Name
Title | ||
|---|---|---|
A least-squares approximation of partial differential equations with high-dimensional random inputs |
Abstract | ||
|---|---|---|
Uncertainty quantification schemes based on stochastic Galerkin projections, with global or local basis functions, and also stochastic collocation methods in their conventional form, suffer from the so called curse of dimensionality: the associated computational cost grows exponentially as a function of the number of random variables defining the underlying probability space of the problem. In this paper, to overcome the curse of dimensionality, a low-rank separated approximation of the solution of a stochastic partial differential (SPDE) with high-dimensional random input data is obtained using an alternating least-squares (ALS) scheme. It will be shown that, in theory, the computational cost of the proposed algorithm grows linearly with respect to the dimension of the underlying probability space of the system. For the case of an elliptic SPDE, an a priori error analysis of the algorithm is derived. Finally, different aspects of the proposed methodology are explored through its application to some numerical experiments. |
Year | DOI | Venue |
|---|---|---|
2009 | 10.1016/j.jcp.2009.03.006 | J. Comput. Physics |
Keywords | Field | DocType |
uncertainty quantification,alternating least-squares,high-dimensional random input data,elliptic spde,partial differential equation,stochastic galerkin projection,curse of dimensionality,stochastic partial differential,proposed algorithm,separated representation,proposed methodology,stochastic partial differential equations,underlying probability space,associated computational cost,least-squares approximation,computational cost,uncertainty quantification separated representation alternating least-squares curse of dimensionality stochastic partial differential equations,stochastic collocation method,collocation method,random variable,least squares approximation,stochastic partial differential equation | Mathematical optimization,Random variable,Uncertainty quantification,Galerkin method,Curse of dimensionality,Partial derivative,Basis function,Stochastic partial differential equation,Partial differential equation,Mathematics | Journal |
Volume | Issue | ISSN |
228 | 12 | Journal of Computational Physics |
Citations | PageRank | References |
33 | 1.79 | 10 |
Authors | ||
2 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Alireza Doostan | 1 | 188 | 15.57 |
| Gianluca Iaccarino | 2 | 229 | 23.37 |