Elements of a function analytic approach to probability

Red-Horse, John; Ghanem, Roger

doi:10.1002/nme.2643

Cited by 5 publications

(5 citation statements)

References 55 publications

(23 reference statements)

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“…The main difficulty when H is infinite dimensional is to ensure the existence of some ξ = {ξ i } i∈I such that ξ, χ H ∼ N (0, ||χ|| 2 H ) for all χ ∈ H. Gaussian measures on infinite dimensional spaces are defined in terms of real measures on their dual space [9,28]. In practice this means that often there is no H-valued Gaussian variable ξ.…”

Section: Extension To Infinite-dimensional Spacesmentioning

confidence: 99%

Reduced Wiener Chaos representation of random fields via basis adaptation and projection

Tsilifis

Ghanem

2017

Journal of Computational Physics

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A new characterization of random fields appearing in physical models is presented that is based on their well-known Homogeneous Chaos expansions. We take advantage of the adaptation capabilities of these expansions where the core idea is to rotate the basis of the underlying Gaussian Hilbert space, in order to achieve reduced functional representations that concentrate the induced probability measure in a lower dimensional subspace. For a smooth family of rotations along the domain of interest, the uncorrelated Gaussian inputs are transformed into a Gaussian process, thus introducing a mesoscale that captures intermediate characteristics of the quantity of interest.

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Section: Extension To Infinite-dimensional Spacesmentioning

confidence: 99%

Reduced Wiener Chaos representation of random fields via basis adaptation and projection

Tsilifis

Ghanem

2017

Journal of Computational Physics

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“…The manner in which second‐order descriptors and their properties are defined for random variables with values in an infinite‐dimensional Hilbert space is more complicated than the manner in which they are defined for random variables with values in a finite‐dimensional Euclidean space. All definitions used here are consistent with those given in references . Now, the mean is defined as the linear function m q from H into

double-struckR

such that

m_{bold-italicq} (bold-italicp) MathClass-rel= {MathClass-op\int}_{Θ} {⟨ bold-italicq MathClass-punc, bold-italicp ⟩}_{H} dP MathClass-punc, MathClass-rel\forall bold-italicp MathClass-rel\in H MathClass-punc,

and the covariance is defined as the bilinear function c q from H × H into

double-struckR

such that

c_{bold-italicq} (bold-italicp MathClass-punc, bold-italicr) MathClass-rel= {MathClass-op\int}_{Θ} ((), {⟨ bold-italicq MathClass-punc, bold-italicp ⟩}_{H} MathClass-bin- m_{bold-italicq} (bold-italicp)) ((), {⟨ bold-italicq MathClass-punc, bold-italicr ⟩}_{H} MathClass-bin- m_{bold-italicq} (bold-italicr)) dP MathClass-punc, MathClass-rel\forall bold-italicp MathClass-punc, bold-italicr MathClass-rel\in H MathClass-punc.

…”

Section: Proposed Adaptation Of the Karhunen–loeve Decompositionmentioning

confidence: 99%

“…The manner in which second-order descriptors and their properties are defined for random variables with values in an infinite-dimensional Hilbert space is more complicated than the manner in which they are defined for random variables with values in a finitedimensional Euclidean space. All definitions used here are consistent with those given in references [18][19][20][21]. Now, the mean is defined as the linear function m q from H into R such that…”

Section: Second-order Descriptorsmentioning

confidence: 99%

“…As we have already mentioned, the structure of this function space can be expected to determine the significance of the function values, the derivatives, and the various other hallmarks of the solution that needs to be reduced. A detailed explanation of the distinction between classical stochastic processes and function space‐valued random variables can be found in and the references therein. This distinction has repercussions on the manner in which the second‐order descriptors are defined and the properties of these second‐order descriptors can be exploited to construct a reduced‐dimensional representation.…”

Section: Proposed Adaptation Of the Karhunen–loeve Decompositionmentioning

confidence: 99%

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Dimension reduction in stochastic modeling of coupled problems

Arnst

Ghanem

Phipps

et al. 2012

Numerical Meth Engineering

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SUMMARY Coupled problems with various combinations of multiple physics, scales, and domains are found in numerous areas of science and engineering. A key challenge in the formulation and implementation of corresponding coupled numerical models is to facilitate the communication of information across physics, scale, and domain interfaces, as well as between the iterations of solvers used for response computations. In a probabilistic context, any information that is to be communicated between subproblems or iterations should be characterized by an appropriate probabilistic representation. Although the number of sources of uncertainty can be expected to be large in most coupled problems, our contention is that exchanged probabilistic information often resides in a considerably lower dimensional space than the sources themselves. This work thus presents an investigation into the characterization of the exchanged information by a reduced‐dimensional representation and in particular by an adaptation of the Karhunen‐Loève decomposition. The effectiveness of the proposed dimension–reduction methodology is analyzed and demonstrated through a multiphysics problem relevant to nuclear engineering. Copyright © 2012 John Wiley & Sons, Ltd.

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“…Indeed, there is a growing awareness that rare failure events require careful study and treatment [4]. It is worth noting that, although this paper focusses only on certification, uncertainty quantification is not limited to certification alone: there is also considerable interest in studying the propagation of uncertainties through models and systems; for some systems, polynomial chaos expansions are well adapted for this task [5,6].…”

Section: Introductionmentioning

confidence: 99%

Uncertainty quantification via codimension‐one partitioning

Sullivan

Topcu

McKerns

et al. 2010

Numerical Meth Engineering

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SUMMARYWe consider uncertainty quantification in the context of certification, i.e. showing that the probability of some 'failure' event is acceptably small. In this paper, we derive a new method for rigorous uncertainty quantification and conservative certification by combining McDiarmid's inequality with input domain partitioning and a new concentration-of-measure inequality. We show that arbitrarily sharp upper bounds on the probability of failure can be obtained by partitioning the input parameter space appropriately; in contrast, the bound provided by McDiarmid's inequality is usually not sharp. We prove an error estimate for the method (Proposition 3.2); we define a codimension-one recursive partitioning scheme and prove its convergence properties (Theorem 4.1); finally, we apply a new concentration-of-measure inequality to give confidence levels when empirical means are used in place of exact ones (Section 5).

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Elements of a function analytic approach to probability

Cited by 5 publications

References 55 publications

Reduced Wiener Chaos representation of random fields via basis adaptation and projection

Reduced Wiener Chaos representation of random fields via basis adaptation and projection

Dimension reduction in stochastic modeling of coupled problems

Uncertainty quantification via codimension‐one partitioning

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