  In this paper, we propose a novel variable-separation (NVS) method for
generic multivariate functions. The idea of NVS is extended to to obtain the
solution in tensor product structure for stochastic partial differential
equations (SPDEs).
  Compared with many widely used variation-separation methods, NVS shares their
merits but has less computation complexity and better efficiency.
  NVS 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 NVS depend on
the previous terms, this impacts on the computation efficiency and brings 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 least angle regression algorithm (LARS) for lasso problems such
that ILARS is much more efficient.
  HSLRTA hierarchically decomposes a high dimensional problem into some low
dimensional problems and brings an accurate approximation for the solution to
SPDEs in high dimensional stochastic spaces using limited computer resource.
  A few numerical examples are presented to illustrate the efficacy of the
proposed methods.
