Simultaneous reduced basis approximation of parameterized elliptic eigenvalue problems

Horger, Thomas; Wohlmuth, Barbara; Dickopf, Thomas

doi:10.1051/m2an/2016025

Cited by 27 publications

(22 citation statements)

References 38 publications

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“…Although stochastic eigenproblems have been of interest in engineering for some time, the mathematical literature is less developed. 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.…”

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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Section: Introductionmentioning

confidence: 99%

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

et al. 2019

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“…There are many options to construct a reduced basis, such as the proper orthogonal decomposition (POD), as well as single-and multi-choice greedy approaches. POD and greedy approaches for parameterised eigenproblems are discussed and compared in [35]. In this paper we focus on the POD.…”

Section: Reduced Basis Approach To Parameterised Eigenproblemsmentioning

confidence: 99%

“…In contrast, parameterised eigenproblems have not been treated extensively with reduced basis ideas. We refer to [27,35,36,43,61,67] for applications and reviews of reduced basis surrogates for parameterised eigenproblems with PDEs. For non-parametric KL eigenproblems we mention that reduced basis methods have been combined with domain decomposition ideas [14].…”

Section: Introductionmentioning

confidence: 99%

Fast sampling of parameterised Gaussian random fields

Latz

Eisenberger

Ullmann

2019

Computer Methods in Applied Mechanics and Engineering

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Gaussian random fields are popular models for spatially varying uncertainties, arising for instance in geotechnical engineering, hydrology or image processing. A Gaussian random field is fully characterised by its mean function and covariance operator. In more complex models these can also be partially unknown. In this case we need to handle a family of Gaussian random fields indexed with hyperparameters. Sampling for a fixed configuration of hyperparameters is already very expensive due to the nonlocal nature of many classical covariance operators. Sampling from multiple configurations increases the total computational cost severely. In this report we employ parameterised Karhunen-Loève expansions for sampling. To reduce the cost we construct a reduced basis surrogate built from snapshots of Karhunen-Loève eigenvectors. In particular, we consider Matérn-type covariance operators with unknown correlation length and standard deviation. We suggest a linearisation of the covariance function and describe the associated online-offline decomposition. In numerical experiments we investigate the approximation error of the reduced eigenpairs. As an application we consider forward uncertainty propagation and Bayesian inversion with an elliptic partial differential equation where the logarithm of the diffusion coefficient is a parameterised Gaussian random field. In the Bayesian inverse problem we employ Markov chain Monte Carlo on the reduced space to generate samples from the posterior measure. All numerical experiments are conducted in 2D physical space, with non-separable covariance operators, and finite element grids with ∼ 10 4 degrees of freedom.

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“…Alternatively to the classical reduced basis approach, component based reduction strategies are considered in [37]. Here, we follow the ideas of [18] where rigorous bounds in the case of multi-query and multiple eigenvalues are given.…”

Section: Introductionmentioning

confidence: 99%

Reduced Basis Isogeometric Mortar Approximations for Eigenvalue Problems in Vibroacoustics

Horger

Wohlmuth

Wunderlich

2017

Model Reduction of Parametrized Systems

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We simulate the vibration of a violin bridge in a multi-query context using reduced basis techniques. The mathematical model is based on an eigenvalue problem for the orthotropic linear elasticity equation. In addition to the nine material parameters, a geometrical thickness parameter is considered. This parameter enters as a 10th material parameter into the system by a mapping onto a parameter independent reference domain. The detailed simulation is carried out by isogeometric mortar methods. Weakly coupled patch-wise tensorial structured isogeometric elements are of special interest for complex geometries with piecewise smooth but curvilinear boundaries. To obtain locality in the detailed system, we use the saddle point approach and do not apply static condensation techniques. However within the reduced basis context, it is natural to eliminate the Lagrange multiplier and formulate a reduced eigenvalue problem for a symmetric positive definite matrix. The selection of the snapshots is controlled by a multi-query greedy strategy taking into account an error indicator allowing for multiple eigenvalues.

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Simultaneous reduced basis approximation of parameterized elliptic eigenvalue problems

Cited by 27 publications

References 38 publications

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

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

Fast sampling of parameterised Gaussian random fields

Reduced Basis Isogeometric Mortar Approximations for Eigenvalue Problems in Vibroacoustics

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