Bounding the Spectral Gap for an Elliptic Eigenvalue Problem with Uniformly Bounded Stochastic Coefficients

Gilbert, Alexander D.; Graham, Ivan G.; Scheichl, Robert; Sloan, Ian H.

doi:10.1007/978-3-030-38230-8_3

Cited by 7 publications

(6 citation statements)

References 11 publications

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“…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].…”

Section: Introductionmentioning

confidence: 92%

Multilevel quasi-Monte Carlo for random elliptic eigenvalue problems I: Regularity and error analysis

Gilbert¹,

Scheichl²

2020

Preprint

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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 , focusses on practical extensions of the MLQMC algorithm to improve efficiency, and presents numerical results.

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Section: Introductionmentioning

confidence: 92%

Multilevel quasi-Monte Carlo for random elliptic eigenvalue problems I: Regularity and error analysis

Gilbert¹,

Scheichl²

2020

Preprint

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“…For the analysis relevant, it requires that the spectral gap λ 2 (y) − λ 1 (y) is bounded away from zero uniformly in U ∞ . In this paper we give a simple proof for the uniform positivity of this spectral gap under a weaker assumption compared to [15,16]. To do this we point out that the set…”

Section: Introductionmentioning

confidence: 96%

“…Gantner et al [12], Gilbert et al [15,16,17,18], Graham et al [20,19,21], Herrmann and Schwab [26,25,27], Kazashi [28], Kuo and Nuyens [29,30], Kuo et al [32,33,31], Lemieux [34], Leobacher and Pillichshammer [35], Nguyen and Nuyens [36,37], Nichols and Kuo [38] to mention just a few.…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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Analyticity of Parametric Elliptic Eigenvalue Problems and Applications to Quasi-Monte Carlo Methods

Nguyen

2022

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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 a 0 (x) + j∈N y j a j (x), where elements of the parameter vector y = (y j ) j∈N ∈ U ∞ are independent and identically uniformly distributed on< ∞ with some positive sequence (ρ j ) j∈N ∈ ℓ p (N) for p ∈ (0, 1] we show that for any y ∈ U ∞ , the elliptic partial differential operator has a countably infinite number of eigenvalues (λ j (y)) j∈N which can be ordered non-decreasingly. Moreover, the spectral gap λ 2 (y) − λ 1 (y) is uniformly positive in U ∞ . From this, we prove the holomorphic extension property of λ 1 (y) to a complex domain in C ∞ and estimate mixed derivatives of λ 1 (y) with respect to the parameters 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 λ 1 (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.

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“…The latter two classes perform poorly for high-dimensional problems, so in order to handle the high-dimensionality of the parameter space, sparse and low-rank versions of those methods have been developed, see e.g., [1,24,17]. Furthermore, to improve upon classical Monte Carlo, while still performing well in high dimensions, the present authors with their colleagues have analysed the use of quasi-Monte Carlo methods [19,20].…”

Section: Introductionmentioning

confidence: 99%

Multilevel quasi-Monte Carlo for random elliptic eigenvalue problems II: Efficient algorithms and numerical results

Gilbert¹,

Scheichl²

2021

Preprint

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Stochastic PDE eigenvalue problems often arise in the field of uncertainty quantification, whereby one seeks to quantify the uncertainty in an eigenvalue, or its eigenfunction. In this paper we present an efficient multilevel quasi-Monte Carlo (MLQMC) algorithm for computing the expectation of the smallest eigenvalue of an elliptic eigenvalue problem with stochastic coefficients. Each sample evaluation requires the solution of a PDE eigenvalue problem, and so tackling this problem in practice is notoriously computationally difficult. We speed up the approximation of this expectation in four ways: 1) we use a multilevel variance reduction scheme to spread the work over a hierarchy of FE meshes and truncation dimensions; 2) we use QMC methods to efficiently compute the expectations on each level; 3) we exploit the smoothness in parameter space and reuse the eigenvector from a nearby QMC point to reduce the number of iterations of the eigensolver; and 4) we utilise a two-grid discretisation scheme to obtain the eigenvalue on the fine mesh with a single linear solve. The full error analysis of a basic MLQMC algorithm is given in the companion paper [Gilbert and Scheichl, 2021], and so in this paper we focus on how to further improve the efficiency and provide theoretical justification of the enhancement strategies 3) and 4). Numerical results are presented that show the efficiency of our algorithm, and also show that the four strategies we employ are complementary.

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Bounding the Spectral Gap for an Elliptic Eigenvalue Problem with Uniformly Bounded Stochastic Coefficients

Cited by 7 publications

References 11 publications

Multilevel quasi-Monte Carlo for random elliptic eigenvalue problems I: Regularity and error analysis

Multilevel quasi-Monte Carlo for random elliptic eigenvalue problems I: Regularity and error analysis

Analyticity of Parametric Elliptic Eigenvalue Problems and Applications to Quasi-Monte Carlo Methods

Multilevel quasi-Monte Carlo for random elliptic eigenvalue problems II: Efficient algorithms and numerical results

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