Abstract: In the field of ocean data assimilation, the SVD (singular value decomposition) method used in the matrix inversion process of ensemble optimal interpolation is very time-consuming. This paper uses matrix LU decomposition, Choleskey decomposition and QR decomposition respectively to replace singular value decomposition to optimize the matrix inversion process in the ensemble optimal interpolation. Firstly, LU (Choleskey or QR) decomposition breaks down some matrices into two products of triangular matrices (or orthogonal matrices). Secondly, the matrices obtained by decomposition process, are used rapidly to calculate the inverse of matrix. As the algorithm complexity of LU (Choleskey or QR) decomposition is much smaller than that of the singular value decomposition, the performance of the assimilation program is improved greatly. The numerical experiments show that the three kinds of matrix decomposition methods used in this paper can improve the computational efficiency of the ensemble at least two times compared with the traditional implementation based on singular value decomposition. Furthermore, it should be noted that the Choleskey decomposition makes the entire assimilation program achieve the best performance.

Key words: ocean data assimilation, ensemble optimal interpolation (EnOI), matrix inversion, matrix decomposition, Choleskey decomposition

摘要： 在海洋数据同化领域，集合最优插值方法中，矩阵求逆过程所使用的奇异值分解（singular value decomposition，SVD）十分耗时。对集合最优插值中逆矩阵的求逆过程进行优化，分别使用LU分解、Choleskey分解、QR分解来替代SVD分解。首先，通过LU分解（Choleskey分解或QR分解）得到相应的三角矩阵（或正交矩阵）；然后，利用分解后的矩阵来实现相关逆矩阵的计算。由于LU分解、Choleskey分解、QR分解的算法复杂度都远小于SVD分解，因此改进后的同化程序能得到大幅度的性能提升。数值结果表明，所采用的三种矩阵分解方法相比于SVD分解，都能将集合最优插值的计算效率提升至少两倍以上。值得一提的是，在四种矩阵分解中Choleskey分解使得整个同化程序的性能达到了最优。

关键词: 海洋数据同化, 集合最优插值（EnOI）, 矩阵求逆, 矩阵分解, Choleskey分解