Mathematics Colloquia and Seminars

Return to Colloquia & Seminar listing

Iterative algorithms for computing eigenpairs of large matrices consist of two main steps: a subspace update step and a Rayleigh-Ritz (RR) step. In this paper, we propose an augmented Rayleigh-Ritz (ARR) step that can provably accelerate convergence under mild conditions. We consider two block (as opposed to Krylov subspace) algorithms by coupling the ARR procedure with two subspace update schemes: (i) the classic power method applied to multiple vectors without periodic orthogonalization, and (ii) a recently proposed Gauss-Newton method. In block algorithms, the RR step is arguably the bottleneck in scalability as the number of computed eigenpairs increases. Our key design objective is for the algorithms to approach a certain optimal scalability under favorable conditions. That is, they should ideally call the augmented Rayleigh-Ritz step once and attain a sufficient accuracy, while the subspace update step is close to being embarrassingly parallel under suitable data mapping schemes. We perform extensive computational experiments in Matlab (without explicit code parallelization) to evaluate the proposed algorithms in comparison to a few state-of-the-art eigensolvers. Numerical results show strong potentials for the proposed algorithms to reach high levels of scalability on a wide range of problems.