October 16-17, 2010
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 caseOctober 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.
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 analysisOctober 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 gridsOctober 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 approximationsOctober 17, 2010 1:30 pm - 2:15 pm
Lecture 5. Numerical examples, numerical comparison of SGM and SCM. Adaptive approximationOctober 17, 2010 2:30 pm - 3:15 pm
A brief review of variational analysisOctober 16, 2010 1:30 pm - 3:00 pm
Functions and their
epigraphs, convexity and semicontinuity.
Set convergence and epigraphical limits. Variational geometry,
and subdifferential calculus.
Random setsOctober 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.
Average Approximations) of random sets. Application to
stochastic variational inequalities
and related variational problems.
Random lsc functions and expectation functionalsOctober 17, 2010 9:00 am - 10:30 am
of random lsc (lower semicontinuous)
functions and calculus. Stochastic processes with lsc paths.
expectation functionals. Almost sure convergence and
convergence in distribution (epigraphical
sense). The Ergodic Theorem for random lsc functions and its
sampled variational problems, approximation, statistical
estimation and homogenization.
Introduction to the calculus of expectation
functionalsOctober 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.
and application to financial valuation.