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In this paper, we propose a novel variable-separation (NVS) method forgeneric multivariate functions. The idea of NVS is extended to to obtain thesolution in tensor product structure for stochastic partial differentialequations (SPDEs).  Compared with many widely used variation-separation methods, NVS shares theirmerits but has less computation complexity and better efficiency.  NVS can be used to get the separated representation of the solution for SPDEin a systematic enrichment manner.  No iteration is performed at each enrichment step. This is a significantimprovement compared with proper generalized decomposition. Because thestochastic functions of the separated representations obtained by NVS depend onthe previous terms, this impacts on the computation efficiency and brings greatchallenge for numerical simulation for the problems in high stochasticdimensional spaces.  In order to overcome the difficulty, we propose an improved least angleregression algorithm (ILARS) and a hierarchical sparse low rank tensorapproximation (HSLRTA) method based on sparse regularization. For ILARS, weexplicitly give the selection of the optimal regularization parameters at eachstep based on least angle regression algorithm (LARS) for lasso problems suchthat ILARS is much more efficient.  HSLRTA hierarchically decomposes a high dimensional problem into some lowdimensional problems and brings an accurate approximation for the solution toSPDEs in high dimensional stochastic spaces using limited computer resource.  A few numerical examples are presented to illustrate the efficacy of theproposed methods.

A novel variable-separation method based on sparse representation for stochastic partial differential equations