Block iterative eigensolvers for sequences of correlated eigenvalue problems

Abstract

In Density Functional Theory simulations based on the LAPW method, each self-consistent field 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 [7], it was demonstrated that, contrary to belief, successive eigenproblems in a sequence are strongly correlated with one another. In particular, by monitoring the 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 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. We provide numerical examples where opportunely selected block iterative eigensolvers benefit from the reuse of eigenvectors by achieving a substantial speed-up. The results presented will eventually open the way to a widespread use of block iterative eigensolvers in ab initio electronic structure codes based on the LAPW approach.

High-performance generation of the Hamiltonian and Overlap matrices in FLAPW methods

“…Listing 5: H += H AB+BAThis algorithm performs a total of 8NA N 2 L N G + 8N A N L N 2 G FLOPs.is Hermitian, so is this entire term. Using this property, we can carefully rewrite H BB in a similar way as we did for H AB+BA in Eq (19). above:…”

An optimized and scalable eigensolver for sequences of eigenvalue problems

“…Because we are provided with as many approximate vectors as the dimension of the sought-after eigenspace, the iterative eigensolver of choice should be able to simultaneously accept multiple vectors as input. Such a feature is in part provided by the class of block iterative eigensolvers [16][17][18][19][20][21][22], among which one has to select the solver that maximizes the number of input vectors and exploits to the maximum extent the approximate guess they provide [23]. Consequently, the class of block solvers is dramatically restricted to subspace iteration algorithms.…”

“…Here, we report only the latter. The interested reader can find details on an initial ** OpenMP-based shared memory implementation in [23,33]. The parallel distributed memory version is implemented, using the MPI, on top of Elemental, a (relatively new) distributed memory dense linear algebra library [34].…”

A Parallel and Scalable Iterative Solver for Sequences of Dense Eigenproblems Arising in FLAPW

“…In particular we need a block iterative eigensolver that accepts at the same time many vectors as input. Among the many choices of block solvers, the Chebyshev Filtered Subspace Iteration method (ChFSI) showed the highest potential to take advantage of approximate eigenvectors [14](see also Fig. 1(b)).…”