Rigorous uncertainty quantification without integral testing

Topcu, Ufuk; Lucas, Leonard J.; Owhadi, Houman; Ortiz, Michael

doi:10.1016/j.ress.2010.07.013

Cited by 9 publications

(11 citation statements)

References 15 publications

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“…We start with a brief review of the uncertainty quantification theory for hierarchical systems [Lucas et al, 2008, Topcu et al, 2011, Sun et al, 2020] that provides the basis for the application to multiscale material modelling pursued in this work.…”

Section: Methodsmentioning

confidence: 99%

“…Conveniently, this graph structure can be exploited to divide the material response into interconnected unit mechanisms, or subsystems, and estimate integral uncertainties of the entire system from a quantification of uncertainties for each subsystem and an appropriate measure of interaction between the subsystems. We specifically follow the approach of Topcu et al [2011], which we briefly summarize next.…”

Section: Hierarchical Uncertainty Quantificationmentioning

confidence: 99%

“…In particular, no integral calculation is required at any stage of the analysis. We specifically follow the approach of Topcu et al [2011], which supplies rigorous upper bounds of integral uncertainties for hierarchies of interconnected subsystems through a systematic computation of moduli of continuity for each individual subsystem. The moduli of continuity supply just the right measure of interaction between the subsystems enabling the propagation of uncertainties across the hierarchy.…”

Section: Introductionmentioning

confidence: 99%

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Hierarchical multiscale quantification of material uncertainty

Liu

Sun

Bhattacharya

et al. 2021

Journal of the Mechanics and Physics of Solids

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The macroscopic behavior of many materials is complex and the end result of mechanisms that operate across a broad range of disparate scales. An imperfect knowledge of material behavior across scales is a source of epistemic uncertainty of the overall material behavior. However, assessing this uncertainty is difficult due to the complex nature of material response and the prohibitive computational cost of integral calculations. In this paper, we exploit the multiscale and hierarchical nature of material response to develop an approach to quantify the overall uncertainty of material response without the need for integral calculations. Specifically, we bound the uncertainty at each scale and then combine the partial uncertainties in a way that provides a bound on the overall or integral uncertainty. The bound provides a conservative estimate on the uncertainty. Importantly, this approach does not require integral calculations that are prohibitively expensive. We demonstrate the framework on the problem of ballistic impact of a polycrystalline magnesium plate. Magnesium and its alloys are of current interest as promising light-weight structural and protective materials. Finally, we remark that the approach can also be used to study the sensitivity of the overall response to particular mechanisms at lower scales in a materials-by-design approach.

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Section: Methodsmentioning

confidence: 99%

Section: Hierarchical Uncertainty Quantificationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Hierarchical multiscale quantification of material uncertainty

Liu

Sun

Bhattacharya

et al. 2021

Journal of the Mechanics and Physics of Solids

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“…As shown in [28], if A is described by independence constraints and inequalities of the form used, then a reduced upper bound on the probability of failure is obtained, but at the cost of solving a higher-dimensional optimization problem. Since, in general, the same methods can be used to provide optimal bounds on E µ [r] for any quantity of interest r, the methods of this paper can be used to optimally propagate uncertainties through a hierarchy (directed acyclic graph) of partially-observed input-output relationships, as in [38]. See Figure 8.1 for a schematic illustration.…”

Section: Generalizationsmentioning

confidence: 99%

Optimal uncertainty quantification for legacy data observations of Lipschitz functions

et al. 2013

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Abstract. We consider the problem of providing optimal uncertainty quantification (UQ) -and hence rigorous certification -for partially-observed functions. We present a UQ framework within which the observations may be small or large in number, and need not carry information about the probability distribution of the system in operation. The UQ objectives are posed as optimization problems, the solutions of which are optimal bounds on the quantities of interest; we consider two typical settings, namely parameter sensitivities (McDiarmid diameters) and output deviation (or failure) probabilities. The solutions of these optimization problems depend non-trivially (even non-monotonically and discontinuously) upon the specified legacy data. Furthermore, the extreme values are often determined by only a few members of the data set; in our principal physically-motivated example, the bounds are determined by just 2 out of 32 data points, and the remainder carry no information and could be neglected without changing the final answer. We propose an analogue of the simplex algorithm from linear programming that uses these observations to offer efficient and rigorous UQ for high-dimensional systems with high-cardinality legacy data. These findings suggest natural methods for selecting optimal (maximally informative) next experiments.

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“…McDiarmid's inequality has usually been applied in the large n limit. The small n properties are of interest because McDiarmid's bound has recently been applied to the problem of deducing margin requirements in complex systems, where n eff is not always large [6,9,5,1,8]. In particular, it has been proposed, along with more general concentration-of-measure (CoM) inequalities, as a way of formalizing the method of quantification of margins and uncertainties (QMU) [3,2,10].…”

mentioning

confidence: 99%

On the Application of McDiarmid's Inequality to Complex Systems

Wallstrom¹

2017

SIAM/ASA J. Uncertainty Quantification

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Abstract.McDiarmid's inequality has recently been proposed as a tool for setting margin requirements for complex systems. If F is the bounded output of a complex system, depending on a vector of n bounded inputs, this inequality provides a bound BF ( ), such that the probability of a deviation exceeding BF ( ) is less than . I compare this bound with the absolute bound, based on the range of F . I show that when n eff , the effective number of independent variates, is small, and when is small, the absolute bound is smaller than BF ( ), while also providing a smaller probability of exceeding the bound, i.e., zero instead of . Thus, for BF ( ) to be useful, the number of inputs must be large, with a small dependence on any single input, which is consistent with the usual guidance for application of concentration-of-measure results. When the number of inputs is small, or when a small number of inputs account for much of the uncertainty, the absolute bounds will provide better results. The use of absolute bounds is equivalent to the original formulation of the method of quantification of margins and uncertainties (QMU).

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Rigorous uncertainty quantification without integral testing

Cited by 9 publications

References 15 publications

Hierarchical multiscale quantification of material uncertainty

Hierarchical multiscale quantification of material uncertainty

Optimal uncertainty quantification for legacy data observations of Lipschitz functions

On the Application of McDiarmid's Inequality to Complex Systems

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