October 16 - 17, 2010
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
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.
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
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.
Functions and their
epigraphs, convexity and semicontinuity.
Set convergence and epigraphical limits. Variational geometry,
subgradients
and subdifferential calculus.
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.
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.