Related papers: Structure Preserving Parallel Algorithms for Solvi…
The Bethe-Salpeter eigenvalue problem is a structured eigenvalue problem arising in many-body physics. In practice, a few of the smallest positive eigenvalues and the corresponding eigenvectors need to be computed. In principle, the LOBPCG…
The discretized Bethe-Salpeter eigenvalue problem arises in the Green's function evaluation in many body physics and quantum chemistry. Discretization leads to a matrix eigenvalue problem for $H \in \mathbb{C}^{2n\times 2n}$ with a…
To solve the Bethe-Salpeter eigenvalue problem with distinct sizes, two efficient methods, called {\Gamma}QR algorithm and {\Gamma}-Lanczos algorithm, are proposed in this paper. Both algorithms preserve the special structure of the initial…
In (relativistic) electronic structure methods, the quaternion matrix eigenvalue problem and the linear response (Bethe-Salpeter) eigenvalue problem for excitation energies are two frequently encountered structured eigenvalue problems.…
The non-Hermitian Bethe-Salpeter eigenvalue problem, in the definite case, is a structured eigenproblem, with real eigenvalues coming in pairs $\{\lambda,-\lambda\}$ where the corresponding pair of eigenvectors are closely related, and…
We analyze the performance of two strategies in solving the structured eigenvalue problem deriving from the Bethe-Salpeter equation (BSE) in condensed matter physics. The BSE matrix is constructed with the Yambo code, and the two strategies…
The linear response eigenvalue problem, which arises from many scientific and engineering fields, is quite challenging numerically for large-scale sparse/dense system, especially when it has zero eigenvalues. Based on a direct sum…
Solving large-scale eigenvalue problems poses a significant challenge due to the computational complexity and limitations on the parallel scalability of the orthogonalization operation, when many eigenpairs are required. In this paper, we…
Optical properties of materials related to light absorption and scattering are explained by the excitation of electrons. The Bethe-Salpeter equation is the state-of-the-art approach to describe these processes from first principles (ab…
A hyperbolic system must have a set of linearly independent eigenvectors and corresponding real eigenvalues. In numerical simulations, however, the eigenvalues can be complex because truncation errors pollute a characteristic polynomial of…
We present a new structure preserving Lanczos algorithm for approximating the optical absorption spectrum in the context of solving full Bethe--Salpeter equation without Tamm--Dancoff approximation. The new algorithm is based on a structure…
The history of research on eigenvalue problems is rich with many outstanding contributions. Nonetheless, the rapidly increasing size of data sets requires new algorithms for old problems in the context of extremely large matrix dimensions.…
Many large-scale scientific computations require eigenvalue solvers in a scaling regime where efficiency is limited by data movement. We introduce a parallel algorithm for computing the eigenvalues of a dense symmetric matrix, which…
A new approach to solving eigenvalue optimization problems for large structured matrices is proposed and studied. The class of optimization problems considered is related to computing structured pseudospectra and their extremal points, and…
We present a high-performance solver for dense skew-symmetric matrix eigenvalue problems. Our work is motivated by applications in computational quantum physics, where one solution approach to solve the so-called Bethe-Salpeter equation…
In Parts I and II of this series of papers, three new methods for the computation of eigenvalues of singular pencils were developed: rank-completing perturbations, rank-projections, and augmentation. It was observed that a straightforward…
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,…
We discuss the effect of structure-preserving perturbations on complex or real Hamiltonian eigenproblems and characterize the structured worst-case effect perturbations. We derive significant expressions for both the structured condition…
In this paper, we study a class of set cover problems that satisfy a special property which we call the {\em small neighborhood cover} property. This class encompasses several well-studied problems including vertex cover, interval cover,…
Energy methods for constructing time-stepping algorithms are of increased interest in application to nonlinear problems, since numerical stability can be inferred from the conservation of the system energy. Alternatively, symplectic…