Some stable methods for calculating inertia and solving symmetric linear systems

Bunch, James R.; Kaufman, Linda

doi:10.1090/s0025-5718-1977-0428694-0

Cited by 310 publications

(254 citation statements)

References 6 publications

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“…This in turn means that one is able to exploit level-3 BLAS applied to the supernodes. Consequently, the classical Bunch-Kaufman pivoting approach [12] need to be performed only inside the supernodes.…”

Section: Sparse Direct Factorization Methodsmentioning

confidence: 99%

On large‐scale diagonalization techniques for the Anderson model of localization

Schenk

BollhÃ¶fer

RÃ¶mer

2007

Proc Appl Math and Mech

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Abstract. We propose efficient preconditioning algorithms for an eigenvalue problem arising in quantum physics, namely the computation of a few interior eigenvalues and their associated eigenvectors for the largest sparse real and symmetric indefinite matrices of the Anderson model of localization. We compare the Lanczos algorithm in the 1987 implementation by Cullum and Willoughby with the shift-and-invert techniques in the implicitly restarted Lanczos method and in the Jacobi-Davidson method. Our preconditioning approaches for the shift-and-invert symmetric indefinite linear system are based on maximum weighted matchings and algebraic multilevel incomplete LDL T factorizations. These techniques can be seen as a complement to the alternative idea of using more complete pivoting techniques for the highly ill-conditioned symmetric indefinite Anderson matrices. We demonstrate the effectiveness and the numerical accuracy of these algorithms. Our numerical examples reveal that recent algebraic multilevel preconditioning solvers can accelerative the computation of a large-scale eigenvalue problem corresponding to the Anderson model of localization by several orders of magnitude.

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Section: Sparse Direct Factorization Methodsmentioning

confidence: 99%

On large‐scale diagonalization techniques for the Anderson model of localization

Schenk

BollhÃ¶fer

RÃ¶mer

2007

Proc Appl Math and Mech

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“…However, this method in its basic form can only decompose a positive definite matrix due to the need to calculate diagonal square roots. To factorise an indefinite matrix efficiently, methods such as the Bunch and Kaufman method [22] or Aasen's method [23] need to be used. These are generally based on an LDL T decomposition [24] and can be used without significant efficiency reduction as compared to a Cholesky decomposition.…”

Section: Factorizing Matricesmentioning

confidence: 99%

Use of negative stiffness in failure analysis of concrete beams

Al‐Sabah

Laefer

2016

Engineering Structures

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A new concrete analysis method is presented entitled continuous, incremental-only, tangential analysis (CITA). CITA employs piece-wise linear stress-strain curve and a tangent elasticity modulus to calculate stiffness including parts with negative values. Since indefinite structure stiffness matrices generally indicate instability, traditionally they have been avoided. However, since CITA analysis involves introducing damage in steps, the full range of concrete behaviour including the softening portion under tensile cracking can be addressed. Herein CITA is verified against numerical and experimental results for concrete beams, thereby showing faster solutions for non-linear problems than sequentially-linear analysis, while reducing self-imposed restrictions against negative stiffness.

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“…For an n-by-n sparse matrix, the LU decomposition generally requires approximately twice as much computational time as the Cholesky decomposition due to the larger number of non-zero entries in the matrix. For EIT, a strictly real valued conductivity problem is symmetric, but complex valued problems are not Hermitian (but are symmetric) and cannot typically take advantage of sparse symmetric solvers (Bunch and Kaufman, 1977). The complex valued problem requires roughly twice the storage and four times the number of multiplications when compared to a real valued problem using the same decomposition due strictly to the handling of the complex numbers.…”

Section: Sparse Solversmentioning

confidence: 99%

Addressing the computational cost of large EIT solutions

2012

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Abstract. Electrical Impedance Tomography (EIT) is a soft field tomography modality based on the application of electric current to a body and measurement of voltages through electrodes at the boundary. The interior conductivity is reconstructed on a discrete representation of the domain using a FEM mesh and a parametrization of that domain. The reconstruction requires a sequence of numerically intensive calculations. There is strong interest in reducing the cost of these calculations.An improvement in the compute time for current problems would encourage further exploration of computationally challenging problems such as the incorporation of time series data, wide-spread adoption of three-dimensional simulations, and correlation of other modalities such as CT and ultrasound. Multicore processors offer an opportunity to reduce EIT computation times but may require some restructuring of the underlying algorithms to maximize the use of available resources.This work profiles two EIT software packages (EIDORS and NDRM) to experimentally determine where the computational costs arise in EIT as problems scale. Sparse matrix solvers, a key component for the FEM forward problem and sensitivity estimates in the inverse problem, are shown to take a considerable portion of the total compute time in these packages. A sparse matrix solver performance measurement tool, Meagre-Crowd, is developed to interface with a variety of solvers and compare their performance over a range of two-and threedimensional problems of increasing node density. Results show that distributed sparse matrix solvers that operate on multiple cores are advantageous up to a limit that increases as the node density increases. We recommend a selection procedure to find a solver and hardware arrangement matched to the problem and provide guidance and tools to perform that selection.

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Some stable methods for calculating inertia and solving symmetric linear systems

Abstract: Several decompositions of, symmetric matrices for calculating inertia and solving systems of linear equations are discussed. New partial pivoting strategies for decomposing symmetric matrices are introduced and analyzed.

Cited by 310 publications

References 6 publications

On large‐scale diagonalization techniques for the Anderson model of localization

On large‐scale diagonalization techniques for the Anderson model of localization

Use of negative stiffness in failure analysis of concrete beams

Addressing the computational cost of large EIT solutions

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