A flexible numerical approach for quantification of epistemic uncertainty

Chen, Xiaoxiao; Park, Eun Jae; Xiu, Dongbin

doi:10.1016/j.jcp.2013.01.018

Cited by 28 publications

(35 citation statements)

References 9 publications

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“…These method have their own advantages, though most do not address efficient numerical implementations. More recent studies employ approximation theory [1,5,13,8].…”

Section: Introductionmentioning

confidence: 99%

“…These method have their own advantages, though most do not address efficient numerical implementations. More recent studies employ approximation theory [1,5,13,8].This paper is largely motivated by the work of [2], where a method utilizing the variational formulas of relative entropy is developed to derive upper bounds for the predictions of epistemic uncertainty computations. In this paper we develop a methodology, similar to that of [2], for failure probability computation.…”

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confidence: 99%

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Computation of Failure Probability Subject to Epistemic Uncertainty

Xiu

2012

SIAM J. Sci. Comput.

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Computing failure probability is a fundamental task in many important practical problems. The computation, its numerical challenges aside, naturally requires knowledge of the probability distribution of the underlying random inputs. On the other hand, for many complex systems it is often not possible to have complete information about the probability distributions. In such cases the uncertainty is often referred to as epistemic uncertainty, and straightforward computation of the failure probability is not available. In this paper we develop a method to estimate both the upper bound and the lower bound of the failure probability subject to epistemic uncertainty. The bounds are rigorously derived using the variational formulas for relative entropy. We examine in detail the properties of the bounds and present numerical algorithms to efficiently compute them. Introduction.Probability of failure is an important quantity in many applications involving system safety, risk management, reliability analysis, etc. Accurate computation of failure probability is thus of fundamental significance. A large amount of literature has been devoted to this task, ranging from more mathematically rigorous studies for model problems to more heuristic ones for engineering systems.This paper seeks to study the problem in a different context. That is, we study how to estimate the probability of failure when complete knowledge of the input probability distribution is not available. When lack of knowledge is the primary source of the uncertainty, it is often referred to as epistemic uncertainty. Though different definitions and classifications exist in the literature, in this paper we will use epistemic uncertainty to refer to the random inputs whose complete information about the probability distribution is not available and use aleatory uncertainty to refer to the random inputs whose probability distribution is fully prescribed.The study of the impacts of epistemic uncertainty is more difficult because many of the existing probabilistic tools do not readily apply. Some of the existing approaches include evidence theory [7], possibility theory [3], and interval analysis [6,12]. These method have their own advantages, though most do not address efficient numerical implementations. More recent studies employ approximation theory [1,5,13,8].This paper is largely motivated by the work of [2], where a method utilizing the variational formulas of relative entropy is developed to derive upper bounds for the predictions of epistemic uncertainty computations. In this paper we develop a methodology, similar to that of [2], for failure probability computation. Most notably we derive both the upper bound and the lower bound for the failure probability subject

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“…These method have their own advantages, though most do not address efficient numerical implementations. More recent studies employ approximation theory [1,5,13,8].…”

Section: Introductionmentioning

confidence: 99%

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confidence: 99%

Computation of Failure Probability Subject to Epistemic Uncertainty

Xiu

2012

SIAM J. Sci. Comput.

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“…The method will proceed from this assumption and eventually reconstruct the correct dependence structure of the original problem in the sampling stage. Such is the foundational results of [20,7] and will be made clear in the following sections.…”

Section: Local Parameterization Of Subdomain Problemsmentioning

confidence: 77%

“…Here we briefly review the numerical method developed in [20,7] to solve this kind of problem. Denote…”

Section: General Procedurementioning

confidence: 99%

Local Polynomial Chaos Expansion for Linear Differential Equations with High Dimensional Random Inputs

Chen¹,

Jakeman²,

Gittelson

et al. 2015

SIAM J. Sci. Comput.

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In this paper we present a localized polynomial chaos expansion for partial differential equations (PDE) with random inputs. In particular, we focus on time independent linear stochastic problems with high dimensional random inputs, where the traditional polynomial chaos methods, and most of the existing methods, incur prohibitively high simulation cost. The local polynomial chaos method employs a domain decomposition technique to approximate the stochastic solution locally. In each subdomain, a subdomain problem is solved independently and, more importantly, in a much lower dimensional random space. In a postprocesing stage, accurate samples of the original stochastic problems are obtained from the samples of the local solutions by enforcing the correct stochastic structure of the random inputs and the coupling conditions at the interfaces of the subdomains. Overall, the method is able to solve stochastic PDEs in very large dimensions by solving a collection of low dimensional local problems and can be highly efficient. In this paper we present the general mathematical framework of the methodology and use numerical examples to demonstrate the properties of the method.

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“…Due to the necessity and importance of treating the aleatory and epistemic uncertainties properly with corresponding mathematical methods rather than simply using the traditional probabilistic methods to treat all the uncertainties as random ones under strong assumptions (Der Kiureghian and Ditlevsen 2009), there emerges increasing literature in recent years to address the reliability analysis problems under both aleatory and epistemic uncertainties, e.g. Fuzzy set theory (Zhang and Huang 2010;Li et al 2014;He et al 2015), random set theory (Oberguggenberger 2015) and probabilistic bounding analysis (Sentz and Ferson 2011), combined probabilistic and interval analysis method (Jiang et al 2013), combined probabilistic and evidence theory method (Du 2008;Eldred et al 2011;Yao et al 2013b), and other numerical approaches such as doubleloop Monte-Carlo Simulation (MCS) (Du et al 2009), perturbation based method (Gao et al 2010(Gao et al , 2011, encapsulation based method (Jakeman et al 2010;Chen et al 2013), families of Johnson distributions based probabilistic method (Urbina et al 2011;Zaman et al 2011), etc. Among these researches, one of the widely used methods is to model the epistemic uncertainties with intervals and generally the interval bounds are fixed.…”

Section: Introductionmentioning

confidence: 99%

An extended probabilistic method for reliability analysis under mixed aleatory and epistemic uncertainties with flexible intervals

Chen

Yao

Zhao

et al. 2016

Struct Multidisc Optim

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The reliability analysis approach based on combined probability and evidence theory is studied in this paper to address the reliability analysis problem involving both aleatory uncertainties and epistemic uncertainties with flexible intervals (the interval bounds are either fixed or variable as functions of other independent variables). In the standard mathematical formulation of reliability analysis under mixed uncertainties with combined probability and evidence theory, the key is to calculate the failure probability of the upper and lower limits of the system response function as the epistemic uncertainties vary in each focal element. Based on measure theory, in this paper it is proved that the aforementioned upper and lower limits of the system response function are measurable under certain circumstances (the system response function is continuous and the flexible interval bounds satisfy certain conditions), which accordingly can be treated as random variables. Thus the reliability analysis of the system response under mixed uncertainties can be directly treated as probability calculation problems and solved by existing welldeveloped and efficient probabilistic methods. In this paper the popular probabilistic reliability analysis method FORM (First Order Reliability Method) is taken as an example to illustrate how to extend it to solve the reliability analysis problem in the mixed uncertainty situation. The efficacy of the proposed method is demonstrated with two numerical examples and one practical satellite conceptual design problem.

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A flexible numerical approach for quantification of epistemic uncertainty

Cited by 28 publications

References 9 publications

Computation of Failure Probability Subject to Epistemic Uncertainty

Computation of Failure Probability Subject to Epistemic Uncertainty

Local Polynomial Chaos Expansion for Linear Differential Equations with High Dimensional Random Inputs

An extended probabilistic method for reliability analysis under mixed aleatory and epistemic uncertainties with flexible intervals

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