Solving stochastic systems with low-rank tensor compression
- 作者：Hermann G. Matthies ; wire@tu-bs.de ; Elmar Zander
- 关键词：Stochastic systems ; Stochastic partial differential equations ; Linear equations on tensor product spaces ; Singular value decomposition (SVD) ; Sparse representation ; Low rank approximation ; Sparse tensor product
- 刊名：Linear Algebra and its Applications
- 出版年：2012
- 期刊代码：38_00243795
- 类别：mt
- 出版时间：15 May, 2012
- 卷：436
- 期：10
- 页码：3819-3838
- 文件大小：697 K
摘要
For parametrised equations, which arise, for example, in equations dependent on random parameters, the solution naturally lives in a tensor product space. The application which we have in mind is a stochastic linear elliptic partial differential equation (SPDE). Usual spatial discretisation leads to a potentially large linear system for each point in the parameter space. Approximating the parametric dependence by a Galerkin 鈥榓nsatz鈥? the already large number of unknowns鈥攆or a fixed parameter value鈥攊s multiplied by the dimension of the Galerkin subspace for the parametric dependence, and thus can be very large. Therefore, we try to solve the total system approximately on a smaller submanifold which is found adaptively through compression during the solution process by alternating iteration and compression. We show that for general linearly converging stationary iterative schemes and general adaptation processes鈥攚hich can be seen as a modification or perturbation of the iteration鈥攖he interlaced procedure still converges. Our proposed modification can be used for most stationary solvers for systems on tensor products. We demonstrate this on an example of a discretised SPDE with a simple preconditioned iteration. We choose a truncated singular value decomposition (SVD) for the compression and give details of the implementation, finishing with some examples.