基于SVD的本征正交分解算法在偏微分方程中的降阶数值模式研究

The Study About Reduced Numerical Model of Proper Orthogonal Decomposition Used in Partial Differential Equation Based on SVD

本文分别论述全矩阵、距平矩阵以及归一化矩阵的奇异正交分解(Singular Value Decomposition,简称SVD)算法的理论基础,推导了任意矩阵的SVD分解过程并且在任意矩阵SVD分解的基础上,给出两种本征正交分解(Proper Orthogonal Decomposition,简称POD)算法,将POD算法与Galerkin投影相结合可以将偏微分方程的高维或者无穷维解投影到POD模态构成的完备空间中进行降阶模拟,进而得到高度近似的低维解,比较用不同阶POD模态降阶前后解的稳定性及精确性.最后给出数值算例分析两种本征正交分解算法的优劣性及适用性.

Foundations of singular value decomposition theory（SVD） about three kinds of matrixes：the common matrix, matrix anomaly and the standard normalized matrix were discussed respectively.The SVD process of arbitrary matrix was deduced. Based on the SVD of arbitrary matrix two proper orthogonal decomposition（POD） algorithms were given. The POD algorithm combined with Galerkin projection the higher dimensional or infinite dimensional solution of partial differential equation can be projected into the complete space which was constituted by POD modes. The low dimensional solution will be obtained and the low dimensional solution will be compared with the solution of partial differential equation. The stability and accuracy of the POD algorithm also will be compared between the solution of partial differential equation and the low dimensional solutions obtained by different POD modes. At last some numerical examples were given to show the pros and cons and its applicability of two kinds of POD algorithms.