Sparse Tensor Approximation of Parametric Eigenvalue Problems

Andreev, Roman; Schwab, Christoph

doi:10.1007/978-3-642-22061-6_7

Cited by 39 publications

(96 citation statements)

References 33 publications

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“…The differential operators A m , A s and A b account for membrane, shear, and bending potential energies, respectively and are independent of t. Finally, M (t) is the inertia operator, which in this case can be split into the sum M (t) = tM l + t 3 M r , with M l (displacements) and M r (rotations) independent of t. Many well-known shell models fall into this framework. Let us next consider the variational formulation of problem (2). Accordingly, we introduce the space V of admissible displacements, and consider the problem: Find:…”

Section: Shell Eigenproblemmentioning

confidence: 99%

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Multiparametric shell eigenvalue problems

Hakula

Laaksonen

2019

Computer Methods in Applied Mechanics and Engineering

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The eigenproblem for thin shells of revolution under uncertainty in material parameters is discussed. Here the focus is on the smallest eigenpairs. Shells of revolution have natural eigenclusters due to symmetries, moreover, the eigenpairs depend on a deterministic parameter, the dimensionless thickness. The stochastic subspace iteration algorithms presented here are capable of resolving the smallest eigenclusters. In the case of random material parameters, it is possible that the eigenmodes cross in the stochastic parameter space. This interesting phenomenon is demonstrated via numerical experiments. Finally, the effect of the chosen material model on the asymptotics in relation to the deterministic parameter is shown to be negligible.

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Section: Shell Eigenproblemmentioning

confidence: 99%

“…The algorithms typically suggested in the literature (e.g. [2,10]) restrict to simple eigenmodes only. As noted, for shells of revolution it does not make sense to assume that the smallest eigenmode is simple.…”

Section: Stochastic Subspacesmentioning

confidence: 99%

Multiparametric shell eigenvalue problems

Hakula

Laaksonen

2019

Computer Methods in Applied Mechanics and Engineering

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“…A common method of tackling these problems is the reduced basis method [27,31,16,22], whereby the full parametric solution (eigenvalue) is approximated in a low-dimensional subspace that is constructed as the span of the solutions at specifically chosen parameter values. For the current work the most relevant paper is [1], where a sparse tensor approximation was used to estimate the expected value of the eigenvalue. A key result there is that simple eigenpairs are analytic with respect to the stochastic parameters, shown using the classical perturbation theory of Kato [24].…”

Section: Introductionmentioning

confidence: 99%

Analysis of quasi-Monte Carlo methods for elliptic eigenvalue problems with stochastic coefficients

et al. 2019

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We consider the forward problem of uncertainty quantification for the generalised Dirichlet eigenvalue problem for a coercive second order partial differential operator with random coefficients, motivated by problems in structural mechanics, photonic crystals and neutron diffusion. The PDE coefficients are assumed to be uniformly bounded random fields, represented as infinite series parametrised by uniformly distributed i.i.d. random variables. The expectation of the fundamental eigenvalue of this problem is computed by (a) truncating the infinite series which define the coefficients; (b) approximating the resulting truncated problem using lowest order conforming finite elements and a sparse matrix eigenvalue solver; and (c) approximating the resulting finite (but high dimensional) integral by a randomly shifted quasi-Monte Carlo lattice rule, with specially chosen generating vector. We prove error estimates for the combined error, which depend on the truncation dimension s, the finite element mesh diameter h, and the number of quasi-Monte Carlo samples N . Under suitable regularity assumptions, our bounds are of the particular form O(h 2 + N −1+δ ), where δ > 0 is arbitrary and the hidden constant is independent of the truncation dimension, which needs to grow as h → 0 and N → ∞. As for the analogous PDE source problem, the conditions under which our error bounds hold depend on a parameter p ∈ (0, 1) representing the summability of the terms in the series expansions of the coefficients. Although the eigenvalue problem is nonlinear, which means it is generally considered harder than the source problem, in almost all cases (p = 1) we obtain error bounds that converge at the same rate as the corresponding rate for the source problem. The proof involves a detailed study of the regularity of the fundamental eigenvalue as a function of the random parameters. As a key intermediate result in the analysis, we prove that the spectral gap (between the fundamental and the second eigenvalues) is uniformly positive over all realisations of the random problem.

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“…The requirement for strict positivity of min (zI − A) in Theorems 1 and 2 is artificial and can be fixed by using "signed" singular values as in the case of the analytic SVD. 35 In practice, sinceÂ(x, ) is an analytic function in x and y, we can expect much faster convergence than the one guaranteed by Theorem 2 (see section 3.4.1 of the work of Sirković et al 30 and section 2.3.2 of the work of Andreev et al 36 ). Numerical experiments shown in Section 4 support this.…”

Section: Interpolation Propertiesmentioning

confidence: 95%

“…In practice, since

\overset{A}{true} false(x, y false)

is an analytic function in x and y , we can expect much faster convergence than the one guaranteed by Theorem (see section 3.4.1 of the work of Sirković et al and section 2.3.2 of the work of Andreev et al). Numerical experiments shown in Section 4 support this.…”

Section: Subspace Acceleration For Pseudospectra Computationmentioning

confidence: 98%

A reduced basis approach to large‐scale pseudospectra computation

Sirković

2018

Numerical Linear Algebra App

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Summary For studying spectral properties of a nonnormal matrix A∈Cn×n, information about its spectrum σ(A) alone is usually not enough. Effects of perturbations on σ(A) can be studied by computing ε‐pseudospectra, i.e. the level sets of the resolvent norm function sans-serifgfalse(zfalse)=false‖false(zI−Afalse)−1‖2. The computation of ε‐pseudospectra requires determining the smallest singular values σminfalse(zI−Afalse) for all z on a portion of the complex plane. In this work, we propose a reduced basis approach to pseudospectra computation, which provides highly accurate estimates of pseudospectra in the region of interest, in particular, for pseudospectra estimates in isolated parts of the spectrum containing few eigenvalues of A. It incorporates the sampled singular vectors of zI − A for different values of z, and implicitly exploits their smoothness properties. It provides rigorous upper and lower bounds for the pseudospectra in the region of interest. In addition, we propose a domain splitting technique for tackling numerically more challenging examples. We present a comparison of our algorithms to several existing approaches on a number of numerical examples, showing that our approach provides significant improvement in terms of computational time.

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Sparse Tensor Approximation of Parametric Eigenvalue Problems

Cited by 39 publications

References 33 publications

Multiparametric shell eigenvalue problems

Multiparametric shell eigenvalue problems

Analysis of quasi-Monte Carlo methods for elliptic eigenvalue problems with stochastic coefficients

A reduced basis approach to large‐scale pseudospectra computation

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