Localized inverse factorization

Abstract

We propose a localized divide and conquer algorithm for inverse factorization S −1 = ZZ * of Hermitian positive definite matrices S with localized structure, e.g. exponential decay with respect to some given distance function on the index set of S. The algorithm is a reformulation of recursive inverse factorization [J. Chem. Phys., 128 (2008), 104105] but makes use of localized operations only. At each level of recursion, the problem is cut into two subproblems and their solutions are combined using iterative refinement [Phys. Rev. B, 70 (2004), 193102] to give a solution to the original problem. The two subproblems can be solved in parallel without any communication and, using the localized formulation, the cost of combining their results is proportional to the cut size, defined by the binary partition of the index set. This means that for cut sizes increasing as o(n) with system size n the cost of combining the two subproblems is negligible compared to the overall cost for sufficiently large systems.We also present an alternative derivation of iterative refinement based on a sign matrix formulation, analyze the stability, and propose a parameterless stopping criterion. We present bounds for the initial factorization error and the number of iterations in terms of the condition number of S when the starting guess is given by the solution of the two subproblems in the binary recursion. These bounds are used in theoretical results for the decay properties of the involved matrices.The localization properties of our algorithms are demonstrated for matrices corresponding to nearest neighbor overlap on one-, two-, and three-dimensional lattices as well as basis set overlap matrices generated using the Hartree-Fock and Kohn-Sham density functional theory electronic structure program Ergo [SoftwareX, 7 (2018), 107].

Parallelization and scalability analysis of inverse factorization using the chunks and tasks programming model

“…Then, I − Z * 0 SZ 0 2 < 1, which implies convergence of iterative refinement with Z 0 as starting guess [23]. This result was recently strengthened and it was shown that [22]. Those convergence results immediately suggest a recursive inverse factorization algorithm where iterative refinement is combined with a recursive binary principal submatrix decomposition of the matrix S.…”

“…In order to minimize the number of parameters, we employ the automatic stopping criterion proposed in [18] and used in [22]. The iterative refinement is stopped as soon as δ i+1 F > δ i m+1 F .…”

“…Eventually the LIF and the IRSI algorithms arrive at very close timings. The number of nonzeros per row is no longer constant, it grows slowly with system size but flattens out for large systems [22]. Note that the RINCH algorithm was not able to handle the largest systems.…”

“…This procedure is applied recursively giving the recursive inverse factorization algorithm. Recently a localized variant of this method was proposed which uses only localized computations for the correction step with cost proportional to the cut size [22]. We refer to this variant as localized inverse factorization.…”

A note on adaptivity in factorized approximate inverse preconditioning

“…This type of factorization has been found useful not only in general parallel iterative solvers but also in solving generalized eigenvalue problems that arise from computation of electronic structures. See, for example, [6], [7] and the recent survey [14].…”