Component mode synthesis and eigenvalues of second order operators : discretization and algorithm

Bourquin, Frédéric

doi:10.1051/m2an/1992260303851

Cited by 62 publications

(50 citation statements)

References 16 publications

(19 reference statements)

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“…We would not in general expect a similar result for the "classical" non-singular Sturm-Liouville choice s m,j = 1, and we note that port reduction approaches within the CMS framework [6,18] typically consider regular rather than singular eigenproblems. Also note that in the case y m,j = 0 a solution to (36) is always (κ 1 m,j = 0, τ 1 m,j = constant), and hence χ m,j,1 is constant for any y m,j (recall in the case y m,j > 0 we set χ m,j,1 = ρ m,j,1 , which is chosen constant).…”

Section: Port Approximationmentioning

confidence: 78%

“…6 We then perform a data compression step: we invoke the proper orthogonal decomposition (POD) [19] (with respect to the L 2 (Γ p * ) inner product). The output from the POD procedure is a set of y m,j − 1 mutually L 2 (Γ p * )-orthonormal empirical modes which have the additional property that they are orthogonal to the constant function over Γ p * .…”

Section: End Formentioning

confidence: 99%

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Port reduction in parametrized component static condensation: approximation and a posteriori error estimation

Eftang

Patera

2013

Numerical Meth Engineering

104

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We introduce a port (interface) approximation and a posteriori error bound framework for a general component-based static condensation method in the context of parameter-dependent linear elliptic partial differential equations. The key ingredients are i) efficient empirical port approximation spaces -the dimensions of these spaces may be chosen small in order to reduce the computational cost associated with formation and solution of the static condensation system, and ii) a computationally tractable a posteriori error bound realized through a non-conforming approximation and associated conditionerthe error in the global system approximation, or in a scalar output quantity, may be bounded relatively sharply with respect to the underlying finite element discretization.Our approximation and a posteriori error bound framework is of particular computational relevance for the static condensation reduced basis element (SCRBE) method. We provide several numerical examples within the SCRBE context which serve to demonstrate the convergence rate of our port approximation procedure as well as the efficacy of our port reduction error bounds.

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Section: Port Approximationmentioning

confidence: 78%

Section: End Formentioning

confidence: 99%

Port reduction in parametrized component static condensation: approximation and a posteriori error estimation

Eftang

Patera

2013

Numerical Meth Engineering

104

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“…However, the eigenfunctions of (5.7) and (5.8) belonging to the smallest eigenvalues are well suited to approximate the sought eigensolutions (λ j , u j ) nes j=1 of the global problem (4.2). This issue is reasoned by various numerical studies (see, e.g., [16]) and is motivated by the error analysis done in [21,22] for a method quite similar to AMLS. Correspondingly, to approximate the sought eigensolutions of problem (4.2), in the third step of AMLS the finite dimensional subspace U k ⊂ H 1 0 (Ω) is defined by…”

Section: The Amls Methods In the Continuous Settingmentioning

confidence: 99%

“…An easy calculation shows that for all α ∈ N d the functions u α (x) are eigenfunctions of the PDE problem (2.17) with the associated eigenvalues 21) and because of u α ∈ H 1 0 (Ω) it follows that (λ α , u α ) is also an eigensolution of (2.18). In particular, all eigensolutions of (2.18) are described by (λ α , u α ) α∈N d .…”

Section: Laplace Eigenvalue Problem On the Unit Cubementioning

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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Bibliography

2017

Dynamics of Large Structures and Inverse Problems

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Component mode synthesis and eigenvalues of second order operators : discretization and algorithm

Cited by 62 publications

References 16 publications

Port reduction in parametrized component static condensation: approximation and a posteriori error estimation

Port reduction in parametrized component static condensation: approximation and a posteriori error estimation

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

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