Figure - available from: Numerische Mathematik
This content is subject to copyright. Terms and conditions apply.
Convergence of the stochastic errors for the eigenfunction and eigenvalue as computed by Algorithm 1. The points represent a log–log plot of the errors as a function of #Aϵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\# \mathcal {A}_{\epsilon }$$\end{document}. The dashed lines represent the rate (#Aϵ)-1.9\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\# \mathcal {A}_{\epsilon })^{-1.9}$$\end{document}

Convergence of the stochastic errors for the eigenfunction and eigenvalue as computed by Algorithm 1. The points represent a log–log plot of the errors as a function of #Aϵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\# \mathcal {A}_{\epsilon }$$\end{document}. The dashed lines represent the rate (#Aϵ)-1.9\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\# \mathcal {A}_{\epsilon })^{-1.9}$$\end{document}

Source publication
Article
Full-text available
We consider and analyze applying a spectral inverse iteration algorithm and its subspace iteration variant for computing eigenpairs of an elliptic operator with random coefficients. With these iterative algorithms the solution is sought from a finite dimensional space formed as the tensor product of the approximation space for the underlying stocha...

Similar publications

Article
Full-text available
In this work we introduce a new method to compute the matrix cosine. It is based on recent new matrix polynomial evaluation methods for the Taylor approximation and a mixed forward and backward error analysis. The matrix polynomial evaluation methods allow to evaluate the Taylor polynomial approximation of the matrix cosine function more efficientl...

Citations

... where E [ · ] is the expectation operator defined by (7). Please note that in our work, we deal with non-degenerated eigenvalue problems, as in [42,56]. In this case, simple eigenvalues are sufficiently well separated from the rest of the spectrum. ...
... As noted above, we deal with the non-degenerate solution with simple eigenvalues of the problem (12), excluding from our analysis the cases of clustered eigenvalues and their associated invariant subspaces. As a result, under certain assumptions concerning second-order random fields, which are fulfilled in our case as well, the existence and uniqueness of a weak solution of (12) can be proved [42,59,56]. Finally, since the variational formulation of (12) involves expectations of the weak formulation in a physical space, the standard finite elements method (FEM) compound with the stochastic collocation method (SCM) can be used to find its solution [50]. ...
... where E [ · ] is the expectation operator defined by (7). Please note that in our work, we deal with non-degenerated eigenvalue problems, as in [42,56]. In this case, simple eigenvalues are sufficiently well separated from the rest of the spectrum. ...
... As noted above, we deal with the non-degenerate solution with simple eigenvalues of the problem (12), excluding from our analysis the cases of clustered eigenvalues and their associated invariant subspaces. As a result, under certain assumptions concerning second-order random fields, which are fulfilled in our case as well, the existence and uniqueness of a weak solution of (12) can be proved [42,59,56]. Finally, since the variational formulation of (12) involves expectations of the weak formulation in a physical space, the standard finite elements method (FEM) compound with the stochastic collocation method (SCM) can be used to find its solution [50]. ...
... A benchmark for stochastic collocation methods for multiparametric eigenvalue problems is the sparse anisotropic collocation algorithm analyzed by Andreev and Schwab in [1]. In the class of stochastic Galerkin methods, on the other hand, many different variants of the have been proposed over the years [11,14,19,26]. Quite recently, low-rank methods have also been introduced [2,7,24]. ...
... For such eigenpairs we therefore have optimal rates of convergence for stochastic collocation algorithms, see [1] for details, and optimal asymptotic rates of convergence for the iterative Galerkin based algorithms considered in [14]. However, these results do not apply to cases where the eigenvalues are of higher multiplicity or where they are allowed to cross within the parameter space. ...
... However, due to possible eigenvalue crossings, this may sometimes be an extremely difficult task to perform computationally, see e.g. [15], [14]. Therefore, we will work under the assumption that the eigenspace of interest is isolated, i.e., the associated eigenvalues are strictly separated from the rest of the spectrum. ...
Article
Full-text available
We consider computing eigenspaces of an elliptic self-adjoint operator depending on a countable number of parameters in an affine fashion. The eigenspaces of interest are assumed to be isolated in the sense that the corresponding eigenvalues are separated from the rest of the spectrum for all values of the parameters. We show that such eigenspaces can in fact be extended to complex-analytic functions of the parameters and quantify this analytic dependence in a way that leads to convergence of sparse polynomial approximations. A stochastic collocation method on an anisoptropic sparse grid in the parameter domain is proposed for computing a basis for the eigenspace of interest. The convergence of this method is verified in a series of numerical examples based on the eigenvalue problem of a stochastic diffusion operator.
... Except for [24] and [30], the first of which considers a tracking technique to detect crossings of eigenvalues and the second of which considers eigenspaces, all of these methods consider the case of eigenvalues of multiplicity one. As an alternative to samplingbased methods, stochastic Galerkin methods where discussed in [6,21,25,31,57], with main emphasis on the acceleration of the eigensolvers. However, crossings, bifurcations, and a rigorous error analysis of the numerical approximations compared to the analytic reality does not appear to have been the subject of investigation. ...
Preprint
We consider generalized operator eigenvalue problems in variational form with random perturbations in the bilinear forms. This setting is motivated by variational forms of partial differential equations such as the diffusion equation or Maxwell's equations with random material laws, for example. The considered eigenpairs can be of higher but finite multiplicity. We investigate stochastic quantities of interest of the eigenpairs and discuss why, for multiplicity greater than 1, only the stochastic properties of the eigenspaces are meaningful, but not the ones of individual eigenpairs. To that end, we characterize the Fr\'echet derivatives of the eigenpairs with respect to the perturbation and provide a new linear characterization for eigenpairs of higher multiplicity. As a side result, we prove local analyticity of the eigenspaces. Based on the Fr\'echet derivatives of the eigenpairs we discuss a meaningful Monte Carlo sampling strategy for multiple eigenvalues and develop an uncertainty quantification perturbation approach. Numerical examples are presented to illustrate the theoretical results.
... The eigenvalue problems of parametric or stochastic elliptic differential operators have been of interest for the past fifty years, see [41,40,13,42,1,43,22,11,23,15,16,17,18] and references therein. These problems appear in many areas of engineering and physics, for example, in nuclear reactor physics; photonics; quantum physics; acoustic; or in electromagnetic. ...
Preprint
In the present paper, we study the analyticity of the leftmost eigenvalue of the linear elliptic partial differential operator with random coefficient and analyze the convergence rate of the quasi-Monte Carlo method for approximation of the expectation of this quantity. The random coefficient is assumed to be represented by an affine expansion a0(x)+jNyjaj(x)a_0(\boldsymbol{x})+\sum_{j\in \mathbb{N}}y_ja_j(\boldsymbol{x}), where elements of the parameter vector y=(yj)jNU\boldsymbol{y}=(y_j)_{j\in \mathbb{N}}\in U^\infty are independent and identically uniformly distributed on U:=[12,12]U:=[-\frac{1}{2},\frac{1}{2}]. Under the assumption jNρjajL(D)< \|\sum_{j\in \mathbb{N}}\rho_j|a_j|\|_{L_\infty(D)} <\infty with some positive sequence (ρj)jNp(N)(\rho_j)_{j\in \mathbb{N}}\in \ell_p(\mathbb{N}) for p(0,1]p\in (0,1] we show that for any yU\boldsymbol{y}\in U^\infty, the elliptic partial differential operator has a countably infinite number of eigenvalues (λj(y))jN(\lambda_j(\boldsymbol{y}))_{j\in \mathbb{N}} which can be ordered non-decreasingly. Moreover, the spectral gap λ2(y)λ1(y)\lambda_2(\boldsymbol{y})-\lambda_1(\boldsymbol{y}) is uniformly positive in UU^\infty. From this, we prove the holomorphic extension property of λ1(y)\lambda_1(\boldsymbol{y}) to a complex domain in C\mathbb{C}^\infty and estimate mixed derivatives of λ1(y)\lambda_1(\boldsymbol{y}) with respect to the parameters y\boldsymbol{y} by using Cauchy's formula for analytic functions. Based on these bounds we prove the dimension-independent convergence rate of the quasi-Monte Carlo method to approximate the expectation of λ1(y)\lambda_1(\boldsymbol{y}). In this case, the computational cost of fast component-by-component algorithm for generating quasi-Monte Carlo N-points scales linearly in terms of integration dimension.
... This is evident from the recent papers in this domain. For example, in three separate works [24][25][26], the authors developed low-rank solutions for stochastic eigenvalue problems. It is worth noting that the complexity associated with the solution of the random eigenvalue problem is significantly enhanced when one deals with non-proportional damping systems; this is because the eigensolutions for such systems become complex. ...
Article
Uncertainties need to be taken into account in the dynamic analysis of complex structures. This is because in some cases uncertainties can have a significant impact on the dynamic response and ignoring it can lead to unsafe design. For complex systems with uncertainties, the dynamic response is characterised by the eigenvalues and eigenvectors of the underlying generalised matrix eigenvalue problem. This paper aims at developing computationally efficient methods for random eigenvalue problems arising in the dynamics of multi-degree-of-freedom systems. There are efficient methods available in the literature for obtaining eigenvalues of random dynamical systems. However, the computation of eigenvectors remains challenging due to the presence of a large number of random variables within a single eigenvector. To address this problem, we project the random eigenvectors on the basis spanned by the underlying deterministic eigenvectors and apply a Galerkin formulation to obtain the unknown coefficients. The overall approach is simplified using an iterative technique. Two numerical examples are provided to illustrate the proposed method. Full-scale Monte Carlo simulations are used to validate the new results.
... The most widely used numerical methods for stochastic EVPs are Monte Carlo methods [35], then more recently stochastic collocation methods [1] and stochastic Galerkin/polynomial chaos methods [15,41,42]. In particular, to deal with the high-dimensionality of the parameter space, sparse and low-rank methods have been considered, see [1,14,22,23]. Additionally, the present authors (along with colleagues) have applied quasi-Monte Carlo methods to (1.1) and proved some key properties of the minimal eigenvalue and its corresponding eigenfunction, see [17,18]. ...
Preprint
Random eigenvalue problems are useful models for quantifying the uncertainty in several applications from the physical sciences and engineering, e.g., structural vibration analysis, the criticality of a nuclear reactor or photonic crystal structures. In this paper we present a simple multilevel quasi-Monte Carlo (MLQMC) method for approximating the expectation of the minimal eigenvalue of an elliptic eigenvalue problem with coefficients that are given as a series expansion of countably-many stochastic parameters. The MLQMC algorithm is based on a hierarchy of discretisations of the spatial domain and truncations of the dimension of the stochastic parameter domain. To approximate the expectations, randomly shifted lattice rules are employed. This paper is primarily dedicated to giving a rigorous analysis of the error of this algorithm. A key step in the error analysis requires bounds on the mixed derivatives of the eigenfunction with respect to both the stochastic and spatial variables simultaneously. An accompanying paper [Gilbert and Scheichl, 2020], focusses on practical extensions of the MLQMC algorithm to improve efficiency, and presents numerical results.
... Power iteration [22] Or power method is an eigenvalue algorithm which may converge slowly Inverse iteration [22] Or inverse power method is an iterative eigenvalue algorithm, the method is similar to power iteration method. (It originally developed for computing the resonance of frequencies in structural analysis) Inverse iteration with spectral shifting [23] In this context the eigenpair of interest is the ground state, i.e., the smallest eigenvalue and the associated eigenfunction of the system Rayliegh quotients [24,25] Known as Rayliegh-Ritz ration, which is defined for Hermitian matrix. For a given matrix, Rayliegh quotients reaches its minimum value (the smallest eigenvalue of Hermitian matrix) Table 2. Subspace construction methods. ...
Article
Thin-walled structures have been widely adopted in engineering projects because of the advantages of good load-bearing capacity, light weight and low cost. In this paper, a refined finite element method (FEM) based on the Carrera Unified Formula (CUF) is performed to analyze the free vibration of thin-walled beams with the variables cross-section, length and boundary. The low-dimensional FEM method could improve the efficiency of numerical analysis and reduce the time consumption. The one-dimensional (1D) CUF is employed in which the models are assumed to be a beam-like axis-oriented structures. In this case, the geometry of the thin-walled beam can be discretized as a limited number of 1D beam elements along the axis, while the Lagrange polynomial expansion may be used to approximate the displacement field on the cross-beam surface. Thus, FEM matrices and vectors can be written in terms of fundamental nuclei whose forms are independent of beam theories. The validity and capabilities of the method presented are examined in some numerical examples, and a comparative study is carried out between the method proposed and the three-dimensional element method with and without warping solutions. The results obtained by CUF 1D models are in close agreement with the reference solutions. In addition , it has been argued that the innovative approach presented in this paper can be used as a precise tool for structural analysis for complex cross-section thin-walled beams to reduce computational costs. ARTICLE HISTORY
... The authors of [23,33] used a quadrature-based normalization of eigenvectors. Normalization based on a solution of a small nonlinear problem was proposed by Hakula et al. [12], and Hakula and Laaksonen [13] also provided an asymptotic convergence theory for the stochastic iteration. In an alternative approach, Ghanem and Ghosh [6,9] proposed two numerical schemes-one based on Newton iteration and another based on an optimization problem (see also [8,10]). ...
... Check for convergence. 13: end for ...
... If the number of random variables is high, this leads to the curse of dimensionality which can be alleviated up to a point with special high-dimensional quadratures, the so-called sparse grids [7][8][9][10]. If the uncertainty is only in the material parameters, one can apply the intrusive approach such as stochastic Galerkin methods, see [11] and references therein. ...
... is issue has previously been illustrated in [11] and considered in the case of shell eigenvalue problems in [15]. ...
Article
Full-text available
Numerical simulation of thin solids remains one of the challenges in computational mechanics. The 3D elasticity problems of shells of revolution are dimensionally reduced in different ways depending on the symmetries of the configurations resulting in corresponding 2D models. In this paper, we solve the multiparametric free vibration of complex shell configurations under uncertainty using stochastic collocation with the p -version of finite element method and apply the collocation approach to frequency response analysis. In numerical examples, the sources of uncertainty are related to material parameters and geometry representing manufacturing imperfections. All stochastic collocation results have been verified with Monte Carlo methods.