50
MSP 02 - Advances in numerical methods and applications II
Equations for the probabilistic moments of the solution of SPDEs Francesca Bonizzoni MOX - Politecnico di Milano Via Bonardi, 9 20133 - Milano Italy francesca1.bonizzoni@mail.polimi.it
Fabio Nobile MOX-Dipartimento di Matematica “Francesco Brioschi”, Politecnico di Milano, Italy and ´ CSQI-MATHICSE, Ecole Polytechnique F´ed´erale de Lausanne, Switzerland. fabio.nobile@epfl.ch
Annalisa Buffa IMATI - CNR Pavia via Ferrata, 1 27100 - Pavia Italy annalisa@imati.cnr.it
The boundary value problems for PDEs which model many natural phenomena and engineering applications are affected by uncertainty in the input data. One way to effectively address this issue is to describe the problem data as random variables or random fields, so that the deterministic problem turns into a stochastic differential equation (SPDE). The solution of a SPDE is itself a random field with values in a suitable function space. The simplest approach is Monte Carlo Method. Generally, its convergence rate is slow, so that this method turns to be costly. An alternative technique is to derive the moment equations, that is the deterministic equations solved by the probabilistic moments of the stochastic solution. See for example [3,4] and the references therein for problems with stochastic loading terms and [2] for problems with stochastic domains. We take into account steady state linear stochastic partial differential equations. We consider both cases of elliptic equations with stochastic loads and random coefficients. Given complete statistical information on the random input data, the aim of our work is to compute the statistics of the random solution. 1. F. Bonizzoni, F. Nobile, A. Buffa: Moment equations for the mixed formulation of the Hodge Laplacian with stochastic data (in preparation). 2. H. Harbrecht, R. Schneider, C. Schwab: Sparse second moment analysis for elliptic problems in stochastic domains. Numer. Math., 109, (2008), 385–414. 3. C. Schwab, R.A. Todor: Sparse finite elements for elliptic problems with stochastic loading. Numer. Math., 95 (2003), 707–734. 4. T. von Petersdorff, C. Schwab: Sparse finite element methods for operator equations with stochastic data. Appl. Math., 51 (2006), 145–180.