A preconditioner based on a low-rank approximation with applications to topology optimization

Probabilistic algorithms for constructing matrix decompositions have recently gained vast popularity due to their ability to handle very large problems. This paper pursues the idea of improving a standard Jacobi preconditioner with a low rank correction, to enhance its performance for the iterative solution of finite element discretizations. This is based on the well-known fact that, in the case of elliptic problems, the Green function is numerically low-rank. We follow the same approach in the context of wave propagation problems. Although the Green function is no longer low-rank, we show that, in the case of moderate wave numbers, a low-rank update can be successfully employed to construct an effective preconditioner. We investigate the behavior of such preconditioner when applied to subsequent discretizations of the Helmholtz equation that arise in the context of topology optimization.