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Optimal uncertainty quantification for legacy data observationsof Lipschitz functions

Published online by Cambridge University Press:  30 August 2013

T.J. Sullivan ,
M. McKerns ,
D. Meyer ,
F. Theil ,
H. Owhadi  and
M. Ortiz
T.J. Sullivan
Affiliation:
Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK.. Tim.Sullivan@warwick.ac.uk
M. McKerns
Affiliation:
Center for Advanced Computing Research, California Institute of Technology, 1200 East California Boulevard, Mail Code 158-79, Pasadena, CA 91125, USA.; mmckerns@caltech.edu
D. Meyer
Affiliation:
Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK.; f.theil@warwick.ac.uk
F. Theil
Affiliation:
Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK.; f.theil@warwick.ac.uk
H. Owhadi
Affiliation:
Applied & Computational Mathematics and Control & Dynamical Systems, California Institute of Technology, Mail Code 9-94, 1200 East California Boulevard, Pasadena, CA 91125, USA.; owhadi@caltech.edu
M. Ortiz
Affiliation:
Division of Engineering and Applied Science, California Institute of Technology, Mail Code 105-50, 1200 East California Boulevard, Pasadena, CA 91125, USA.; ortiz@caltech.edu
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Abstract

We consider the problem of providing optimal uncertainty quantification (UQ) – and hencerigorous certification – for partially-observed functions. We present a UQ frameworkwithin which the observations may be small or large in number, and need not carryinformation about the probability distribution of the system in operation. The UQobjectives are posed as optimization problems, the solutions of which are optimal boundson the quantities of interest; we consider two typical settings, namely parametersensitivities (McDiarmid diameters) and output deviation (or failure) probabilities. Thesolutions of these optimization problems depend non-trivially (even non-monotonically anddiscontinuously) upon the specified legacy data. Furthermore, the extreme values are oftendetermined by only a few members of the data set; in our principal physically-motivatedexample, the bounds are determined by just 2 out of 32 data points, and the remaindercarry no information and could be neglected without changing the final answer. We proposean analogue of the simplex algorithm from linear programming that uses these observationsto offer efficient and rigorous UQ for high-dimensional systems with high-cardinalitylegacy data. These findings suggest natural methods for selecting optimal (maximallyinformative) next experiments.

Keywords

Uncertainty quantificationprobability inequalitiesnon-convex optimizationLipschitz functionslegacy datapoint observations
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 6 , November 2013 , pp. 1657 - 1689
Copyright
© EDP Sciences, SMAI, 2013

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References

Adams, M., Lashgari, A., Li, B., McKerns, M., Mihaly, J.M., Ortiz, M., Owhadi, H., Rosakis, A.J., Stalzer, M. Sullivan, T.J., Rigorous model-based uncertainty quantification with application to terminal ballistics. Part II: Systems with uncontrollable inputs and large scatter. J. Mech. Phys. Solids 60 (2011) 1002–1019. Google Scholar
J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Modern Birkhäuser Classics, Birkhäuser Boston Inc., Boston, MA (2009), Reprint of the 1990 edition [MR1048347].
Babuška, I., Nobile, F. and Tempone, R., Reliability of computational science. Numer. Methods Partial Differ. Eq. 23 (2007) 753–784. Google Scholar
R. E. Barlow and F. Proschan, Mathematical Theory of Reliability, in vol. 17 of Classics in Applied Mathematics. Society Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1996). With contributions by L. C. Hunter, Reprint of the 1965 original [MR 0195566].
Bertsimas, D. and Popescu, I., Optimal inequalities in probability theory: a convex optimization approach. SIAM J. Optim. 15 (2005) 780–804. Google Scholar
P. Billingsley, Convergence of Probability Measures, 2nd edn., Wiley Series in Probability and Statistics: Probability and Statistics. John Wiley and Sons Inc., New York (1999). http://dx.doi.org/10.1002/9780470316962. MR 1700749 (2000e:60008)
S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge University Press, Cambridge (2004).
H. Federer, Geometric Measure Theory, Die Grundlehren der Mathematischen Wissenschaften, Band 153. Springer-Verlag New York Inc., New York (1969).
W. Hoeffding, The role of assumptions in statistical decisions. Proc. of the Third Berkeley Symposium on Mathematical Statistics and Probability, vol. I, 1954–1955 (Berkeley and Los Angeles). University of California Press (1956) 105–114.
A. Holder, Mathematical Programming Glossary, INFORMS Computing Society, http://glossary.computing.society.informs.org (2006). Originally authored by H. J. Greenberg, 1999–2006.
Isbell, J.R., Six theorems about injective metric spaces, Comment. Math. Helv. 39 (1964), 65–76. Google Scholar
Jones, D.R., Perttunen, C.D. and Stuckman, B.E., Lipschitzian optimization without the Lipschitz constant. J. Optim. Theory Appl. 79 (1993) 157–181. Google Scholar
Kidane, A.A., Lashgari, A., Li, B., McKerns, M., Ortiz, M., Owhadi, H., Ravichandran, G., Stalzer, M. and Sullivan, T.J., Rigorous model-based uncertainty quantification with application to terminal ballistics. Part I: Systems with controllable inputs and small scatter. J. Mech. Phys. Solids 60 (2011) 983–1001. Google Scholar
Kirszbraun, M.D., Über die zusammenziehende und Lipschitzsche Transformationen. Fund. Math. 22 (1934) 77–108. Google Scholar
V. Klee and G.J. Minty, How good is the simplex algorithm?, Inequalities, III, in Proc. Third Sympos. (Univ. California, Los Angeles, Calif., 1969; dedicated to the memory of Theodore S. Motzkin). Academic Press, New York (1972) 159–175.
P. Limbourg, Multi-objective optimization of problems with epistemic uncertainty, Evolutionary Multi-Criterion Optimization, in Lect. Notes Comput. Sci., of vol. 3410, edited by C.A. Coello Coello, A. Hernández Aguirre and E. Zitzler. Springer Berlin/Heidelberg (2005) 413–427.
Lucas, L.J., Owhadi, H. and Ortiz, M., Rigorous verification, validation, uncertainty quantification and certification through concentration-of-measure inequalities. Comput. Methods Appl. Mech. Engrg. 197 (2008) 51–52, 4591–4609. Google Scholar
C. McDiarmid, On the method of bounded differences, Surveys in combinatorics, London Math. Soc. in vol. 141 of Lecture Note Ser. Cambridge Univ. Press, Cambridge (1989) 148–188.
McDiarmid, C., Centering sequences with bounded differences, Combin. Probab. Comput. 6 (1997) 79–86, Google Scholar
C. McDiarmid, Concentration, Probabilistic Methods for Algorithmic Discrete Mathematics. In vol. 16 of Algorithms Combin. Springer, Berlin (1998) 195–248.
M. McKerns, P. Hung and M. Aivazis, Mystic: A simple model-independent inversion framework (2009).
M. McKerns, H. Owhadi, C. Scovel, T.J. Sullivan and M. Ortiz, The optimal uncertainty algorithm in the mystic framework, Caltech CACR Technical Report, August 2010, available at http://arxiv.org/pdf/1202.1055v1.
M.M. McKerns, L. Strand, T.J. Sullivan, A. Fang and M.A.G. Aivazis, Building a framework for predictive science. Proc. of the 10th Python in Science Conference (SciPy 2011), edited by S. van der Walt and J. Millman (2011) 67–78. Available at http://jarrodmillman.com/scipy2011/pdfs/mckerns.pdf.
McShane, E.J., Extension of range of functions. Bull. Amer. Math. Soc. 40 (1934) 837–842. Google Scholar
R. Morrison, C. Bryant, G. Terejanu, K. Miki and S. Prudhomme, Optimal data split methodology for model validation, Proc. of World Congress on Engrg and Comput. Sci. (2011) vol. II, 1038–1043.
Oberkampf, W.L., Helton, J.C., Joslyn, C.A., Wojtkiewicz, S.F. and Ferson, S., Challenge problems: Uncertainty in system response given uncertain parameters. Reliab. Eng. Sys. Safety 85 (2004) 11–19. Google Scholar
Oberkampf, W.L., Trucano, T.G. and Hirsch, C., Verification, validation and predictive capability in computational engineering and physics. Appl. Mech. Rev. 57 (2004) 345–384. Google Scholar
H. Owhadi, C. Scovel, T. J. Sullivan, M. McKerns and M. Ortiz, Optimal Uncertainty Quantification. SIAM Rev. To appear.
K.V. Price, R.M. Storn and J.A. Lampinen, Differential Evolution: A Practical Approach to Global Optimization, Natural Comput. Ser. Springer-Verlag, Berlin (2005).
C.J. Roy and W.L. Oberkampf, A complete framework for verification, validation and uncertainty quantification in scientific computing, 48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition (2010).
L. Schwartz, Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures, Published for the Tata Institute of Fundamental Research, Bombay by Oxford University Press, London (1973). Tata Institute of Fundamental Research Studies in Mathematics, No. 6.
Skorohod, A.V., Limit theorems for stochastic processes, Teor. Veroyatnost. i Primenen. (Theor. Probab. Appl.) 1 (1956), 289–319. Google Scholar
L.A. Steen and J.A. Seebach, Jr., Counterexamples in Topology, 2nd edn. Springer-Verlag, New York (1978).
Storn, R. and Price, K., Differential evolution—a simple and efficient heuristic for global optimization over continuous spaces. J. Global Optim. 11 (1997) 341–359. Google Scholar
Stuart, A.M., Inverse problems: a Bayesian perspective. Acta Numer. 19 (2010) 451–559. Google Scholar
Sullivan, T. J., Topcu, U., McKerns, M. and Owhadi, H., Uncertainty quantification via codimension-one partitioning. Int. J. Numer. Meth. Engng. 85 (2011) 1499–1521. Google Scholar
M. Talagrand, Concentration of measure and isoperimetric inequalities in product spaces. Inst. Hautes É tudes Sci. Publ. Math. (1995) 73–205.
Topcu, U., Lucas, L. J., Owhadi, H. and Ortiz, M., Rigorous uncertainty quantification without integral testing. Reliab. Eng. Sys. Safety 96 (2011) 1085–1091. Google Scholar
Valentine, F.A., A Lipschitz condition preserving extension for a vector function. Amer. J. Math. 67 (1945) 83–93. Google Scholar
Vu, V.H., Concentration of non-Lipschitz functions and applications, Random Structures Algorithms 20 (2002) 262–316. Google Scholar
Wage, M.L., The product of Radon spaces, Uspekhi Mat. Nauk 35 (1980) 151–153, International Topology Conference (Moscow State Univ., Moscow, 1979), Translated from the English by A.V. Arhangel′skiĭ.Google Scholar