×

zbMATH — the first resource for mathematics

Rigorous verification, validation, uncertainty quantification and certification through concentration-of-measure inequalities. (English) Zbl 1194.74550
Summary: We apply concentration-of-measure inequalities to the quantification of uncertainties in the performance of engineering systems. Specifically, we envision uncertainty quantification in the context of certification, i.e., as a tool for deciding whether a system is likely to perform safely and reliably within design specifications. We show that concentration-of-measure inequalities rigorously bound probabilities of failure and thus supply conservative certification criteria. In addition, they supply unambiguous quantitative definitions of terms such as margins, epistemic and aleatoric uncertainties, verification and validation measures, confidence factors, and others, as well as providing clear procedures for computing these quantities by means of concerted simulation and experimental campaigns. We also investigate numerically the tightness of concentration-of-measure inequalities with the aid of an imploding ring example. Our numerical tests establish the robustness and viability of concentration-of-measure inequalities as a basis for certification in that particular example of application.

MSC:
74S60 Stochastic and other probabilistic methods applied to problems in solid mechanics
74K99 Thin bodies, structures
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Sharp, D.H.; Wood-Schultz, M.M., QMU and nuclear weapons certification: what’s under the Hood, Los alamos sci., 28, 47-53, (2003)
[2] D. Eardley (Study Leader), Quantification of margins and uncertainties (QMU), Technical Report JSR-04-330, JASON, The MITRE Corporation, 7515 Colshire Drive McLean, Virginia 22102, March 2005.
[3] M. Pilch, T.G. Trucano, J.C. Helton, Ideas underlying quantification of margins and uncertainties (QMU): A white paper, Unlimited Release SAND2006-5001, Sandia National Laboratory, Albuquerque, New Mexico 87185 and Livermore, California 94550, September 2006.
[4] Oberkampf, W.L.; Trucano, T.G., Verification and validation in computational fluid dynamics, Progr. aerospace sci., 38, 209-272, (2002)
[5] Oberkampf, W.L.; Trucano, T.G.; Hirsch, C., Verification, validation and predictive capability in computational engineering and physics, Appl. mech. rev., 57, 5, 345-384, (2004)
[6] Michel Ledoux, The concentration of measure phenomenon, volume 89 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2001. · Zbl 0995.60002
[7] Boucheron, S.; Bousquet, O.; Lugosi, G., Concentration inequalities, (), 208-240 · Zbl 1120.68427
[8] Lawrence C. Evans, An introduction to stochastic differential equations, 2004. Available at <http://math.berkeley.edu/ evans/SDE.course.pdf>.
[9] Hoeffding, Wassily, Probability inequalities for sums of bounded random variables, J. amer. statist. assoc., 58, 13-30, (1963) · Zbl 0127.10602
[10] Colin McDiarmid, On the method of bounded differences, in: Surveys in Combinatorics, 1989 (Norwich, 1989), vol. 141 of London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, 1989, pp. 148-188. · Zbl 0712.05012
[11] McDiarmid, Colin, Centering sequences with bounded differences, Combin. probab. comput., 6, 1, 79-86, (1997) · Zbl 0869.60040
[12] Hughes, T.J.R., Analysis of transient algorithms with particular reference to stability behavior, (), 67-155
[13] Paul Lévy, Problèmes concrets d’analyse fonctionnelle, Avec un complément sur les fonctionnelles analytiques par F. Pellegrino, 2nd ed., Gauthier-Villars, Paris, 1951. · Zbl 0043.32302
[14] Milman, V.D., Geometric theory of Banach spaces. II. geometry of the unit ball, Uspehi mat. nauk, 26, 6(162), 73-149, (1971) · Zbl 0229.46017
[15] Milman, V.D., A certain property of functions defined on infinite-dimensional manifolds, Dokl. akad. nauk SSSR, 200, 781-784, (1971)
[16] Milman, V.D., Asymptotic properties of functions of several variables that are defined on homogeneous spaces, Dokl. akad. nauk SSSR, 199, 1247-1250, (1971)
[17] Milman, V.D., A new proof of A. dvoretzky’s theorem on cross-sections of convex bodies, Funk. anal. priložen., 5, 4, 28-37, (1971)
[18] Gromov, M., Isoperimetry of waists and concentration of maps, Geom. funct. anal., 13, 1, 178-215, (2003) · Zbl 1044.46057
[19] M. Gromov, CAT(κ)-spaces: construction and concentration, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 280 (Geom. i Topol. 7): 100-140 (2001) 299-300.
[20] M. Gromov, V.D. Milman, Brunn theorem and a concentration of volume phenomena for symmetric convex bodies, in: Israel Seminar on Geometrical Aspects of Functional Analysis (1983/84), pages V, 12. Tel Aviv Univ., Tel Aviv, 1984.
[21] Chernoff, Herman, A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations, Ann. math. stat., 23, 493-507, (1952) · Zbl 0048.11804
[22] Smirnov, N.V., Approximate laws of distribution of random variables from empirical data, Uspehi matem. nauk, 10, 179-206, (1944)
[23] Dvoretzky, A.; Kiefer, J.; Wolfowitz, J., Asymptotic minimax character of the sample distribution function and of the classical multinomial estimator, Ann. math. stat., 27, 642-669, (1956) · Zbl 0073.14603
[24] Cantelli, F., Sulla probabilita come limita Della frequenza, Rend. accad. lincei, 26, 1, (1933) · JFM 46.0779.02
[25] Glivenko, V., Sulla determinazione empirica delle leggi di probabilita, Giornale dell’istituta italiano degli attuari, 4, (1933) · JFM 59.1166.04
[26] Talagrand, Michel, Concentration and influences, Israel J. math., 111, 275-284, (1999) · Zbl 0937.28004
[27] Talagrand, Michel, New concentration inequalities in product spaces, Invent. math., 126, 3, 505-563, (1996) · Zbl 0893.60001
[28] Talagrand, Michel, Nouvelles inégalités de concentration à q points, C. R. acad. sci. Paris Sér. I math., 321, 11, 1505-1507, (1995) · Zbl 0846.60016
[29] Talagrand, Michel, Nouvelles inégalités de concentration “convexifiées”, C. R. acad. sci. Paris Sér. I math., 321, 10, 1367-1370, (1995) · Zbl 0840.60023
[30] Talagrand, Michel, Concentration of measure and isoperimetric inequalities in product spaces, Inst. hautes études sci. publ. math., 81, 73-205, (1995) · Zbl 0864.60013
[31] Michel Talagrand, A new isoperimetric inequality and the concentration of measure phenomenon, in: Geometric Aspects of functional analysis (1989-1990), vol. 1469 of Lecture Notes in Math., Springer, Berlin, 1991, pp. 94-124. · Zbl 0818.46047
[32] WanSoo, T., Rhee and Michel talagrand. A concentration inequality for the K-Median problem, Math. oper. res., 14, 2, 189-202, (1989) · Zbl 0682.90036
[33] Boucheron, Stéphane, Gábor lugosi, and Pascal massart. concentration inequalities using the entropy method, Ann. probab., 31, 3, 1583-1614, (2003) · Zbl 1051.60020
[34] Pascal Massart, Some applications of concentration inequalities to statistics, Ann. Fac. Sci. Toulouse Math. (6), 9(2):245-303, 2000. Probability theory. · Zbl 0986.62002
[35] Massart, Pascal, About the constants in talagrand’s concentration inequalities for empirical processes, Ann. probab., 28, 2, 863-884, (2000) · Zbl 1140.60310
[36] Boucheron, Stéphane; Lugosi, Gábor; Massart, Pascal, A sharp concentration inequality with applications, Random structures algorithms, 16, 3, 277-292, (2000) · Zbl 0954.60008
[37] Michel Ledoux, Concentration of measure and logarithmic Sobolev inequalities, in: Séminaire de Probabilités, XXXIII, vol. 1709 of Lecture Notes in Math., Springer, Berlin, 1999, pp. 120-216. · Zbl 0957.60016
[38] Bobkov, S.; Ledoux, M., Poincaré’s inequalities and talagrand’s concentration phenomenon for the exponential distribution, Probab. theory related fields, 107, 3, 383-400, (1997) · Zbl 0878.60014
[39] Ledoux, M., A heat semigroup approach to concentration on the sphere and on a compact Riemannian manifold, Geom. funct. anal., 2, 2, 221-224, (1992) · Zbl 0752.53039
[40] Ledoux, M., A remark on hypercontractivity and the concentration of measure phenomenon in a compact Riemannian manifold, Israel J. math., 69, 3, 361-370, (1990) · Zbl 0711.53037
[41] Talagrand, Michel, A new look at independence, Ann. probab., 24, 1, 1-34, (1996) · Zbl 0858.60019
[42] Gábor Lugosi, Concentration-of-measure inequalities, Lecture notes, 2006, <http://www.econ.upf.edu/ lugosi/anu.pdf>.
[43] Samson, Paul-Marie, Concentration of measure inequalities for Markov chains and φ-mixing processes, Ann. probab., 28, 1, 416-461, (2000) · Zbl 1044.60061
[44] Marton, K., A measure concentration inequality for contracting Markov chains, Geom. funct. anal., 6, 3, 556-571, (1996) · Zbl 0856.60072
[45] Marton, Katalin, Measure concentration for a class of random processes, Probab. theory related fields, 110, 3, 427-439, (1998) · Zbl 0927.60050
[46] Houdré, C.; Tetali, P., Concentration of measure for products of Markov kernels and graph products via functional inequalities, Combin. probab. comput., 10, 1, 1-28, (2001) · Zbl 0986.28004
[47] Gross, Leonard, Logarithmic Sobolev inequalities, Amer. J. math., 97, 4, 1061-1083, (1975) · Zbl 0318.46049
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.