Efficient computation of matrix-vector products with full observation weighting matrices in data assimilation

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.