A numerical method for solving linear systems in the preconditioned Crankxe2x80x93Nicolson algorithm Abstract   The preconditioned Crankxe2x80x93Nicolson (pCN) algorithm speedups the convergence of MarkovChainMonteCarlo methods to high probability zones of target distributions. This method involves the solution of linear systems to propose candidates, which can be critical for a large number of variables to estimate. Thus, this paper proposes an iterative method that avoids the direct inversion of large matrices. The convergence of the proposed numerical method is theoretically proven. Besides, an error bound is provided to approximate candidates with a predefined accuracy. Experimental tests are performed on a nonlinear Data Assimilation problem and the Lorenz96 model. The results reveal that the use of our proposed iterative method does not impact the quality of candidates in the pCN method and therefore, posterior errors can be decreased by order of magnitudes for full observational errors.

A Maximum Likelihood Ensemble Filter via a Modified Cholesky Decomposition for NonGaussian Data Assimilation This paper proposes an efficient and practical implementation of the Maximum Likelihood Ensemble Filter via a Modified Cholesky decomposition (MLEFMC). The method works as follows: via an ensemble of model realizations, a wellconditioned and fullrank squareroot approximation of the background error covariance matrix is obtained. This squareroot approximation serves as a control space onto which analysis increments can be computed. These are calculated via LineSearch (LS) optimization. We theoretically prove the convergence of the MLEFMC. Experimental simulations were performed using an Atmospheric General Circulation Model (ATGCM) and a highly nonlinear observation operator. The results reveal that the proposed method can obtain posterior error estimates within reasonable accuracies in terms of     l  2     error norms. Furthermore, our analysis estimates are similar to those of the MLEF with large ensemble sizes and full observational networks.