Preconditioner Design via Bregman Divergences

We study a preconditioner for a Hermitian positive definite linear system, which is obtained as the solution of a matrix nearness problem based on the Bregman log determinant divergence. The preconditioner is of the form of a Hermitian positive definite matrix plus a low-rank matrix. For this choice of structure, the generalized eigenvalues of the preconditioned matrix are easily calculated, and we show under which conditions the preconditioner minimizes the ℓ2 condition number of the preconditioned matrix. We develop practical numerical approximations of the preconditioner based on the randomized singular value decomposition (SVD) and the Nyström approximation and provide corresponding approximation results. Furthermore, we prove that the Nyström approximation is in fact also a matrix approximation in a range-restricted Bregman divergence and establish several connections between this divergence and matrix nearness problems in different measures. Numerical examples are provided to support the theoretical results.