There is growing awareness that errors in the model equations cannot be ignored in data assimilation methods such as four-dimensional variational assimilation (4D-Var). If allowed for, more information can be extracted from observations, longer time windows are possible, and the minimisation process is easier, at least in principle. Weak constraint 4D-Var estimates the model error and minimises a series of linear least-squares cost functions using the conjugate gradient (CG) method; minimising each cost function is called an inner loop. CG needs preconditioning to improve its performance. In previous work, limited memory preconditioners (LMPs) have been constructed using approximations of the eigenvalues and eigenvectors of the Hessian in the previous inner loop. If the Hessian changes significantly in consecutive inner loops, the LMP may be of limited usefulness. To circumvent this, we propose using randomised methods for low rank eigenvalue decomposition and use these approximations to cheaply construct LMPs using information from the current inner loop. Three randomised methods are compared. Numerical experiments in idealized systems show that the resulting LMPs perform better than the existing LMPs. Using these methods may allow more efficient and robust implementations of incremental weak constraint 4D-Var.

翻译：人们日益认识到,模型方程式中的错误不能在四维变异同化(4D-Var)等数据同化方法中被忽视,模型方程式中的错误不能被忽略。如果允许的话,可以从观测中提取更多的信息,可以延长时间窗口,最小化过程至少原则上比较容易。 弱点限制 4D-Var 估计模型错误,并使用共振梯度(CG) 方法将一系列线性最小方方程式成本功能最小化; 最小化每个成本函数称为内部循环。 CG 需要以改进其性能为先决条件。 在以往的工作中, 有限的记忆先决条件(LMP) 已经用前一内部循环中赫萨赫萨的埃根值和精子精度近似值来构建。 如果赫萨在连续的内环中发生重大变化, LMP 作用可能有限。 为了规避这一点, 我们提议使用随机化方法来降低低级的egen值分值分解, 并使用这些近似方法来廉价地构建LMP 。 三个随机化的方法是比较了前内循环中赫萨亚的希萨亚氏精准值, 在理想化系统中进行较强的LMP4的递化实验。