Block Low-Rank Matrices with Shared Bases: Potential and Limitations of the BLR$^2$ Format

Abstract

We investigate a special class of data sparse rank-structured matrices that combine a flat block low-rank (BLR) partitioning with the use of shared (called nested in the hierarchical case) bases. This format is to H 2 matrices what BLR is to H matrices: we therefore call it the BLR 2 matrix format. We present algorithms for the construction and LU factorization of BLR 2 matrices, and perform their cost analysis-both asymptotically and for a fixed problem size. With weak admissibility, BLR 2 matrices reduce to block separable matrices (the flat version of HBS/HSS). Our analysis and numerical experiments reveal some limitations of BLR 2 matrices with weak admissibility, which we propose to overcome with two approaches: strong admissibility, and the use of multiple shared bases per row and column.

O(N) distributed direct factorization of structured dense matrices using runtime systems.

“…Format Algorithm Compute complexity Distributed paradigm Comm. complexity DPLASMA [6] Dense Tile Cholesky 𝑂 (𝑁 3 ) Asynchronous 𝑂 (𝑁 3 ) SLATE [9] Dense Panel Cholesky 𝑂 (𝑁 3 ) Fork-join 𝑂 (𝑁 3 ) LORAPO [7] BLR Tile Cholesky 𝑂 (𝑁 2 ) Asynchronous 𝑂 (𝑁 3 ) H -LU [4] H -matrix H -LU 𝑂 (𝑁𝑙𝑜𝑔(𝑁 )) Asynchronous 𝑂 (𝑁𝑙𝑜𝑔(𝑁 )) STRUMPACK [13] HSS ULV 𝑂 (𝑁 ) Fork-join 𝑂 (𝑁 2 ) Ma et. al.…”

“…In order to make it convenient to represent the ULV factorization, we permute the skeleton and redundant parts as shown in Eq. (3).…”

“…The complexity of the factorization is determined by the format of the hierarchical matrix and the algorithm used for the factorization. Various formats such as BLR [2], BLR 2 [3], HODLR [1], H -matrix, HSS [8] and H 2 -matrix [5] have been proposed, varying by conditions of admissibility and the use of nested basis.…”

Computing the k-th Eigenvalue of Symmetric H2-Matrices

“…This kind of matrix is often encountered in computational science, notably from the discretization of elliptic partial differential operators that govern a wide range of application areas [18]. There are several kinds of structured low-rank formats, such as BLR [2], BLR 2 [4], HODLR [1], quasiseparable/semiseparable [13,36], HSS [12], H [17], and H 2 -matrices [16]. Many studies have been conducted to obtain fast algorithms by adapting well-known dense eigenvalue solvers to these formats.…”

“…This method relies on Sylverster's inertia theorem that allows one to compute the number of eigenvalues of a matrix A that are smaller than a value µ by evaluating the LDL factorization of the shifted matrix A − µI. It has been studied with HODLR matrices in [6], allowing the computation of the k-th eigenvalue in O(n log 4 2 (n) log 2 ((b − a)/ϵev)) operations, where [a, b] is the bisection starting interval and ϵev is the desired accuracy. Generalized LDL factorization of HSS-matrices has also been used to compute the inertia in O(n), which ultimately reduces the cost down to O(n log 2 ((b − a)/ϵev)) per eigenvalue [37].…”