Accelerating an ensemble of variational data assimilations with randomized preconditioning
For practitioners of variational data assimilation, this provides a computationally efficient preconditioning method that leverages ensemble structure to reduce iterative solver costs.
The paper accelerates ensembles of variational data assimilations by using a randomized preconditioner built from Hessian eigenpairs, exploiting EDA structure to construct sketching matrices. Idealized Lorenz-96 experiments show the preconditioner accelerates all ensemble members, even when built from the control member only.
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.