Householder orthogonalization with a non-standard inner product

Householder orthogonalization plays an important role in numerical linear algebra. It attains perfect orthogonality regardless of the conditioning of the input. However, in the context of a non-standard inner product, it becomes difficult to apply Householder orthogonalization due to the lack of an initial orthogonal basis. We propose strategies to overcome this obstacle and discuss algorithms and variants of Householder orthogonalization with a non-standard inner product. Rounding error analysis and numerical experiments demonstrate that our approach is numerically stable

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 − λ B $A-\lambda B$ , that is, A, B are Hermitian and there is a shift λ 0 $\lambda _{0}$ such that A − λ 0 B $A-\lambda _{0} B$ 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

Self-patterning Gd nano-fibers in Mg-Gd alloys

Manipulating the shape and distribution of strengthening units, e.g. particles, fibers, and precipitates, in a bulk metal, has been a widely applied strategy of tailoring their mechanical properties. Here, we report self-assembled patterns of Gd nano-fibers in Mg-Gd alloys for the purpose of improving their strength and deformability. 1-nm Gd nano-fibers, with a 〈c〉-rod shape, are formed and hexagonally patterned in association with Gd segregations along dislocations that nucleated during hot extrusion. Such Gd-fiber patterns are able to regulate the relative activities of slips and twinning, as a result, overcome the inherent limitations in strength and ductility of Mg alloys. This nano-fiber patterning approach could be an effective method to engineer hexagonal metals

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