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Abstracts and Talk Materials
Computing with Uncertainty
October 16-17, 2010

Christoph Schwab - ETH Zürich

Lecture 1.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
October 16, 2010 9:00 am - 9:45 am


Lecture 6. The infinite dimensional case
October 17, 2010 3:30 pm - 4:15 pm

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.

Raul Tempone - King Abdullah University of Science & Technology

Lecture 2. 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
October 16, 2010 10:00 am - 10:45 am


Lecture 3. 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
October 16, 2010 11:00 am - 11:45 am


Lecture 4. Elliptic equations with random input parameters: regularity results; convergence analysis for Galerkin and Collocation approximations. Anisotropic approximations
October 17, 2010 1:30 pm - 2:15 pm


Lecture 5. Numerical examples, numerical comparison of SGM and SCM. Adaptive approximation
October 17, 2010 2:30 pm - 3:15 pm

Roger Wets - University of California, Davis

A brief review of variational analysis
October 16, 2010 1:30 pm - 3:00 pm

Functions and their epigraphs, convexity and semicontinuity. Set convergence and epigraphical limits. Variational geometry, subgradients and subdifferential calculus.


Random sets
October 16, 2010 3:15 pm - 4:15 pm

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.


Random lsc functions and expectation functionals
October 17, 2010 9:00 am - 10:30 am

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.


Introduction to the calculus of expectation functionals
October 17, 2010 11:00 am - 12:00 pm

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.