Toward genuine efficiency and cluster robustness of preconditioned CG-like eigensolvers

The performance of eigenvalue problem solvers (eigensolvers) depends on various factors such as preconditioning and eigenvalue distribution. Developing stable and rapidly converging vectorwise eigensolvers is a crucial step in improving the overall efficiency of their blockwise implementations. The present paper is concerned with the locally optimal block preconditioned conjugate gradient (LOBPCG) method for Hermitian eigenvalue problems, and motivated by two recently proposed alternatives for its single-vector version LOPCG. A common basis of these eigensolvers is the well-known CG method for linear systems. However, the optimality of CG search directions cannot perfectly be transferred to CG-like eigensolvers. In particular, while computing clustered eigenvalues, LOPCG and its alternatives suffer from frequent delays, leading to a staircase-shaped convergence behavior which cannot be explained by the existing estimates. Keeping this in mind, we construct a class of cluster robust vector iterations where LOPCG is replaced by asymptotically equivalent two-term recurrences and the search directions are timely corrected by selecting a far previous iterate as augmentation. The new approach significantly reduces the number of required steps and the total computational time.