资料同化中具有全观测加权矩阵的矩阵-向量乘积的有效计算
Efficient computation of matrix-vector products with full observation weighting matrices in data assimilation
摘要：
最近的研究表明，通过在数据同化系统中使用空间相关的观测误差协方差信息，数值天气预报的技能得到了提高。在这种情况下，同化中使用的观测加权矩阵（逆误差协方差阵）可以是全矩阵而不是对角矩阵。因此，变分最小化问题中的矩阵-向量积的计算可能非常耗时，特别是如果矩阵-向量积的并行计算需要处理元件之间的高度通信。因此，我们引入了一个著名的数值逼近方法，称为快速多极子方法（FMM），以加快数据同化中的矩阵向量乘法。我们探讨了一种特殊类型的FMM，使用奇异值分解（SVD-FMM），并调整它，以适应我们的新应用数据同化。与标准方法相比，SVD-FMM技术通过近似大部分矩阵-向量积的计算，大大降低了计算复杂性。我们开发了一种新的可能的并行化方案的SVD-FMM为我们的应用程序，它可以减少通信成本。我们调查的SVD-FMM技术在几个数值实验的准确性：我们首先使用协方差矩阵，使用不同的相关函数和lengthscales创建的准确性进行评估;然后调查的影响，重新调整的协方差阵的准确性;最后检查的可行性，该技术在存在缺失的观察。我们还对一些数值结果提供了理论解释。我们的研究结果表明，SVD-FMM技术有潜力作为一个有效的技术同化大量的观测数据在一个很短的时间间隔。
英文摘要：
Recent studies have demonstrated improved skill in numerical weather prediction via the use of spatially correlated observation error covariance information in data assimilation systems. In this case, the observation weighting matrices (inverse error covariance matrices) used in the assimilation may be full matrices rather than diagonal. Thus, the computation of matrix-vector products in the variational minimization problem may be very time-consuming, particularly if the parallel computation of the matrix-vector product requires a high degree of communication between processing elements. Hence, we introduce a well-known numerical approximation method, called the fast multipole method (FMM), to speed up the matrix-vector multiplications in data assimilation. We explore a particular type of FMM that uses a singular value decomposition (SVD-FMM) and adjust it to suit our new application in data assimilation. By approximating a large part of the computation of the matrix-vector product, the SVD-FMM technique greatly reduces the computational complexity compared with the standard approach. We develop a novel possible parallelization scheme of the SVD-FMM for our application, which can reduce the communication costs. We investigate the accuracy of the SVD-FMM technique in several numerical experiments: we first assess the accuracy using covariance matrices that are created using different correlation functions and lengthscales; then investigate the impact of reconditioning the covariance matrices on the accuracy; and finally examine the feasibility of the technique in the presence of missing observations. We also provide theoretical explanations for some numerical results. Our results show that the SVD-FMM technique has potential as an efficient technique for assimilation of a large volume of observational data within a short time interval.
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