Data assimilation algorithms combine information from observations and prior model information to obtain the most likely state of a dynamical system. The linearised weak-constraint four-dimensional variational assimilation problem can be reformulated as a saddle point problem, in order to exploit highly-parallel modern computer architectures. In this setting, the choice of preconditioner is crucial to ensure fast convergence and retain the inherent parallelism of the saddle point formulation. We propose new preconditioning approaches for the model term and observation error covariance term which lead to fast convergence of preconditioned Krylov subspace methods, and many of these suggested approximations are highly parallelisable. In particular our novel approach includes model information in the model term within the preconditioner, which to our knowledge has not previously been considered for data assimilation problems. We develop new theory demonstrating the effectiveness of the new preconditioners for a specific class of problems. Linear and non-linear numerical experiments reveal that our new approach leads to faster convergence than existing state-of-the-art preconditioners for a broader range of problems than indicated by the theory alone. We present a range of numerical experiments performed in serial, with further improvements expected if the highly parallelisable nature of the preconditioners is exploited.

翻译：数据同化算法结合了来自观测和先前模型信息的信息,以获得最有可能的动态系统状态。线性弱度限制四维变异同化问题可以重新拟订为支撑点问题,以便利用高度平行的现代计算机结构。在这种背景下,选择先决条件对于确保快速趋同和保留马鞍配方固有的平行性至关重要。我们提出了示范术语和观察误差的新的先决条件性办法,导致先质Krylov子空间方法迅速趋同,其中许多建议近似值高度平行。特别是,我们的新办法包括了先质中模型术语中的模型信息,而据我们所知,在数据同化问题上我们以前没有考虑过。我们制定了新的理论,表明新的先质对特定类别问题的有效性。线性和非线性数字实验表明,我们的新办法比现有的状态和先质的前提更快地趋同了比理论本身所显示的更广泛的问题范围。我们提出了一系列的数值实验,在序列中作了进一步的改进,如果高度平行性的先决条件是可加以利用的话,则预期会得到进一步的改进。