基于稀疏表示的新型变量分离方法用于随机偏微分方程
A novel variable-separation method based on sparse representation for stochastic partial differential equations
AI总结：
本文提出一种新型变量分离方法，用于求解随机偏微分方程的张量积结构解，相比传统方法，计算复杂度更低且效率更高，通过改进的least angle regression算法和分层稀疏低秩张量近似方法提高求解精度。
摘要：
本文提出了一种新型变量分离（NVS）方法，用于求解通用多变量函数。NVS的思想被扩展以在张量积结构中获得随机偏微分方程（SPDEs）的解。与许多广泛使用的变量分离方法相比，NVS继承了其优点，但计算复杂度更低且效率更高。NVS可以系统地获得SPDE解的分离表示。在每个丰富步骤中不进行迭代，这与proper generalized decomposition相比是一个显著改进。由于NVS获得的分离表示的随机函数依赖于之前的项，这影响了计算效率，并为高随机维数空间中的数值模拟带来了巨大挑战。为克服这一困难，我们提出了基于稀疏正则化的改进least angle regression算法（ILARS）和分层稀疏低秩张量近似（HSLRTA）方法。对于ILARS，我们明确给出了每个步骤中最优正则化参数的选择，基于least angle regression算法（LARS）用于lasso问题，使得ILARS更加高效。HSLRTA将高维问题分解为一些低维问题，并利用有限的计算机资源，在高维随机空间中对SPDEs的解提供准确的近似。一些数值示例被提出以说明所提方法的有效性。
英文摘要：
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.
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