A Random Line-Search Optimization Method via Modified Cholesky Decomposition for Non-linear Data Assimilation

This paper proposes a line-search optimization method for non-linear data assimilation via random descent directions. The iterative method works as follows: at each iteration, quadratic approximations of the Three-Dimensional-Variational (3D-Var) cost function are built about current solutions. These approximations are employed to build sub-spaces onto which analysis increments can be estimated. We sample search-directions from those sub-spaces, and for each direction, a line-search optimization method is employed to estimate its optimal step length. Current solutions are updated based on directions along which the 3D-Var cost function decreases faster. We theoretically prove the global convergence of our proposed iterative method. Experimental tests are performed by using the Lorenz-96 model, and for reference, we employ a Maximum-Likelihood-Ensemble-Filter (MLEF) whose ensemble size doubles that of our implementation. The results reveal that, as the degree of observational operators increases, the use of additional directions can improve the accuracy of results in terms of \(\ell _2\)-norm of errors, and even more, our numerical results outperform those of the employed MLEF implementation.