Name
Title | ||
|---|---|---|
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data |
Abstract | ||
|---|---|---|
Abstract In this paper we propose and analyze a Stochastic-Collocation method to solve elliptic Partial Difierential Equations with random,coe‐cients and forcing terms (input data of the model). The input data are assumed to depend on a flnite number of random,variables. The method consists in a Galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space and naturally leads to the solution of uncoupled deterministic prob- lems as in the Monte Carlo approach. It can be seen as a generalization of the Stochastic Galerkin method proposed in [Babu• ska -Tempone-Zouraris, SIAM J. Num. Anal. 42(2004)] and allows one to treat easily a wider range of situations, such as: input data that depend non-linearly on the random variables, difiusivity coe‐cients with unbounded second moments , random variables that are correlated or have unbounded support. We provide a rigorous convergence analysis and demonstrate exponential con- vergence of the \probability error" with respect of the number of Gauss points in each direction in the probability space, under some regularity assumptions on the random,input data. Numerical examples show the efiectiveness of the method. Key words: Collocation method, stochastic PDEs, flnite elements, un- certainty quantiflcation, exponential convergence. AMS subject classiflcation: 65N35, 65N15, 65C20 |
Year | DOI | Venue |
|---|---|---|
2007 | 10.1137/100786356 | SIAM Review |
Keywords | Field | DocType |
input data,probability space,effective collocation strategy,elliptic partial differential equations,random input data,gauss point,finite number,sparse grid stochastic collocation,random variable,stochastic collocation method,random coefficient,finite elements,uncertainty quantification,stochastic partial differential equations,elliptic operators,collocation method,elliptic partial differential equation,polynomial chaos | Convergence of random variables,Random element,Mathematical optimization,Random field,Orthogonal collocation,Mathematical analysis,Multivariate random variable,Sum of normally distributed random variables,Collocation method,Mathematics,Random function | Journal |
Volume | Issue | ISSN |
52 | 2 | 0036-1445 |
Citations | PageRank | References |
168 | 12.10 | 16 |
Authors | ||
3 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Ivo Babuška | 1 | 660 | 118.05 |
| Fabio Nobile | 2 | 336 | 29.63 |
| Raul | 3 | 477 | 54.12 |