptic case

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

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

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

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

Report Nr. 2010-40
R. Andreev and Ch. Schwab
Sparse tensor approximation of parametric eigenvalue problems

Report Nr. 2010-34
H. Harbrecht and Ch. Schwab
Sparse tensor finite elements for elliptic multiple scale problems

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

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

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

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

Report Nr. 2010-12
H. Brandsmeier, K. Schmidt and Ch. Schwab
A multiscale hp-FEM for 2D photonic crystal band

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

Report Nr. 2010-09
B. Pentenrieder and Ch. Schwab
hp-FEM for second moments of elliptic PDEs with stochastic data.
Part 2: Exponential convergence

Report Nr. 2010-08
B. Pentenrieder and Ch. Schwab
hp-FEM for second moments of elliptic PDEs with stochastic data.
Part 1: Analytic regularity

Report Nr. 2010-06
C. Schwab and O. Reichmann
Numerical analysis of additive, Lévy and Feller processes with applications to option pricing

Report Nr. 2010-05
C. Schwab and R. Stevenson
Fast evaluation of nonlinear functionals of tensor product wavelet expansions

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

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

Report Nr. 2010-01
N. Hilber, S. Kehtari, C. Schwab and C. Winter
Wavelet finite element method for option pricing in highdimensional
diffusion market models