关于预条件化加权线性最小二乘的注记，及其对弱约束变分数据同化的影响

加速变分数据同化集合的随机预处理方法

Accelerating an ensemble of variational data assimilations with randomized preconditioning

针对变分数据同化集合（EDA）求解线性方程组效率低的问题，提出利用随机低秩矩阵近似构造有限内存预处理器，通过利用右端项差异构造草图矩阵，有效加速所有集合成员的求解过程。

Randomised preconditioning for the forcing formulation of weak constraint 4D-Var

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 functionsfunctions, which can be achieved using the conjugate gradient (CG) method; minimising each cost function is called an inner loop. CG needs preconditioning …

On time-parallel preconditioning for the state formulation of incremental weak constraint 4D-Var

Using a high degree of parallelism is essential to perform data assimilation efficiently. The state formulation of the incremental weak constraint four-dimensional variational data assimilation method allows parallel calculations in the time dimension. In this approach, the solution is approximated by minimising a series of quadratic cost functions using the conjugate gradient method. To use this method in practice, effective preconditioning strategies that maintain the potential for parallel calculations are needed. We examine approximations to the control variable transform (CVT) technique w…

关于在弱约束4D-VAR数据同化中使用鞍点公式的应用

On the use of the saddle formulation in weakly-constrained 4D-VAR data assimilation

本文讨论了在数据同化中弱约束4D-VAR方法中使用鞍点变分公式在实际中的应用。研究表明，原始形式的方法由于生成的目标函数值缺乏单调性，可能导致结果异常或发散。随后提出了收敛且变分一致的算法变体，并将其性能与其他公式进行比较。该比较在两个数据同化实例（Burgers方程和准地转模型）中进行，使用了两种不同的并行计算环境假设。由于这些变体本质上保留了原始提案的并行化优势，因此它们通常但不总是表现最好，即使在中等数量的计算过程中。

A parallel-in-time two-sided preconditioning for all-at-once system from a non-local evolutionary equation with weakly singular kernel

In this paper, we study a parallel-in-time (PinT) algorithm for all-at-once system from a non-local evolutionary equation with weakly singular kernel where the temporal term involves a non-local convolution with a weakly singular kernel and the spatial term is the usual Laplacian operator with variable coefficients. We propose to use a two-sided preconditioning technique for the all-at-once discretization of the equation. Our preconditioner is constructed by replacing the variable diffusion coefficients with a constant coefficient to obtain a constant-coefficient all-at-once matrix. We split a…

针对具有弱奇异核的非局部演化方程线性系统的单侧全预条件

A single-sided all-at-once preconditioning for linear system from a non-local evolutionary equation with weakly singular kernels

本文将非局部演化方程的双侧预条件合并为单侧预条件，通过快速可对角化矩阵近似空间矩阵，理论上证明单侧预条件GMRES方法收敛速率不劣于双侧，且总计算时间更少。