• Numerical techniques for PDEs with random input data
• Stochastic Variational Analysis

When building a mathematical model to describe the behavior of a physical system, one has often to face a certain level of uncertainty in the proper characterization of the model parameters and input data. Examples appear in the description of flows in porous media, behavior of living tissues, combustion problems, deformation of composite materials, meteorology and atmospheric models, etc.

The increasing computer power and the need for reliable predictions have pushed researchers to include uncertainty models, often in a probabilistic setting, for the input parameters of otherwise deterministic mathematical models.

In this series of lectures we focus on mathematical models based on Partial Differential Equations with stochastic input parameters (coefficients, forcing terms, boundary conditions, shape of the physical domain, etc.), and review the most used numerical techniques for propagating the input random data into to the solution of the problem.

Plan (6 lectures of 45min)

Lecture 1 (Christoph Schwab)

Problem formulation; examples of elliptic, parabolic, hyperbolic equations with stochastic data; well posedness; the case of infinite dimensional input data (random field); data representation; expansions using a countable number of random variables; truncation and convergence results.

Lecture 2 (Raul F. Tempone)

Mathematical problems parametrized by a finite number of input random variables (finite dimensional case). Perturbation techniques and second order moment analysis. Sampling methods: Monte Carlo and variants; convergence analysis.

Lecture 3 (Raul F. Tempone)

Approximation of functions using polynomial or piecewise polynomial functions either by projection or interpolation. Stochastic Galerkin method (SGM): derivation; algorithmic aspects; preconditioning of the global system. Stochastic Collocation Method (SCM): collocation on tensor grids; sparse grid approximation; construction of generalized sparse grids.

Lecture 4 (Raul F. Tempone)

Elliptic equations with random input parameters: regularity results; convergence analysis for Galerkin and Collocation approximations. Anisotropic approximations.

Lecture 5 (Raul F. Tempone)

Numerical examples, numerical comparison of SGM and SCM. Adaptive approximation.

Raul F. Tempone's References for Lectures 2 – 5:

1. J. Bäck, F. Nobile, L. Tamellini and R. Tempone. Stochastic Spectral Galerkin and Collocation methods for PDEs with random coefficients: a numerical comparison. To appear in the Proceedings of ICOSAHOM 09, Lecture Notes in Computational Science and Engineering, Springer. ICES report 09-33, Institute for Computational Engineering and Sciences, University of Texas at Austin, USA, 2009 MOX-Report 23-2010, Department of Mathematics, Politecnico di Milano, Italy, 2008.


2. F. Nobile and R. Tempone. Analysis and implementation issues for the numerical approximation of parabolic equations with random coefficients. Int. J. Num. Methods Engrg. Published online, June 12, 2009, DOI 10.1002/nme.2656. MOX-Report 22-2008, Department of Mathematics, Politecnico di Milano, Italy, 2008.


3. F. Nobile, R. Tempone and C.G. Webster. An anisotropic sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J. Numer. Anal., 46(5):2411–2442, 2008.


4. F. Nobile, R. Tempone and C.G. Webster. A sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J. Numer. Anal., 46(5):2309–2345, 2008.


5. I. Babuška, F. Nobile and R. Tempone. A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal., 45(3):1005–1034, 2007.


6. I. Babuška, R. Tempone and G.E. Zouraris. Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulation. Computer Methods in Applied Mechanics and Engineering 2005; 194(12–16):1251–1294.


7. I. Babuška, F. Nobile, and R. Tempone, Worst-case scenario analysis for elliptic problems with uncertainty, Numer. Math., 101 (2005), pp. 185–219.


8. I. Babuška, R. Tempone and G.E. Zouraris. Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J. Numer. Anal., 42(2):800–825, 2004.


9. I. Babuška, K-M. Liu, and R. Tempone, Solving stochastic partial differential equations based on the experimental data, Math. Models Methods Appl. Sci., 13 (2003), pp. 415–444.


10. I. Babuška, F. Nobile, R. Tempone, A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data, SIAM Review, Volume 52, Issue 2, pp. 317-355, 2010.


11. I. Babuška, F. Nobile and R. Tempone, A Stochastic Collocation method for elliptic Partial Differential Equations with Random Input Data, SIAM Review (Vol.52, 2010).

Lecture 6 (Christoph Schwab)

The infinite dimensional case. We review representation results of the random solutions by so-called "generalized polynomial chaos" (gpc) expansions in countably many variables. We present recent mathematical results on regularity of such solutions as well as computational approaches for the adaptive numerical Galerkin and Collocation approximations of the infinite dimensional parametric, deterministic solution. A key principle are new sparsity estimates of gpc expansions of the parametric solution. We present such estimates for elliptic, parabolic and hyperbolic problems with random coefficients, as well as eigenvalue problems.

We compare the possible convergence rates with the best convergence results on Monte Carlo Finite Element Methods (MCFEM) and on MLMCFEM.

References:

1. R. Andreev and Ch. Schwab, Sparse Tensor Discretization of High-dimensional Eigenvalue problems (in preparation 09/2010).


2. M. Bieri, Sparse tensor discretization of elliptic sPDEs with random input data, Ph.D. dissertation ETH Zürich, 2009.


3. M. Bieri, A sparse composite collocation finite element method for elliptic sPDEs, SIAM J. Numer. Anal. (2010).


4. . M. Bieri, R. Andreev and Ch. Schwab, Sparse Tensor Discretization of Elliptic sPDEs SIAM J. Sci. Comput. 31(6) 4281-4304 (2009).


5. A. Cohen, R.A. DeVore and Ch. Schwab, Convergence Rates of best N-term approximations for a class of elliptic sPDEs (to appear in Journ. Found. Comp. Math. 2010).


6. A. Cohen, R.A. DeVore and Ch. Schwab, Analytic regularity and polynomial approximation of parametric and stochastic elliptic PDEs, Report 2010-03, Seminar for Applied Mathematics, ETH Zürich (to appear in Journal for Analysis and its Applications)


7. V. H. Hoang and Ch. Schwab, Sparse Tensor Galerkin Discretizations for parametric and random parabolic PDEs I: Analytic regularity and gpc-Approximation, Report 2010-11, Seminar for Applied Mathematics, ETH Zürich (in review)


8. V. H. Hoang and Ch. Schwab, Analytic regularity and gpc approximation for parametric and random 2nd order hyperbolic PDEs, Report 2010-19, Seminar for Applied Mathematics, ETH Zürich (in review)


9. Ch. Schwab and C.J. Gittelson, Acta Numerica 2011 (to appear).


10. R.A. Todor and Ch. Schwab, Convergence Rates of Sparse Chaos Approximations of Elliptic Problems with stochastic coefficients, IMA Journ. Numer. Anal. 44 (2007) 232-261.

Part II: Stochastic Variational Analysis (Lectures by Roger Wets)

This is a series of four lectures designed to introduce, in a unified framework, a broad spectrum of mathematical theory that has grown in connection with the study of problems of optimization, equilibrium, control, and stability of linear and nonlinear systems that naturally arise in a stochastic environment. With the advent of computers, there has been a tremendous expansion of interest in new problem formulations that demand new modes of analysis but are far from being covered by classical concepts and classical results. For those problems, finite-dimensional spaces alongside of function spaces, and theoretical concerns go hand in hand with the practical ones of mathematical modeling and the design of numerical procedures.

The presentation will touch on a variety of applications: stochastic programming problems, equilibrium problems in a stochastic environment, stochastic variational inequalities, stochastic homogenization, financial valuation, flow problems in heterogeneous media, etc. However, the primordial goal will be to provide an introduction to the mathematical foundations.

Plan

1. A brief review of variational analysis. Functions and their epigraphs, convexity and semicontinuity. Set convergence and epigraphical limits. Variational geometry, subgradients and subdifferential calculus.


2. Random sets. Definition and properties of random sets, selections. The distribution function (∼ Choquet capacity) of a random set and convergence in distribution. The expectation of a random set and the law of large numbers for random sets. SAA (Sample. Average Approximations) of random sets. Application to stochastic variational inequalities and related variational problems.


3. Random lsc functions and expectation functionals. Definition of random lsc (lower semicontinuous) functions and calculus. Stochastic processes with lsc paths. Properties of expectation functionals. Almost sure convergence and convergence in distribution (epigraphical sense). The Ergodic Theorem for random lsc functions and its applications: sampled variational problems, approximation, statistical estimation and homogenization.


4. Introduction to the calculus of expectation functionals. Decomposable spaces. Fatou’s lemma for random set and random lsc functions. Interchange of minimization and (conditional) expectation. Subdifferentiation of expectation functionals. Martingale integrands and application to financial valuation.