8 research outputs found
Structure Preserving Parallel Algorithms for Solving the Bethe-Salpeter Eigenvalue Problem
The Bethe-Salpeter eigenvalue problem is a dense structured eigenvalue
problem arising from discretized Bethe-Salpeter equation in the context of
computing exciton energies and states. A computational challenge is that at
least half of the eigenvalues and the associated eigenvectors are desired in
practice. We establish the equivalence between Bethe-Salpeter eigenvalue
problems and real Hamiltonian eigenvalue problems. Based on theoretical
analysis, structure preserving algorithms for a class of Bethe-Salpeter
eigenvalue problems are proposed. We also show that for this class of problems
all eigenvalues obtained from the Tamm-Dancoff approximation are overestimated.
In order to solve large scale problems of practical interest, we discuss
parallel implementations of our algorithms targeting distributed memory
systems. Several numerical examples are presented to demonstrate the efficiency
and accuracy of our algorithms
Convergence of the Eberlein diagonalization method under the generalized serial pivot strategies
The Eberlein method is a Jacobi-type process for solving the eigenvalue
problem of an arbitrary matrix. In each iteration two transformations are
applied on the underlying matrix, a plane rotation and a non-unitary elementary
transformation. The paper studies the method under the broad class of
generalized serial pivot strategies. We prove the global convergence of the
Eberlein method under the generalized serial pivot strategies with permutations
and present several numerical examples.Comment: 16 pages, 3 figure
Fast iterative solution of the Bethe–Salpeter eigenvalue problem using low-rank and QTT tensor approximation
In this paper, we propose and study two approaches to approximate the solution of the Bethe–Salpeter equation (BSE) by using structured iterative eigenvalue solvers. Both approaches are based on the reduced basis method and low-rank factorizations of the generating matrices. We also propose to represent the static screen interaction part in the BSE matrix by a small active sub-block, with a size balancing the storage for rank-structured representations of other matrix blocks. We demonstrate by various numerical tests that the combination of the diagonal plus low-rank plus reduced-block approximation exhibits higher precision with low numerical cost, providing as well a distinct two-sided error estimate for the smallest eigenvalues of the Bethe–Salpeter operator. The complexity is reduced to O(Nb 2) in the size of the atomic orbitals basis set, Nb, instead of the practically intractable O(Nb 6) scaling for the direct diagonalization. In the second approach, we apply the quantized-TT (QTT) tensor representation to both, the long eigenvectors and the column vectors in the rank-structured BSE matrix blocks, and combine this with the ALS-type iteration in block QTT format. The QTT-rank of the matrix entities possesses almost the same magnitude as the number of occupied orbitals in the molecular systems, Nob, hence the overall asymptotic complexity for solving the BSE problem by the QTT approximation is estimated by O(log(No)No 2). We confirm numerically a considerable decrease in computational time for the presented iterative approaches applied to various compact and chain-type molecules, while supporting sufficient accuracy.</p
Implicit QR algorithms for palindromic and even eigenvalue problems
In the spirit of the Hamiltonian QR algorithm and other bidirectional chasing algorithms, a structure-preserving variant of the implicit QR algorithm for palindromic eigenvalue problems is proposed. This new palindromic QR algorithm is strongly backward stable and requires less operations than the standard QZ algorithm, but is restricted to matrix classes where a preliminary reduction to structured Hessenberg form can be performed. By an extension of the implicit Q theorem, the palindromic QR algorithm is shown to be equivalent to a previously developed explicit version. Also, the classical convergence theory for the QR algorithm can be extended to prove local quadratic convergence. We briefly demonstrate how even eigenvalue problems can be addressed by similar techniques. © 2008 Springer Science+Business Media, LLC
Fast iterative solution of the Bethe–Salpeter eigenvalue problem using low-rank and QTT tensor approximation
In this paper, we study and implement the structural iterative eigensolvers
for the large-scale eigenvalue problem in the Bethe-Salpeter equation (BSE)
based on the reduced basis approach via low-rank factorizations in generating
matrices, introduced in the previous paper. The approach reduces numerical
costs down to in the size of atomic orbitals basis set,
, instead of practically intractable complexity
scaling for the direct diagonalization of the BSE matrix. As an alternative to
rank approximation of the static screen interaction part of the BSE matrix, we
propose to restrict it to a small active sub-block, with a size balancing the
storage for rank-structured representations of other matrix blocks. We
demonstrate that the enhanced reduced-block approximation exhibits higher
precision within the controlled numerical cost, providing as well a distinct
two-sided error estimate for the BSE eigenvalues. It is shown that further
reduction of the asymptotic computational cost is possible due to ALS-type
iteration in block tensor train (TT) format applied to the quantized-TT (QTT)
tensor representation of both long eigenvectors and rank-structured matrix
blocks. The QTT-rank of these entities possesses almost the same magnitude as
the number of occupied orbitals in the molecular systems, , hence the
overall asymptotic complexity for solving the BSE problem can be estimated by
. We confirm numerically a considerable
decrease in computational time for the presented iterative approach applied to
various compact and chain-type molecules, while supporting sufficient accuracy.Comment: 23 pages, 11 figure
ON ASYMPTOTIC CONVERGENCE OF NONSYMMETRIC JACOBI ALGORITHMS
Abstract. The asymptotic convergence behavior of cyclic versions of the nonsymmetric Jacobi algorithm for the computation of the Schur form of a general complex matrix is investigated. Similar to the symmetric case, the nonsymmetric Jacobi algorithm proceeds by applying a sequence of rotations that annihilate a pivot element in the strict lower triangular part of the matrix until convergence to the Schur form of the matrix is achieved. In this paper, it is shown that the cyclic nonsymmetric Jacobi method converges locally and asymptotically quadratically under mild hypotheses if special ordering schemes are chosen, namely ordering schemes that lead to so-called northeast directed sweeps. The theory is illustrated by the help of numerical experiments. In particular, it is shown that there are ordering schemes that lead to asymptotic quadratic convergence for the cyclic symmetric Jacobi method, but only to asymptotic linear convergence for the cyclic nonsymmetric Jacobi method. Finally, a generalization of the nonsymmetric Jacobi method to the computation of the Hamiltonian Schur form for Hamiltonian matrices is introduced and investigated