科研之友
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摘要
In this paper, we propose a novel variable-separation (VS) method for generic multivariate functions. The idea of the novel VS is extended to obtain the solution in tensor product structure for stochastic partial differential equations (SPDEs). Compared with many widely used variation-separation methods, the novel VS shares their merits but has less computation complexity and better efficiency. The novel VS can be used to get the separated representation of the solution for SPDE in a systematic enrichment manner. No iteration is performed at each enrichment step. This is a significant improvement compared with proper generalized decomposition. Because the stochastic functions of the separated representations obtained by the novel VS depend on the previous terms, this impacts on the computation efficiency and brings a great challenge for numerical simulation for the problems in high stochastic dimensional spaces. In order to overcome the difficulty, we propose an improved least angle regression algorithm (ILARS) and a hierarchical sparse low rank tensor approximation (HSLRTA) method based on sparse regularization. For ILARS, we explicitly give the selection of the optimal regularization parameters at each step based on a least angle regression algorithm for lasso problems such that ILARS is much more efficient. HSLRTA hierarchically decomposes a high-dimensional problem into low-dimensional problems and brings an accurate approximation for the solution to SPDEs in high-dimensional stochastic spaces using limited computer resources. Four examples are presented to illustrate the efficacy of the proposed methods.
- 出版日期2017
- 单位湖南大学
