In some Density Functional Theory based simulations each self-consistent cycle comprises dozens of large dense generalized eigenproblems. In  a recent study  , it was proposed to  consider simulations as made of dozens of sequences of eigenvalue problems , where each sequence groups together eigenproblems with equal {\bf k}-vectors and an increasing outer-iteration cycle index i. It was then demonstrated that successive eigenproblems in a sequence are strongly correlated to one another. In particular, by tracking the evolution of subspace angles between eigenvectors of successive eigenproblems, it was shown that these angles decrease noticeably after the first few iterations and become close to collinear  . This last result suggests that we could manipulate the eigenvectors, solving for a specific eigenproblem in a sequence, as an approximate solution for the following eigenproblem. In the present work we present a set of preliminary results confirming this initial intuition. We provide numerical examples where opportunely selected block iterative solvers benefit from the reuse of eigenvectors by achieving a substantial speed-up.  All the numerical tests are run employing sequences of eigenproblems extracted from simulations of real-world materials . The results presented here could eventually open the way to a widespread use of block iterative solvers in ab initio electronic structure codes even when dealing with dense eigenproblems .
In Density Functional Theory simulations based on the LAPW method, each self-consistent cycle comprises dozens of large dense generalized eigenproblems. In contrast to real-space methods, eigenpairs solving for problems at distinct cycles have either been believed to be independent or at most very loosely connected. In a recent study \mbox{%DIFAUXCMD DBB , it was proposed to revert this point of view and consider simulations as made of dozens of sequences of eigenvalue problems ; each sequence groups together eigenproblems with equal {\bf k}-vectors and an increasing outer-iteration cycle index \ell. From this different standpoint it was possible to demonstrate that, contrary to belief, successive eigenproblems in a sequence are strongly correlated with one another. In particular, by tracking the evolution of subspace angles between eigenvectors of successive eigenproblems, it was shown that these angles decrease noticeably after the first few iterations and become close to collinear : the closer to convergence the stronger the correlation becomes . This last result suggests that we can manipulate the eigenvectors, solving for a specific eigenproblem in a sequence, as an approximate solution for the following eigenproblem. In this work we present results that are in line with this intuition. First, we provide numerical examples where opportunely selected block iterative solvers benefit from the reuse of eigenvectors by achieving a substantial speed-up. We then develop a C language version of one of these algorithms and run a series of tests specifically focused on performance and scalability. All the numerical tests are carried out employing sequences of eigenproblems extracted from simulations of solid-state physics crystals . The results presented here could eventually open the way to a widespread use of block iterative solvers in ab initio electronic structure codes based on the LAPW approach .
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