Solution of the k-th eigenvalue problem in large-scale electronic structure calculations

Abstract

We consider computing the k-th eigenvalue and its corresponding eigenvector of a generalized Hermitian eigenvalue problem of n × n large sparse matrices. In electronic structure calculations, several properties of materials, such as those of optoelectronic device materials, are governed by the eigenpair with a material-specific index k. We present a three-stage algorithm for computing the k-th eigenpair with validation of its index. In the first stage of the algorithm, we propose an efficient way of finding an interval containing the k-th eigenvalue (1 k n) with a non-standard application of the Lanczos method. In the second stage, spectral bisection for large-scale problems is realized using a sparse direct linear solver to narrow down the interval of the k-th eigenvalue. In the third stage, we switch to a modified shift-and-invert Lanczos method to reduce bisection iterations and compute the k-th eigenpair with validation. Numerical results with problem sizes up to 1.5 million are reported, and the results demonstrate the accuracy and efficiency of the three-stage algorithm.

Numerical aspect of large-scale electronic state calculation for flexible device material

“…3 of Ref. [20], for example. Two examples are explained; The matrix data of 'APF4686' stems from an organic polymer system, poly-(9,9 dioctyl-fluorene), in a disordered structure with 2076 atoms [17], [21].…”

“…Here we discusses the potential need of purpose-specific solvers suitable to the present problem. The need is the one for the solver of internal eigenpairs, like z-PARES [36], [37], FEAST [38], [39], the filtering method [40], k-ep [20], [41], because we would like to calculate only internal eigenpairs with the eigenenergies λ k near the highest occupied one λ HO (λ k ≤ λ HO ). Internal eigenpair solvers are desirable both for a solution of large problems, or a faster solution of middle-size problems.…”

A Mixed Precision, Multi-GPU Design for Large-scale Top-K Sparse Eigenproblems

“…To the best of our knowledge, no other work optimizes large Top-K sparse eigencomputations using FPGAs or Domain Specific Architectures (DSAs). There are numerous implementations of large-scale Top-K sparse eigenproblem solver for CPUs [15]- [17]. However, none is as well-known as ARPACK [18], a multi-core Fortran library that implements the Implicitly Restarted Arnoldi Method (IRAM), a variation of the Lanczos algorithm with support for non-Hermitian matrices.…”

Solving Large Top-K Graph Eigenproblems with a Memory and Compute-optimized FPGA Design

“…ARPACK implements the Implicitly Restarted Arnoldi Method (IRAM), a variation of the Lanczos algorithm that supports non-Hermitian matrices. Other sparse eigensolvers provide techniques optimized for specific domains or matrix types, although none is as common as ARPACK [11]- [14].…”