A Fast Parallel Algorithm for Selected Inversion of Structured Sparse Matrices with Application to 2D Electronic Structure Calculations

Abstract

An efficient parallel algorithm is presented and tested for computing selected components of H −1 where H has the structure of a Hamiltonian matrix of two-dimensional lattice models with local interaction. Calculations of this type are useful for several applications, including electronic structure analysis of materials in which the diagonal elements of the Green's functions are needed. The algorithm proposed here is a direct method based on an LDL T factorization. The elimination tree is used to organize the parallel algorithm. Synchronization overhead is reduced by passing the data level by level along this tree using the technique of local buffers and relative indices. The performance of the proposed parallel algorithm is analyzed by examining its load balance and communication overhead, and is shown to exhibit an excellent weak scaling on a large-scale high performance parallel machine with distributed memory.

Element orbitals for Kohn-Sham density functional theory

“…The future improvement includes treating C and H as sparse matrices so that the construction of the Hamiltonian matrix C t HC and the mass matrix C t C is of linear scaling. By treating C and H as sparse matrices, we can also incorporate the recently developed pole expansion and selected inversion type fast algorithms [25][26][27][28][29][30] to reduce the asymptotic scaling for solving the generalized eigenvalue problem (10) from cubic scaling to at most quadratic scaling for 3D bulk systems. We also remark that the current procedure for constructing the orbitals from adaptive local basis functions is still a costly procedure inside each element.…”

“…However, notice that one only needs the knowledge of the diagonal blocks of the Gram matrix G to construct the electron density. This allows us to use the recently developed pole expansion and selected inversion type fast algorithms [25][26][27][28][29][30] to reduce the asymptotic scaling for solving the generalized eigenvalue problem (10) from cubic scaling to at most quadratic scaling for 3D bulk systems. For simplicity we employ a cubic scaling implementation within the current work, as described in more detail in Section IV.…”

Shift‐and‐invert parallel spectral transformation eigensolver: Massively parallel performance for density‐functional based tight‐binding

“…Both the factorization and inversion steps scale with where is 3 for bulk systems, 2 for slabs and surfaces, and 1 for long and thin systems. The number of iterations for the chemical potential varies significantly depending on the initial guess for the chemical potential, band gap, and the required accuracy . The number of poles (typically 40‐80) in PEXSI is independent of the system size, but depends on the given electronic temperature and spectral width (the difference between max/min eigenvalues) of the Kohn‐Sham Hamiltonian.…”

“…The number of iterations for the chemical potential varies significantly depending on the initial guess for the chemical potential, band gap, and the required accuracy . The number of poles (typically 40‐80) in PEXSI is independent of the system size, but depends on the given electronic temperature and spectral width (the difference between max/min eigenvalues) of the Kohn‐Sham Hamiltonian. On the other hand, the computational cost for SIPs is , where is the total number of shifts and is the combined cost of one numerical factorization and the solution/orthogonalization step per shift.…”

Accelerating atomic orbital-based electronic structure calculation via pole expansion and selected inversion

“…To obtain these selected elements, we need to compute the corresponding elements of (H − (z l + µ)S) −1 for all z l . The recently developed selected inversion method [17][18][19] provides an efficient way of computing the selected elements of an inverse matrix. For a symmetric matrix of the form A = H − zS, the selected inversion algorithm first constructs an LDL T factorization of A, where L is a block lower diagonal matrix called the Cholesky factor, and D is a block diagonal matrix.…”