Meiyue Shao

Structure Preserving Parallel Algorithms for Solving the Bethe{Salpeter Eigenvalue Problem

Abstract 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.

A Supernodal Approach to Incomplete LU Factorization with Partial Pivoting

We present a new supernode-based incomplete LU factorization method to construct a preconditioner for solving sparse linear systems with iterative methods. The new algorithm is primarily based on the ILUTP approach by Saad, and we incorporate a number of techniques to improve the robustness and performance of the traditional ILUTP method. These include new dropping strategies that accommodate the use of supernodal structures in the factored matrix and an area-based fill control heuristic for the secondary dropping strategy. We present numerical experiments to demonstrate that our new method is competitive with the other ILU approaches and is well suited for modern architectures with memory hierarchy.

An indefinite variant of LOBPCG for definite matrix pencils

In this paper, we propose a novel preconditioned solver for generalized Hermitian eigenvalue problems. More specifically, we address the case of a definite matrix pencil A−λB

Factorization-based sparse solvers and preconditioners

A-\lambda B

On aggressive early deflation in parallel variants of the QR algorithm

, that is, A, B are Hermitian and there is a shift λ0

Aggressively Truncated Taylor Series Method For Accurate Computation Of Exponentials Of Essentially Nonnegative Matrices

\lambda _{0}

A Structure Preserving Lanczos Algorithm for Computing the Optical Absorption Spectrum

such that A−λ0B

Accelerating Nuclear Configuration Interaction Calculations through a Preconditioned Block Iterative Eigensolver

A-\lambda _{0} B

Algorithm 953: Parallel Library Software for the Multishift QR Algorithm with Aggressive Early Deflation

is definite. Our new method can be seen as a variant of the popular LOBPCG method operating in an indefinite inner product. It also turns out to be a generalization of the recently proposed LOBP4DCG method by Bai and Li for solving product eigenvalue problems. Several numerical experiments demonstrate the effectiveness of our method for addressing certain product and quadratic eigenvalue problems.

A robust and efficient implementation of LOBPCG

Ecient solution of large-scale, ill-conditioned and highly-indefinite algebraic equations often relies on high quality preconditioners together with iterative solvers. Because of their robustness, the factorization-based algorithms could play a significant role when they are combined with iterative methods, particularly in the development of scalable solvers. We present our recent work in using the direct solver SuperLU code base to develop a new supernode-based ILU preconditioner and a domain-decomposition hybrid solver. Our ILU preconditioner is a modification of the classic ILUTP approach, incorporating a number of techniques to improve robustness and performance, which include new dropping strategies that accommodate the use of supernodal structure in the factored matrix. Our hybrid solver is based on the Schur complement method. We use parallel graph partitioning to obtain hierarchical interface/domain decomposition, and multiple parallel direct solvers to solve the subdomain problems simultaneously, and parallel preconditioned iterative solvers to solve the interface problem. We will demonstrate the eectiveness