S. Gürol

S. Gratton, S. Gürol, E. Simon et al.

This paper discusses the practical use of the saddle variational formulation for the weakly-constrained 4D-VAR method in data assimilation. It is shown that the method, in its original form, may produce erratic results or diverge because of the inherent lack of monotonicity of the produced objective function values. Convergent, variationaly coherent variants of the algorithm are then proposed whose practical performance is compared to that of other formulations. This comparison is conducted on two data assimilation instances (Burgers equation and the Quasi-Geostrophic model), using two different assumptions on parallel computing environment. Because these variants essentially retain the parallelization advantages of the original proposal, they often --- but not always --- perform best, even for moderate numbers of computing processes.

I. Daužickaitė, M. A. Freitag, S. Gürol et al.

Variational data assimilation is a technique for combining measured data with dynamical models. It is a key component of Earth system state estimation and is commonly used in weather and ocean forecasting. The approach involves a large-scale generalized nonlinear least-squares problem. Solving the resulting sequence of sparse linear subproblems requires the use of sophisticated numerical linear algebra methods. In practical applications, the computational demands severely limit the number of iterations of a Krylov subspace solver that can be performed and so high-quality preconditioners are vital. In this paper, we present a numerical linear algebra perspective on variational data assimilation and discuss contemporary solution methods for the challenges posed by large-scale geophysical applications. The principal contribution is a focused treatment of the underlying linear algebraic subproblems, accompanied by a concise and clear introduction to the essential concepts of variational data assimilation and an extensive bibliography.

I. Daužickaitė, S. Gürol, M. Destouches et al.

Ensembles of variational data assimilations (EDA) require solving systems of linear equations with iterative methods. The solution process can be accelerated using a limited memory preconditioner constructed with approximations of the leading eigenpairs of the Hessian matrix. Randomized methods for low-rank matrix approximations provide a feasible approach for computing these eigenpairs. These methods use a random sketching matrix to obtain a low-rank representation of the Hessian matrix, which is then used for computing the eigendecomposition. The sketching matrix highly influences the quality of the approximation. In this paper, we show how the structure of the EDA can be exploited to construct a suitable sketching matrix, i.e., using the differences of the right-hand sides of the linear systems of equations. Idealised numerical experiments with the Lorenz-96 model show that the resulting preconditioner is able to accelerate the EDA solution process for all ensemble members, even if constructed from the control member only.