发表机构
School of Geography, Earth and Atmospheric Sciences, University of Melbourne; Bureau of Meteorology（墨尔本大学地理、地球与大气科学学院; 澳大利亚气象局）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种快速可扩展算法，通过奇异值分解计算水平局地化集合协方差矩阵平方根的左逆，支持并行计算，在合成实验中精度高，比传统算法快9个数量级，并应用于构建新型正定混合协方差模型。
AI 中文摘要
使用已知算法计算矩阵逆（或左逆）在高维情况下（如业务天气预报背景）通常计算代价高昂。本文介绍了一种快速、可扩展的算法，通过奇异值分解技术计算水平局地化集合协方差矩阵平方根的左逆。在该算法中，涉及不同网格垂直列中模型变量的计算可以分开进行。这一特性使得算法可扩展，允许并行计算。该算法在集合成员数等于每个网格垂直列中模型变量数（对于当前高分辨率模型，该数量约为10^3）的情况下开发。算法在三维网格上的合成实验中进行了测试，发现其具有非常好的精度。此外，还介绍了该算法的一个版本，输出通用向量与上述左逆的左乘结果。对于业务应用，如果在单处理器上执行，该版本算法已比传统算法快9个数量级，并且使用多个并行处理器时，可扩展到处理器数量达到约10^6。此外，指出了该算法的一个应用：执行从模型变量到替代变量集（本文定义为标准化变量）的变换，这些变量在气候学上不相关，且每个变量具有单位气候误差方差。进一步指出，标准化变量允许构建新的混合协方差模型，该模型是正定的、气候学上无偏的，并且具有地理上依赖的气候误差协方差矩阵。
英文摘要
Computing matrix inverses (or, e.g., left inverses) using known algorithms is in general computationally prohibitive in high dimension, such as in an operational weather forecasting context. In this article, a swift, scalable algorithm to calculate the left inverse (through singular values decompositions techniques) of a square root of a horizontally localized ensemble covariance matrix is introduced. In this algorithm, calculations involving model variables located in different grid vertical columns can be performed separately. This is the aspect that makes this algorithm scalable by allowing for parallel computing. The algorithm is developed in the case when the number of ensemble members equals the number of model variables in each grid vertical column (this number is of order $10^3$ for current high-resolution models). The algorithm is tested in a synthetic experiment on a three-dimensional grid, and it is found to have very good accuracy. Additionally, a version of the algorithm outputting the left-multiplication of a generic vector by that above-mentioned left inverse is also introduced. For operational applications, if executed on a single processor, that version of the algorithm is already $9$ orders of magnitude faster than a traditional algorithm, and, with multiple, parallel processors, it scales until the number of processors reaches the order of $10^6$. In addition, an application of the algorithm is pointed out: performing the transform from model variables to an alternative set of variables, defined herein as standardized variables, being climatologically uncorrelated and each having unit climatological error variance. Furthermore, it is pointed out that the standardized variables allow constructing a new Hybrid covariance model, which is positive definite and climatologically unbiased, and it features a geographically dependent climatological error covariance matrix.
CommentsSubmitted to the Quarterly Journal of the Royal Meteorological Society
