Stochastic Polynomial Approximations for Partial Diff erential Equations with Random Input Data

Prof. Dr. Fabio Nobile, Politecnico di Milano

8 Jul 2010, 17:00; Location: S4|10-1

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.

An example is given by the study of groundwater flow, where the subsurface permeability is largely unknown and often reconstructed from few available measurements via geostatistical techniques.

In this talk we focus on models based on Partial Diff erential Equations with random coefficients or forcing terms, where randomness is used to model our insufficient knowledge or intrinsic variability of the physical system.

We fi rst parametrize the random input data by a fi nite number or random variables. Then we approximate the functional dependence of the solution of the PDE on the random variables by global multivariate polynomials, exploiting the fact that such functional dependence is often highly smooth (even analytic).

We will review both Galerkin and Collocation type approximations, based either on full or sparse tensor product polynomial spaces. In particular, we will consider approximations in total degree polynomial spaces, hyperbolic cross type polynomial spaces (as the ones induced by the standard Smolyak construction for sparse grid approximation), as well as other polynomial spaces adapted to the speci fic stochastic model. For each of them we will also introduce an anisotropic version which takes into account the diff erent influence that each random variable might have on the solution.

We show some theoretical and numerical results comparising the Galerkin and Collocation approaches in terms of error versus computational cost. The numerical results for a linear elliptic SPDE indicate a slight computational work advantage of Collocation over Galerkin.

Also, for anisotropic problems, an optimal tuning of anisotropy ratios in the polynomial space lead to highly improved convergence rates both for Galerkin and Collocation approaches.

Category: CE Seminar


Technische Universität Darmstadt

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