Rigorous verification, validation, uncertainty quantification and certification through concentration-of-measure inequalities

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

doi:10.1016/j.cma.2008.06.008

“…Most of the novel computational methods for fracture are based on deterministic approaches. There are far less contributions on statistical computational methods for fracture [309][310][311][312][313][314]. The development of efficient and reliable stochastic computational methods is one of the key challenges in the future.…”

Section: Future Perspectives and Conclusionmentioning

confidence: 99%

Computational Methods for Fracture in Brittle and Quasi-Brittle Solids: State-of-the-Art Review and Future Perspectives

Rabczuk

¹

2013

ISRN Applied Mathematics

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An overview of computational methods to model fracture in brittle and quasi-brittle materials is given. The overview focuses on continuum models for fracture. First, numerical difficulties related to modelling fracture for quasi-brittle materials will be discussed. Different techniques to eliminate or circumvent those difficulties will be described subsequently. In that context, regularization techniques such as nonlocal models, gradient enhanced models, viscous models, cohesive zone models, and smeared crack models will be discussed. The main focus of this paper will be on computational methods for discrete fracture (discrete cracks). Element erosion technques, inter-element separation methods, the embedded finite element method (EFEM), the extended finite element method (XFEM), meshfree methods (MMs), boundary elements (BEMs), isogeometric analysis, and the variational approach to fracture will be reviewed elucidating advantages and drawbacks of each approach. As tracking the crack path is of major concern in computational methods that preserve crack path continuity, one section will discuss different crack tracking techniques. Finally, cracking criteria will be reviewed before the paper ends with future research perspectives.

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“…McDiarmid's inequality is by no means the strongest concentration-of-measure inequality in the literature, but is useful because of its simple hypotheses and proof. McDiarmid's inequality and its variants have been used for uncertainty quantification in the context of certification [2,17,22].…”

Section: Notation and Backgroundmentioning

confidence: 99%

Distances and Diameters in Concentration Inequalities: From Geometry to Optimal Assignment of Sampling Resources

Sullivan

¹

,

Owhadi

²

2012

Int J Uncertainty Quantification

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Abstract. This note reviews, compares and contrasts three notions of "distance" or "size" that arise often in concentration-of-measure inequalities. We review Talagrand's convex distance and McDiarmid's diameter, and consider in particular the normal distance on a topological vector space X , which corresponds to the method of Chernoff bounds, and is in some sense "natural" with respect to the duality structure on X . We show that, notably, with respect to this distance, concentration inequalities on the tails of linear, convex, quasiconvex and measurable functions on X are mutually equivalent. We calculate the normal distances that correspond to families of Gaussian and of bounded random variables in R N , and to functions of N empirical means. As an application, we consider the problem of estimating the confidence that one can have in a quantity of interest that depends upon many empirical -as opposed to exact -means and show how the normal distance leads to a formula for the optimal assignment of sampling resources.

show abstract

“…McDiarmid's inequality is by no means the strongest concentration-of-measure inequality in the literature, but is useful because of its simple hypotheses and proof. McDiarmid's inequality and its variants have been used for uncertainty quantification in the context of certification [2,17,22].…”

Section: Notation and Backgroundmentioning

confidence: 99%

“…Concentration inequalities have found many applications beyond pure mathematics, e.g. in fields such as uncertainty quantification [22], machine learning [31] and distributed computing [12].…”

Section: Introductionmentioning

confidence: 99%

Distances and Diameters in Concentration Inequalities: From Geometry to Optimal Assignment of Sampling Resources

Sullivan

¹

,

Owhadi

²

2012

Int J Uncertainty Quantification

0

View full text Add to dashboard Cite

Abstract. This note reviews, compares and contrasts three notions of "distance" or "size" that arise often in concentration-of-measure inequalities. We review Talagrand's convex distance and McDiarmid's diameter, and consider in particular the normal distance on a topological vector space X , which corresponds to the method of Chernoff bounds, and is in some sense "natural" with respect to the duality structure on X . We show that, notably, with respect to this distance, concentration inequalities on the tails of linear, convex, quasiconvex and measurable functions on X are mutually equivalent. We calculate the normal distances that correspond to families of Gaussian and of bounded random variables in R N , and to functions of N empirical means. As an application, we consider the problem of estimating the confidence that one can have in a quantity of interest that depends upon many empirical -as opposed to exact -means and show how the normal distance leads to a formula for the optimal assignment of sampling resources.

show abstract

Rigorous verification, validation, uncertainty quantification and certification through concentration-of-measure inequalities

Cited by 56 publications

References 25 publications

Computational Methods for Fracture in Brittle and Quasi-Brittle Solids: State-of-the-Art Review and Future Perspectives

Computational Methods for Fracture in Brittle and Quasi-Brittle Solids: State-of-the-Art Review and Future Perspectives

Distances and Diameters in Concentration Inequalities: From Geometry to Optimal Assignment of Sampling Resources

Distances and Diameters in Concentration Inequalities: From Geometry to Optimal Assignment of Sampling Resources

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