Computing entries of the inverse of a sparse matrix using the FIND algorithm

Li, S.; Ahmed, Shaikh; Klimeck, Gerhard; Darve, Eric

doi:10.1016/j.jcp.2008.06.033

Cited by 61 publications

(107 citation statements)

References 20 publications

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“…We would like to mention that Li et al [17] have recently proposed an algorithm that has a similar spirit to our approach. However, our algorithm is derived from a different viewpoint, and the numerical results suggest that our approach is more efficient for the problems considered here.…”

Section: Main Observation and Contribution Of This Workmentioning

confidence: 97%

Fast algorithm for extracting the diagonal of the inverse matrix with application to the electronic structure analysis of metallic systems

Lin¹,

Lu²,

Ying³

et al. 2009

Communications in Mathematical Sciences

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Abstract. We propose an algorithm for extracting the diagonal of the inverse matrices arising from electronic structure calculation. The proposed algorithm uses a hierarchical decomposition of the computational domain. It first constructs hierarchical Schur complements of the interior points for the blocks of the domain in a bottom-up pass and then extracts the diagonal entries efficiently in a top-down pass by exploiting the hierarchical local dependence of the inverse matrices. The overall cost of our algorithm is O(N 3/2 ) for a two dimensional problem with N degrees of freedom. Numerical results in electronic structure calculation illustrate the efficiency and accuracy of the proposed algorithm.

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Section: Main Observation and Contribution Of This Workmentioning

confidence: 97%

Fast algorithm for extracting the diagonal of the inverse matrix with application to the electronic structure analysis of metallic systems

Lin¹,

Lu²,

Ying³

et al. 2009

Communications in Mathematical Sciences

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“…These geometrical regions or clusters have associated matrices and factors. The central mathematical objects used in this procedure are antisymmetric "cluster" matrices U A (J ) and U A (J ) [14,20] which depend both on the set of spin couplings J = {J ij } and the region A. The dependence of U on the spin couplings J that define the given realization of a sample will be implicit in the remainder of this paper and so we will write U A for U A (J ).…”

Section: Cluster Matrices and Their Operationsmentioning

confidence: 99%

“…We simplify the sampling technique used in our previous work [11] by simplifying the matrices and by using a different approach to maintain computed correlation functions as spins are sampled. Many of these improvements are based on the FIND (fast inverse using nested dissection) algorithm [14] which computes desired elements of a matrix inverse quickly and was developed to compute nonequilibrium Green's function applications in nanodevices. This particular flavor of hierarchical decomposition is very well suited to the geometry of the mapping between two-dimensional Ising models and dimer coverings.…”

Section: Introductionmentioning

confidence: 99%

Numerically exact correlations and sampling in the two-dimensional Ising spin glass

Thomas

Middleton

2013

Phys. Rev. E

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A powerful existing technique for evaluating statistical mechanical quantities in two-dimensional Ising models is based on constructing a matrix representing the nearest neighbor spin couplings and then evaluating the Pfaffian of the matrix. Utilizing this technique and other more recent developments in evaluating elements of inverse matrices and exact sampling, a method and computer code for studying two-dimensional Ising models is developed. The formulation of this method is convenient and fast for computing the partition function and spin correlations. It is also useful for exact sampling, where configurations are directly generated with probability given by the Boltzmann distribution. These methods apply to Ising model samples with arbitrary nearest-neighbor couplings and can also be applied to general dimer models. Example results of computations are described, including comparisons with analytic results for the ferromagnetic Ising model, and timing information is provided.

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“…The main idea used here is very general. It dates back to [12], and is discussed in other recent work [22,24,34]. Before we describe the algorithm, it will be helpful to first review the major operations involved including the LDL T factorization.…”

Section: Selected Inversion Based On Ldlmentioning

confidence: 99%

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

Lin¹,

Yang²,

Lu³

et al. 2011

SIAM J. Sci. Comput.

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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.

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Computing entries of the inverse of a sparse matrix using the FIND algorithm

Cited by 61 publications

References 20 publications

Fast algorithm for extracting the diagonal of the inverse matrix with application to the electronic structure analysis of metallic systems

Fast algorithm for extracting the diagonal of the inverse matrix with application to the electronic structure analysis of metallic systems

Numerically exact correlations and sampling in the two-dimensional Ising spin glass

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

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