A multilevel approach for computing the limited-memory Hessian and its inverse in variational data assimilation

Use of data assimilation techniques is becoming increasingly common across many application areas. The inverse Hessian (and its square root) plays an important role in several different aspects of these processes. In geophysical and engineering applications, the Hessian-vector product is typically defined by sequential solution of a tangent linear and adjoint problem; for the inverse Hessian, however, no such definition is possible. Frequently, the requirement to work in a matrix-free environment means that compact representation schemes are employed. In this paper, we propose an enhanced approach based on a new algorithm for constructing a multilevel eigenvalue decomposition of a given operator, which results in a much more efficient compact representation of the inverse Hessian (and its square root). After introducing these multilevel approximations, we investigate their accuracy and demonstrate their efficiency (in terms of reducing memory requirements and/or computational time) using the example of preconditioning a Gauss-Newton minimisation procedure.