Hierarchical Interpolative Factorization for Elliptic Operators: Differential Equations

Ho, Kenneth L.; Ying, Lexing

doi:10.1002/cpa.21582

Cited by 133 publications

(116 citation statements)

References 51 publications

Supporting

Mentioning

116

Contrasting

Order By: Relevance

“…Closely related to the concept of H-matrices and its arithmetic are "hierarchically semiseparable matrices", [Xia13,XCGL09,LGWX12] and the idea of "recursive skeletonization", [HG12,GGMR09,HY13a]; for discretizations of PDEs, we mention [HY13a,GM13,SY12,Mar09], and particular applications to boundary integral equations are [MR05,CMZ13,HY13b]. These factorization algorithms aim to exploit that some off-diagonal blocks of certain Schur complements are low rank.…”

Section: Introductionmentioning

confidence: 99%

Existence of $\mathcal {H}$-matrix approximants to the inverses of BEM matrices: The simple-layer operator

2015

View full text Add to dashboard Cite

Abstract. We consider the question of approximating the inverse W = V −1 of the Galerkin stiffness matrix V obtained by discretizing the simple-layer operator V with piecewise constant functions. The block partitioning of W is assumed to satisfy any one of several standard admissibility criteria that are employed in connection with clustering algorithms to approximate the discrete BEM operator V. We show that W can be approximated by blockwise low-rank matrices such that the error decays exponentially in the block rank employed. Similar exponential approximability results are shown for the Cholesky factorization of V.

show abstract

Section: Introductionmentioning

confidence: 99%

Existence of $\mathcal {H}$-matrix approximants to the inverses of BEM matrices: The simple-layer operator

2015

View full text Add to dashboard Cite

show abstract

“…These algorithms may be extended to volumes in 2D and surfaces in 3D, producing direct solvers with complexity O N 3/2 [Gil+12, HG12, Gil11]. More recently, approaches that aim to extend linear complexity to Hierarchically Semi-Separable (HSS) matrices have been developed [Cor+14,HY15]. Furthermore, a general inverse algorithm has been proposed for FMM matrices [AD14].…”

Section: Aσ(x) + ω B(x)k(x Y)c( Y)σ( Y) Dω Y = F (X)mentioning

confidence: 99%

“…Ho and Ying [HY15] proposed an alternate approach, the Hierarchical Interpolative Factorization (HIF), using additional skeletonization levels and implemented it for both 2D volume and 3D boundary integral equations using standard direct solvers for sparse matrices in an augmented system. While the structure of the algorithms suggests linear scaling, for 3D problems the observed behavior is still above linear.…”

Section: Direct Solvers For Integral Equationsmentioning

confidence: 99%

See 1 more Smart Citation

A Tensor-Train accelerated solver for integral equations in complex geometries

Corona

Rahimian

Zorin

2017

Journal of Computational Physics

View full text Add to dashboard Cite

We present a framework using the Quantized Tensor Train (QTT) decomposition to accurately and efficiently solve volume and boundary integral equations in three dimensions. We describe how the QTT decomposition can be used as a hierarchical compression and inversion scheme for matrices arising from the discretization of integral equations. For a broad range of problems, computational and storage costs of the inversion scheme are extremely modest O log N and once the inverse is computed, it can be applied in O N log N .We analyze the QTT ranks for hierarchically low rank matrices and discuss its relationship to commonly used hierarchical compression techniques such as FMM and HSS. We prove that the QTT ranks are bounded for translation-invariant systems and argue that this behavior extends to nontranslation invariant volume and boundary integrals.For volume integrals, the QTT decomposition provides an efficient direct solver requiring significantly less memory compared to other fast direct solvers. We present results demonstrating the remarkable performance of the QTT-based solver when applied to both translation and non-translation invariant volume integrals in 3D.For boundary integral equations, we demonstrate that using a QTT decomposition to construct preconditioners for a Krylov subspace method leads to an efficient and robust solver with a small memory footprint. We test the QTT preconditioners in the iterative solution of an exterior elliptic boundary value problem (Laplace) formulated as a boundary integral equation in complex, multiply connected geometries.

show abstract

“…In particular, this variant was shown to lead to a bounded number of iterations irrespective of problem size and condition number (under certain assumptions). In [16] a fast sparse solver was introduced based on interpolative decomposition and skeletonization. It was optimized for meshes that are perturbations of a structured grid.…”

Section: Introductionmentioning

confidence: 99%

Sparse supernodal solver using block low-rank compression: Design, performance and analysis

Pichon

Darve

Faverge

et al. 2018

Journal of Computational Science

View full text Add to dashboard Cite

Abstract:This paper presents two approaches using a Block Low-Rank (BLR) compression technique to reduce the memory footprint and/or the time-to-solution of the sparse supernodal solver PaStiX. This flat, non-hierarchical, compression method allows to take advantage of the low-rank property of the blocks appearing during the factorization of sparse linear systems, which come from the discretization of partial differential equations. The first approach, called Minimal Memory, illustrates the maximum memory gain that can be obtained with the BLR compression method, while the second approach, called Just-In-Time, mainly focuses on reducing the computational complexity and thus the time-to-solution. Singular Value Decomposition (SVD) and Rank-Revealing QR (RRQR), as compression kernels, are both compared in terms of factorization time, memory consumption, as well as numerical properties. Experiments on a single node with 24 threads and 128 GB of memory are performed to evaluate the potential of both strategies. On a set of matrices from real-life problems, we demonstrate a memory footprint reduction of up to 4 times using the Minimal Memory strategy and a computational time speedup of up to 3.5 times with the Just-In-Time strategy. Then, we study the impact of configuration parameters of the BLR solver that allowed us to solve a 3D laplacian of 36 million unknowns a single node, while the full-rank solver stopped at 8 million due to memory limitation. Key-words:Sparse linear solver, block low-rank compression, PaStiX direct solver, multithreaded architectures * Inria Bordeaux -Sud-Ouest, Talence, France † University of Bordeaux, Talence, France ‡ CNRS (Labri UMR 5800), Talence, France § Mechanical Engineering Department, Stanford University, United States ¶ Bordeaux INP, Talence, France Un solveur supernodal creux utilisant une compression de rang faible par bloc: conception, étude de performance et analyse des résultats Résumé :Ce papier présente deux approches utilisant les techniques de compression de rang faible par bloc (BLR) afin de réduire l'empreinte mémoire et/ou le temps de résolution du solveur superndal PaStiX. Cette technique de compression à plat, non hiérarchique, permet de tirer parti des propriétés de rang faible dans les blocs obtenus lors de la factorisation du système linéaire provenant par exemple de la discrétisation des équations différentielles partielles. La première approche, appelée Minimal Memory, montre le gain mémoire maximum qu'il est possible d'obtenir avec une compression BLR, alors que la seconde approche, appelée Just-InTime, se concentre principalement sur la réduction du temps de calcul pour la résolution du système. Dans cette étude, nous comparons, en termes de temps de calcul et de consommation mémoire, les noyaux de compression qui utilisent soit la technique de décomposition en valeurs propres singulières (SVD) soit la factorisation QR avec détermination du rang (RRQR). Nous avons réalisé les expériences sur un noeud composé de 24 coeurs avec 128 GB de mémoire sur une collec...

show abstract

Hierarchical Interpolative Factorization for Elliptic Operators: Differential Equations

Cited by 133 publications

References 51 publications

Existence of $\mathcal {H}$-matrix approximants to the inverses of BEM matrices: The simple-layer operator

Existence of $\mathcal {H}$-matrix approximants to the inverses of BEM matrices: The simple-layer operator

A Tensor-Train accelerated solver for integral equations in complex geometries

Sparse supernodal solver using block low-rank compression: Design, performance and analysis

Contact Info

Product

Resources

About