Numerical Analysis of Stochastic Partial Differential Equations

Lectures begin: Wednesday, February 24

Testat requirement: 70% of exercises reasonably solved

Exam: oral, 30 minutes

Office hours: Thursday, 12:00 noon, HG G 56.1

Problem Sets

Problem sets can be turned in during the lecture or in the mailbox at HG G 53.  For programming exercises, print out and hand in plots and other output, and send your code to Claude Gittelson.

Prerequisites

• Numerical solution of elliptic and parabolic PDEs
• Linear Functional Analysis
• Probability Theory

Contents

1. Mathematical Foundations
   

   We briefly recapitulate the foundations.
   It is assumed that students have taken the above
   three courses.

   1. Linear Functional Analysis (recapitulation)

      Hilbert spaces, Lax-Milgram Lemma,
      Spectral Theory of compact, self-adjoint operators,
      Tensor-Product spaces and measures.

   2. Probability Theory (recapitulation)

      Probability Spaces,
      Stochastic processes,
      Random fields,
      Karhunen-Loeve Expansion,
      Gaussian Measures on Hilbert spaces,
      Bochner-Minlos Theorem.

   3. Numerical Analysis (recapitulation)

      h-, p- and hp-FEMs:
      basic approximation results and implementation,
      hierarchic bases, multilevel decompositions of FE-spaces.

2. Elliptic sPDEs with random sources
   
   1. Basic examples, well-posedness, unique solvability
   2. First and higher moments
   3. Regularity of random solution and its moments
   4. Approximation of second and higher moments by sparse tensor FEM
3. Elliptic sPDEs with random coefficients
   
   1. Basic examples, well-posedness, unique solvability
   2. Representations of random fields: Karhunen-Loeve and Multiresolution expansions
   3. Regularity and approximations spaces
   4. Sparse tensor Finite Element discretizations: implementation, error analysis
4. Parabolic sPDEs
   
   1. Random forcing: white and colored
   2. Random coefficients

Literature

Required:

• Introduction to Infinite Dimensional Analysis
  Giuseppe da Prato
  Springer Universitext (2006)

Supplementary Reading:

• Stochastic Equations in Infinite Dimensions
  G. da Prato and J. Zabczyk
  Cambridge Univ. Press (1992)
• Numerical Solution of Stochastic Differential Equations
  P.E. Kloeden and E. Platen
  Springer Verlag (1992)
• Stochastic partial differential equations with Levy Noise
  S. Peszat and J. Zabcyk
  Cambridge University Press (2007)
• Numerical Analysis of Wavelet Methods
  A. Cohen
  North-Holland (2003)
• Nonlinear Approximation
  R. DeVore
  Acta Numerica 7, Cambridge University Press (1997)
  

The following research articles will be used and referenced in the course of the lecture.

• Sparse Second Moment Analysis for Elliptic Problems in Stochastic Domains
  (H. Harbrecht R. Schneider Ch. Schwab), Numer. Math. 109(3)(2008), 385-414.
• Multilevel Frames for Sparse Tensor Product Spaces
  (H. Harbrecht R. Schneider Ch. Schwab), Numerische Mathematik 110(2) (2008) 199-220.
• Sparse finite elements for stochastic elliptic problems - higher order moments
  (Ch. Schwab and R.A. Todor) Computing 71(2003) 43-63.