Graduate course: Numerical techniques for PDEs with random input data

Teacher: Raul Tempone, KAUST University.

Schedule: 10.15-12.00 and 13.15-15 Monday October 14 to Friday October 18th and week 22, i.e. May 26-30 2014 in room 3418 Lindtedsvagen 25, KTH, except Wednesday 13-15 room 3424. Here is a map. Enter from Lindstedsvägen 25, take the stairs one floor down (to the previous library). You may send an sms to Anders 0702-876598 for help (i.e. to open locked doors).

Course load: Two intensive weeks, one in Fall 2013 and one in Spring 2014. Each week will have an intended course load of 1.5 hrs/day (lectures) and some extra lab work at 13.15-15.00 in room 3434 Monday and Friday, room 3418 Tuesday and Thursday, and room 3733 Wednesday.

Examination: the examination will be done based on the solution of the labs provided by the student. Working in groups of two or three students is allowed and encouraged.

Introduction: 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 mostly 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 onto the solution of the problem.

Topics to be covered:

Introduction and motivating examples; PDEs with random coefficients (Darcy, acoustic, elasticity) and differential equations driven by noise. Random fields; mean, variance, covariance, stationarity, smoothness; gaussian random fields / Spectral decomposition / Karhunen Loeve expansion (review if Blanca has already introduced it) Approximation of high dimensional functions by Monte Carlo / MLMC Quasi Monte Carlo Presentation of Lab1. Exercises on: high dimensional integration of functions, random fields, Monte Carlo, MLMC, QMC Review of 1D Legendre polynomials and L^2 projection results; finite regularity case; analytic case multivariate L^2 projection: isotropic case: TP/TD/HC spaces and corresponding implied regularity of the function - curse of dimensionality Non-linear approximation - Stechkin infinite dimensional case: analytic case with different analyticity regions and corresponding polynomial spaces. Random discrete L^2 projection. Collocation - 1D result / nD result by tensorization; curse of dimensionality Sparse grid construction Standard Smolyak approximation - convergence result for functions with mixed derivatives optimal sparse grid - generalized Stechkin Hyperbolic equations with random coefficients; well posedness and regularity results Adjoint problem and smoothness of QoI. Design of experiments


The students are expected to have basic knowledge on probability theory, approximation theory, Partial Differential Equations, numerical analysis in general and finite element analysis in particular. Programming knowledge and familiarity with Matlab or any of its free clones (Octave, Scilab, etc) is assumed for the Lab sessions.

-- Course slides and a set of lab exercises prepared by F. Nobile and R. Tempone. -- Some papers that serve as references (not a complete list yet):

1. J. Baeck, F. Nobile, L. Tamellini, R. Tempone, On the optimal polynomial approximation of stochastic PDEs by Galerkin and Collocation methods, MOX Report 23/2011, Politecnico di Milano, Italy.

2. J. Baeck, F. Nobile, L. Tamellini and R. Tempone. Stochastic Spectral Galerkin and Collocation methods for PDEs with random coefficients: a numerical comparison. In Spectral and High Order Methods for Partial Differential Equations, Lecture Notes in Computational Science and Engineering, Volume 76, pp. 43-62, 2011.

3. I. Babuska, 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.

4. F. Nobile and R. Tempone. Analysis and implementation issues for the numerical approximation of parabolic equations with random coefficients. Int. J. Numer. Methods Engrg, 2009, vol. 80/6-7, pp. 979-1006.

5. 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.

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

7. I. Babuska, 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.

8. "Approximation of quantities of interest in stochastic PDES by the random discrete L2 projection on polynomial spaces", by G. Migliorati, F. Nobile, E. von Schwerin and R. Tempone. To appear in SIAM Journal of Scientific Computing, 2013.

9. "A stochastic collocation method for the second order wave equation with a discon- tinuous random speed", by M. Motamed, F. Nobile and R. Tempone. In Numerische Mathematik, vol. 123, Issue 3 (2013), pp. 493-536.


Anders Szepessy