求解具有张量积结构系统的算法研究

项目名称： 求解具有张量积结构系统的算法研究

项目编号： No.11261012

项目类型： 地区科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 卢琳璋

作者单位： 贵州师范大学

项目金额： 45万元

中文摘要： 科学、技术和工程中的许多数学模型都需要用高维方程、特别是高维的微分方程来描述。如何数值求解这些高维问题，是当今大规模科学与工程计算所遇到的挑战之一。本项目主要研究一些高维问题离散化所产生的具有张量积结构的系统的高效数值解法, 特别关注预处理子和并行算法的设计。在本项目中，我们将做如下的研究工作：将Tucker分解与PARAFAC分解相结合设计基于张量格式的预处理投影算法，特别是共轭梯度法（CG）及其预处理技巧；从数值分析的角度研究高维问题离散化后的低秩张量结构的表示，并尝试给出其误差估计；从数值代数的角度设计张量截断的算法，并给出误差估计；尝试设计基于张量运算的并行算法。在本项目中涉及的所有新设计、改进的算法都将用数值试验来检验其有效性及理论分析的正确性。

中文关键词： 张量积结构；高阶奇异值分解；张量分解；；

英文摘要： Description of a lot of mathematical models in science, technology and engineering needs to use equations, especially differential equations of high dimension. How to solve numerically the problems of the high dimensions is one of challenges for large scale of scientific and engineering computation. This project studies mainly high performance algorithms for the systems with tensor product structure arising from discretization of some high dimension problems, especially focus in designs of preconditioners and parallel algorithms. In this project, we will do the following studies: Combining Tucker decomposition with PARAFAC decomposition to design preconditioned projection algorithms based on tensor form, especially the CG and its preconditioning technology；studying expressions of structures of tensors of low rank which are obtained by discretizating the high dimension problems from the view of numerical analysis and trying to give its error estimations;designing algorithms about tensor truncation from the view of numerical algebra and giving corresponding error estimations; try to design parallel algorithms based on tensor operations. Numerical experiments will be done to demonstrate efficiency of all new designed or improved algorithms involved in the project and verify correct of theoretic analysis.

英文关键词： tensor product structure；high-order SVD；tensor decomposition；；