Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations. (English) Zbl 1088.65002
Authors’ abstract: Stationary systems modelled by elliptic partial differential equations – linear as well as nonlinear – with stochastic coefficients (random fields) are considered. The mathematical setting as a variational problem, existence theorems, and possible discretisations – in particular with respect to the stochastic part – are given and investigated with regard to stability.
Different and increasingly sophisticated computational approaches involving both Wiener’s polynomial chaos as well as the Karhunen-Loève expansion are addressed in conjunction with stochastic Galerkin procedures, and stability within the Galerkin framework is established. New and effective algorithms to compute the mean and covariance of the solution are proposed.
The similarities and differences with better known Monte Carlo methods are exhibited, as well as alternatives to integration in high-dimensional spaces. Hints are given regarding the numerical implementation and parallelisation. Numerical examples serve as illustration.
Different and increasingly sophisticated computational approaches involving both Wiener’s polynomial chaos as well as the Karhunen-Loève expansion are addressed in conjunction with stochastic Galerkin procedures, and stability within the Galerkin framework is established. New and effective algorithms to compute the mean and covariance of the solution are proposed.
The similarities and differences with better known Monte Carlo methods are exhibited, as well as alternatives to integration in high-dimensional spaces. Hints are given regarding the numerical implementation and parallelisation. Numerical examples serve as illustration.
Reviewer: Dominique Lepingle (Orléans)
MSC:
| 65C30 | Numerical solutions to stochastic differential and integral equations |
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 35R60 | PDEs with randomness, stochastic partial differential equations |
| 60H35 | Computational methods for stochastic equations (aspects of stochastic analysis) |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 35J65 | Nonlinear boundary value problems for linear elliptic equations |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
| 65C05 | Monte Carlo methods |
Keywords:
linear and nonlinear elliptic stochastic partial differential equations; Galerkin methods; Karhunen-Loeve expansion; Wiener’s polynomial chaos; white noise analysis; sparse Smolyak quadrature; Monte Carlo methods; stochastic finite elements; parallel computation; stability; algorithms; numerical examplesSoftware:
SmolpackReferences:
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