Abstract of Claude Jeffrey Gittelson

Infinite dimensional tensor product bases for stochastic operator equations
Stochastic operator equations, arising for example from boundary value problems with stochastic coefficients in the differential operator, can be recast as parametric operator equations depending on a sequence of parameters. We consider a weak formulation on the parameter domain, and approximate the parameter dependence of the solution by Galerkin's method. To this end, we construct tensor product basis functions on the infinite dimensional parameter domain. These generalize the polynomial chaos bases often used in stochastic Galerkin methods, and encompass, for example, wavelet bases for arbitrary continuous probability distributions. The hierarchic structure of the infinite dimensional bases permits anisotropic sparse tensor product approximation. Furthermore, we avoid truncation errors in the representation of the input data by discretizing the full infinite dimensional parameter domain.
These results are part of a PhD thesis under the supervision of Prof. Ch. Schwab, supported in part by the Swiss National Science Foundation under grant No. 200021-120290/1.


Lars Grasedyck (MPI Leipzig, Germany)
Wolfgang Hackbusch (MPI Leipzig, Germany)
Boris Khoromskij (MPI Leipzig, Germany)