Improving eigenpairs of automated multilevel substructuring with subspace iterations

Yin, Jiacong; Voß, Heinrich; Chen, Pu

doi:10.1016/j.compstruc.2013.01.004

Cited by 15 publications

(17 citation statements)

References 30 publications

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“…For these two reasons the subspace iteration is a good choice for the iterative improvement of the H-AMLS approximations. The efficiency of this approach has already been demonstrated in [68] in a purely algebraic setting. There a basic version of the subspace iteration (without any acceleration techniques) has been used to improve the eigenpair approximations of the classical AMLS method.…”

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confidence: 89%

“…There a basic version of the subspace iteration (without any acceleration techniques) has been used to improve the eigenpair approximations of the classical AMLS method. It is shown in [68] for several problems that the accuracy of the eigenvalue and eigenvector approximations of AMLS can be significantly improved already within one or two iteration steps. The same results can be expected when H-AMLS is combined with the subspace iteration.…”

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confidence: 99%

“…[68]) motivates to introduce a variation of the in Section 7.1 presented H-AMLS method: Instead of applying task (T9) it is proposed to apply Algorithm 4 right after finishing task (T8). Let S ∈ R N ×nes be the matrix containing column-wise the H-AMLS eigenvector approximations y j which have been computed in task (T8), then Algorithm 4 performs one (approximative) iteration step of the subspace iteration using S as initial subspace.…”

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confidence: 99%

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Solving an elliptic PDE eigenvalue problem via automated multi-level substructuring and hierarchical matrices

Gerds

Grasedyck

2013

Comput. Visual Sci.

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To solve an elliptic PDE eigenvalue problem in practice, typically the finite element discretisation is used. From approximation theory it is known that only the smaller eigenvalues and their corresponding eigenfunctions can be well approximated by the finite element discretisation because the approximation error increases with increasing size of the eigenvalue. The number of well approximable eigenvalues or eigenfunctions, however, is unknown. In this work asymptotic estimates of these quantities are derived. For example, it is shown that for three-dimensional problems under certain smoothness assumptions on the data only the smallest Θ(N 2/5 ) eigenvalues and only the eigenfunctions associated to the smallest Θ(N 1/4 ) eigenvalues can be well approximated by the finite element discretisation, when the N -dimensional finite element spaces of piecewise affine functions with uniform mesh refinement are used.To solve the discretised elliptic PDE eigenvalue problem and to compute all well approximable eigenvalues and eigenfunctions, a new method is introduced which combines a recursive version of the automated multi-level substructuring (short AMLS) method with the concept of hierarchical matrices (short H-matrices). AMLS is a domain decomposition technique for the solution of elliptic PDE eigenvalue problems where, after some transformation, a reduced eigenvalue problem is derived whose eigensolutions deliver approximations of the sought eigensolutions of the original problem.Whereas the classical AMLS method is very efficient for elliptic PDE eigenvalue problems posed in two dimensions, it is getting very expensive for three-dimensional problems, due to the fact that it computes the reduced eigenvalue problem via dense matrix operations. This problem is resolved by the use of hierarchical matrices. H-matrices are a data-sparse approximation of dense matrices which, e.g., result from the inversion of the stiffness matrix that is associated to the finite element discretisation of an elliptic PDE operator. The big advantage of H-matrices is that they provide matrix arithmetic with almost linear complexity. This fast H-matrix arithmetic is used for the computation of the reduced eigenvalue problem. Beside this, the size of the reduced eigenvalue problem is bounded by a new recursive version of AMLS which further reduces the costs for the computation and the solution of this problem. Altogether this leads to a new method which is well-suited for three-dimensional problems and which allows us to compute a large amount of eigenpair approximations in optimal complexity.

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Solving an elliptic PDE eigenvalue problem via automated multi-level substructuring and hierarchical matrices

Gerds

Grasedyck

2013

Comput. Visual Sci.

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“…Compared with directly using Kron's subspace iteration to improve the result of the first-order approximated Kron's method, as those have been done for the fixed-interface CMS approximation, [32][33][34] the proposed method does not need the modal transformation (the terms related to the high-order modal group) before the subspace iteration, and therefore, it (i) needs less operations and simpler programming and (ii) can employ enhancements such as the shifting and restarting throughout the iterations, especially for the initial guess.…”

Section: Initialization Of the Iterationsmentioning

confidence: 99%

An eigenvector‐based iterative procedure for the free‐interface component modal synthesis method

Cui

Wang

Xing

et al. 2018

Numerical Meth Engineering

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A novel eigenvector-based iteration procedure is developed for the free-interface component modal synthesis (CMS) method. To derive the iteration formula, Kron's substructuring is employed to distribute the computations of subspace iteration, and the free-interface component modes are chosen as the initial guess. Then, the modal transformation matrix of the first-order approximated Kron's CMS method is proved to be the free-interface component modes with one step of Kron's inverse iteration. The proposed CMS method has the advantages of both free-interface CMS approximation and subspace iteration: on one hand, the CMS approximation provides a high-quality initial guess and distributes the computational cost of the subspace iteration; on the other hand, the subspace iteration provides a more efficient way for truncation compensation and is compatible with using deflation, shifting, and restarting for further enhancements on the efficiency. Numerical examples show that the efficiency of the proposed method is higher than that of the conventional simultaneous iterative Kron's CMS method, especially for obtaining a large number of high-precision modes. Moreover, the proposed method is as efficient as the global Lanczos method for first-time analysis without parallelization while retaining the advantages of CMS methods for reanalysis tasks, parallelization, etc. KEYWORDScomponent modal synthesis, iterative, subspace iteration, substructure Int J Numer Methods Eng. 2018;116:723-740. wileyonlinelibrary.com/journal/nme

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“…In the current version we can treat symmetric generalized eigenvalue problems of the form (12) with positive definite stiffness matrices and positive (semi-)definite mass matrices, thus we can also deal with problems having eigenvalues at or close to infinity. Moreover, we complement the AMLS method with a subspace iteration method (SIM) [8,73] to turn the subspace generated by the AMLS method into a good approximation of the invariant subspace [73]. Our implementation does not require any additional assumptions on the matrices beyond symmetry and positive (semi-)definiteness, i.e., they do not need to be necessarily finite element matrices.…”

Section: Numerical Experimentsmentioning

confidence: 99%

Adaptive numerical solution of eigenvalue problems arising from finite element models. AMLS vs. AFEM

Conrads¹,

Mehrmann²,

Międlar³

2016

A Panorama of Mathematics: Pure and Applied

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We discuss adaptive numerical methods for the solution of eigenvalue problems arising either from the finite element discretization of a partial differential equation (PDE) or from discrete finite element modeling. When a model is described by a partial differential equation, the adaptive finite element method starts from a coarse finite element mesh which, based on a posteriori error estimators, is adaptively refined to obtain eigenvalue/eigenfunction approximations of prescribed accuracy. This method is well established for classes of elliptic PDEs, but is still in its infancy for more complicated PDE models. For complex technical systems, the typical approach is to directly derive finite element models that are discrete in space and are combined with macroscopic models to describe certain phenomena like damping or friction. In this case one typically starts with a fine uniform mesh and computes eigenvalues and eigenfunctions using projection methods from numerical linear algebra that are often combined with the algebraic multilevel substructuring to achieve an adequate performance. These methods work well in practice but their convergence and error analysis is rather difficult. We analyze the relationship between these two extreme approaches. Both approaches have their pros and cons which are discussed in detail. Our observations are demonstrated with several numerical examples.

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Improving eigenpairs of automated multilevel substructuring with subspace iterations

Cited by 15 publications

References 30 publications

Solving an elliptic PDE eigenvalue problem via automated multi-level substructuring and hierarchical matrices

Solving an elliptic PDE eigenvalue problem via automated multi-level substructuring and hierarchical matrices

An eigenvector‐based iterative procedure for the free‐interface component modal synthesis method

Adaptive numerical solution of eigenvalue problems arising from finite element models. AMLS vs. AFEM

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