Computing several eigenpairs of Hermitian problems by conjugate gradient iterations

Ovtchinnikov, Evgueni

doi:10.1016/j.jcp.2008.06.038

Cited by 13 publications

(9 citation statements)

References 29 publications

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“…BEAn uses the pre-release version of the Python package RALEIGH, a general-purpose eigenvalue problem solver that implements the block conjugate gradient algorithm [19], to compute eigenvectors of the data covariance matrix. Unlike most other eigensolvers, RALEIGH does not require the number of desired eigenvalues to be specified by the user.…”

Section: Principal Component Analysismentioning

confidence: 99%

Developments towards Bragg edge imaging on the IMAT beamline at the ISIS Pulsed Neutron and Muon Source: BEAn software

et al. 2019

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The BEAn (Bragg Edge Analysis) software has been developed as a toolkit for analysis of Bragg edges and strain maps from data obtained at the time-of-flight imaging instrument IMAT or other compatible instruments. The code is built primarily using Python 3 and the Qt framework, and includes tools useful for neutron imaging such as principal component analysis. This paper introduces BEAn and its features, briefly discuses the scientific concepts behind them, and concludes with planned future work on the code.

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Section: Principal Component Analysismentioning

confidence: 99%

Developments towards Bragg edge imaging on the IMAT beamline at the ISIS Pulsed Neutron and Muon Source: BEAn software

et al. 2019

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“…These features facilitate efficient exploitation of highly optimized matrix–matrix multiplication subroutines from the BLAS library and modern multicore computer architectures. The BJCG algorithm is based on the block conjugate gradient method of and is implemented as the software package SPRAL_SSMFE, which is available within the SPRAL mathematical software library .…”

Section: Solving the Generalized Eigenvalue Problemmentioning

confidence: 99%

“…In , various suggestions for σ i have been studied thoroughly to arrive at the conclusion that most of them are asymptotically equivalent and make the value ψ ( v i + 2 ) ‘nearly’ the smallest possible for a given v i + 1 and u i . It should be noted that the value of σ i that makes ψ ( v i + 2 ) the smallest possible can be computed by finding the minimum of ψ ( v ) among all linear combinations of v i + 1 ,∇ ψ ( v i + 1 ) and u i ; however, this locally optimal version of CG is numerically unstable .…”

Section: Solving the Generalized Eigenvalue Problemmentioning

confidence: 99%

Level‐set topology optimization with many linear buckling constraints using an efficient and robust eigensolver

Dunning

Ovtchinnikov

Scott

et al. 2016

Numerical Meth Engineering

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SUMMARYLinear buckling constraints are important in structural topology optimization for obtaining designs that can support the required loads without failure. During the optimization process, the critical buckling eigenmode can change; this poses a challenge to gradient-based optimization and can require the computation of a large number of linear buckling eigenmodes. This is potentially both computationally difficult to achieve and prohibitively expensive. In this paper, we motivate the need for a large number of linear buckling modes and show how several features of the block Jacobi conjugate gradient (BJCG) eigenvalue method, including optimal shift estimates, the reuse of eigenvectors, adaptive eigenvector tolerances and multiple shifts, can be used to efficiently and robustly compute a large number of buckling eigenmodes. This paper also introduces linear buckling constraints for level-set topology optimization. In our approach, the velocity function is defined as a weighted sum of the shape sensitivities for the objective and constraint functions. The weights are found by solving an optimization sub-problem to reduce the mass while maintaining feasibility of the buckling constraints. The effectiveness of this approach in combination with the BJCG method is demonstrated using a 3D optimization problem.

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“…Generalizations of spectral approximations for holomorphic Fredholm operator functions are derived in the papers [5,6]. Preconditioned iterative methods for solving linear spectral problems are proposed and investigated in the papers [7][8][9][10][11][12][13][14]. Iterative methods for solving spectral problems with nonlinear parameter are proposed and investigated in the papers [15][16][17][18][19][20][21][22][23][24][25][26].…”

Section: Introductionmentioning

confidence: 99%

Eigenvibrations of a beam with two mechanical resonators attached to the ends

et al. 2020

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The fourth-order ordinary differential spectral problem describing vertical eigenvibrations of a beam with two mechanical resonators attached to the ends is studied. This problem has positive simple eigenvalues and corresponding eigenfunctions. We define limit differential spectral problem and establish the convergence of the eigenvalues and eigenfunctions of the original spectral problem to the eigenvalues and eigenfunctions of the limit spectral problem as parameters of the attached resonators tending to infinity. The initial fourth-order ordinary differential spectral problem is approximated by the finite difference method. Theoretical error estimates for approximate eigenvalues and eigenfunctions are derived. Obtained theoretical results are illustrated by computations for model problem with constant coefficients. Theoretical and experimental results of this paper can be developed for the problems on eigenvibrations of complex mechanical constructions with systems of resonators.

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Computing several eigenpairs of Hermitian problems by conjugate gradient iterations

Cited by 13 publications

References 29 publications

Developments towards Bragg edge imaging on the IMAT beamline at the ISIS Pulsed Neutron and Muon Source: BEAn software

Developments towards Bragg edge imaging on the IMAT beamline at the ISIS Pulsed Neutron and Muon Source: BEAn software

Level‐set topology optimization with many linear buckling constraints using an efficient and robust eigensolver

Eigenvibrations of a beam with two mechanical resonators attached to the ends

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