A weighted reduced basis method for elliptic partial differential equations with random input data. (English) Zbl 1288.65007
Authors’ abstract: We propose and analyze a weighted reduced basis method to solve elliptic partial differential equations (PDEs) with random input data. The PDEs are first transformed into a weighted parametric elliptic problem depending on a finite number of parameters. Distinctive importance of the solution at different values of the parameters is taken into account by assigning different weights to the samples in the greedy sampling procedure. A priori convergence analysis is carried out by constructive approximation of the exact solution with respect to the weighted parameters. Numerical examples are provided for the assessment of the advantages of the proposed method over the reduced basis method and the stochastic collocation method in both univariate and multivariate stochastic problems.
Reviewer: Grigori N. Milstein (Yekaterinburg)
MSC:
65C30 | Numerical solutions to stochastic differential and integral equations |
65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |
60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
60H35 | Computational methods for stochastic equations (aspects of stochastic analysis) |
35R60 | PDEs with randomness, stochastic partial differential equations |