On the Density Functions of Integrals of Gaussian Random Fields

Liu, Jingchen; Xu, Gongjun

doi:10.1017/s0001867800006388

Cited by 3 publications

(2 citation statements)

References 21 publications

(25 reference statements)

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“…For heavy-tailed stochastic systems, some recent works are [5][6][7]. In the context of Gaussian processes and random fields, the most well studied events are the high level excursions (tail events of the supremum) [1]; the tail events of other convex functionals of Gaussian random fields are also of interest [15][16][17][18]. The method in this paper is in part built on the results in this literature.…”

Section: Introductionmentioning

confidence: 99%

Efficient Rare Event Simulation for Failure Problems in Random Media

Liu

Zhou

2015

SIAM J. Sci. Comput.

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In this paper we study rare events associated to the solutions of an elliptic partial differential equation with a spatially varying random coefficient. The random coefficient follows the lognormal distribution, which is determined by a Gaussian process. This model is employed to study the failure problem of elastic materials in random media in which the failure is characterized by the criterion that the strain field exceeds a high threshold. We propose an efficient importance sampling scheme to compute the small failure probability in the high threshold limit. The change of measure in our scheme is parametrized by two density functions. The efficiency of the importance sampling scheme is validated by numerical examples. Introduction.The study and computation of rare events in stochastic systems have received intensive attention in recent years. Rare events, though they do not occur often, represent the most severe consequence of uncertainty and random effects. The study of these rare events hence gives crucial understanding and has important applications. However, due to the small probabilities of occurrence of such events, quantification casts a serious challenge for conventional probabilistic methods. For example, a direct Monte Carlo strategy to estimate the vanishing small probability will require a huge number of sample points to give estimates with small relative error; in other words, the huge relative variance of these estimators makes them incapable of accurate prediction.In this work, we aim at developing an efficient important sampling strategy to study the rare events associated with a materials failure problem. The method we develop in this work applies to the general linear elasticity model. For simplicity, we will restrict our discussions here to a scalar model in two dimensions, which can be viewed as a model for out-of-plane deformation of an elastic membrane under external forcing. Similar equations also arise from other contexts, such as groundwater hydraulics and electrostatic response of a planar media. Let D ⊂ R 2 be an open domain with smooth boundary, which is the equilibrium configuration of the membrane. We consider an out-of-plane displacement field u given by the following boundary

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Section: Introductionmentioning

confidence: 99%

Efficient Rare Event Simulation for Failure Problems in Random Media

Liu

Zhou

2015

SIAM J. Sci. Comput.

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“…Importance sampling based efficient simulation procedures have been proposed in the literature to estimate the tail probabilities. Numerical methods for rare-event analysis of the suprema were studied in [1,2]; see also [8,20,24,[26][27][28]34] for related studies.…”

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Uniformly efficient simulation for extremes of Gaussian random fields

2018

J. Appl. Probab.

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This paper considers the problem of simultaneously estimating rare-event probabilities for a class of Gaussian random fields. A conventional rareevent simulation method is usually tailored to a specific rare event and consequently would lose estimation efficiency for different events of interest, which often results in additional computational cost in such simultaneous estimation problem. To overcome this issue, we propose a uniformly efficient estimator for a general family of Hölder continuous Gaussian random fields.We establish the asymptotic and uniform efficiency of the proposed method and also conduct simulation studies to illustrate its effectiveness.

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Rare-event simulation and efficient discretization for the supremum of Gaussian random fields

Liu

2015

Advances in Applied Probability

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In this paper, we consider a classic problem concerning the high excursion probabilities of a Gaussian random field f living on a compact set T . We develop efficient computational methods for the tail probabilities P (sup T f (t) > b) and the conditional expectationsFor each ε positive, we present Monte Carlo algorithms that run in constant time and compute the interesting quantities with ε relative error for arbitrarily large b. The efficiency results are applicable to a large class of Hölder continuous Gaussian random fields. Besides computations, the proposed change of measure and its analysis techniques have several theoretical and practical indications in the asymptotic analysis of extremes of Gaussian random fields.

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On the Density Functions of Integrals of Gaussian Random Fields

Cited by 3 publications

References 21 publications

Efficient Rare Event Simulation for Failure Problems in Random Media

Efficient Rare Event Simulation for Failure Problems in Random Media

Uniformly efficient simulation for extremes of Gaussian random fields

Rare-event simulation and efficient discretization for the supremum of Gaussian random fields

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