We propose a rigorous framework for uncertainty quantification (UQ) in which the UQ objectives and its assumptions/information set are brought to the forefront. This framework, which we call optimal uncertainty quantification (OUQ), is based on the observation that, given a set of assumptions and information about the problem, there exist optimal bounds on uncertainties: these are obtained as values of well-defined optimization problems corresponding to extremizing probabilities of failure, or of deviations, subject to the constraints imposed by the scenarios compatible with the assumptions and information. In particular, this framework does not implicitly impose inappropriate assumptions, nor does it repudiate relevant information. Although OUQ optimization problems are extremely large, we show that under general conditions they have finite-dimensional reductions. As an application, we develop optimal concentration inequalities (OCI) of Hoeffding and McDiarmid type. Surprisingly, these results show that uncertainties in input parameters, which propagate to output uncertainties in the classical sensitivity analysis paradigm, may fail to do so if the transfer functions (or probability distributions) are imperfectly known. We show how, for hierarchical structures, this phenomenon may lead to the nonpropagation of uncertainties or information across scales. In addition, a general algorithmic framework is developed for OUQ and is tested on the Caltech surrogate model for hypervelocity impact and on the seismic safety assessment of truss structures, suggesting the feasibility of the framework for important complex systems. The introduction of this paper provides both an overview of the paper and a self-contained minitutorial on the basic concepts and issues of UQ.


  1. uncertainty quantification
  2. concentration inequalities
  3. sensitivity analysis
  4. Markov--Krein-type reduction theorems for generalized Chebyshev optimization problems

MSC codes

  1. 60E15
  2. 62G99
  3. 65C50
  4. 90C26
  5. 60-08
  6. 28E99

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M. Adams, A. Lashgari, B. Li, M. McKerns, J. Mihaly, M. Ortiz, H. Owhadi, A. Rosakis, M. Stalzer, and T. J. Sullivan, Rigorous model-based uncertainty quantification with application to terminal ballistics. Part II: Systems with uncontrollable inputs and large scatter, J. Mech. Phys. Solids, 60 (2012), pp. 1002--1019.
C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis: A Hitchhiker's Guide, 3rd ed., Springer, Berlin, 2006.
T. W. Anderson, The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities, Proc. Amer. Math. Soc., 6 (1955), pp. 170--176.
I. Babuška, F. Nobile, and R. Tempone, A stochastic collocation method for elliptic partial differential equations with random input data, SIAM Rev., 52 (2010), pp. 317--355.
R. E. Barlow and F. Proschan, Mathematical Theory of Reliability, SIAM, Philadelphia, 1996.
J. L. Beck, Bayesian system identification based on probability logic, Struct. Control Health Monit., 17 (2010), pp. 825--–847.
J. L. Beck and L. S. Katafygiotis, Updating models and their uncertainties: Bayesian statistical framework, J. Engrg. Mech., 124 (1998), pp. 455--461.
A. Bensoussan, J.-L. Lions, and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, Stud. Math. Appl. 5, North-Holland, Amsterdam, 1978.
V. Bentkus, A remark on the inequalities of Bernstein, Prokhorov, Bennett, Hoeffding, and Talagrand, Liet. Mat. Rink., 42 (2002), pp. 332--342.
V. Bentkus, On Hoeffding's inequalities, Ann. Probab., 32 (2004), pp. 1650--1673.
V. Bentkus, G. D. C. Geuze, and M. C. A. van Zuijlen, Optimal Hoeffding-like inequalities under a symmetry assumption, Statistics, 40 (2006), pp. 159--164.
S. N. Bernstein, Collected Works, Izdat. “Nauka”, Moscow, 1964.
D. Bertsimas and I. Popescu, Optimal inequalities in probability theory: A convex optimization approach, SIAM J. Optim., 15 (2005), pp. 780--804.
M. Bieri and C. Schwab, Sparse high order FEM for elliptic sPDEs, Comput. Methods Appl. Mech. Engrg., 198 (2009), pp. 1149--1170.
Boeing, Statistical Summary of Commercial Jet Airplane Accidents Worldwide Operations 1959--2009, Tech. Report, Aviation Safety Boeing Commercial Airplanes, Seattle, WA, 2010.
S. Boucheron, G. Lugosi, and P. Massart, A sharp concentration inequality with applications, Random Structures Algorithms, 16 (2000), pp. 277--292.
P. Diaconis and D. Freedman, On the consistency of Bayes estimates, Ann. Statist., 14 (1986), pp. 1--67.
P. W. Diaconis and D. Freedman, Consistency of Bayes estimates for nonparametric regression: Normal theory, Bernoulli, 4 (1998), pp. 411--444.
A. Doostan and H. Owhadi, A non-adapted sparse approximation of PDEs with stochastic inputs, J. Comput. Phys., 230 (2011), pp. 3015--3034.
R. F. Drenick, Model-free design of aseismic structures, J. Eng. Mech. Div. Am. Soc. Civ. Eng., 96 (1970), pp. 483--493.
R. F. Drenick, Aseismic design by way of critical excitation, J. Eng. Mech. Div. Am. Soc. Civ. Eng., 99 (1973), pp. 649--667.
R. F. Drenick, On a class of non-robust problems in stochastic dynamics, in Stochastic Problems in Dynamics, B. L. Clarkson, ed., Pitman, London, 1977, pp. 237--255.
R. F. Drenick, P. C. Wang, C. B. Yun, and A. J. Philippacopoulos, Critical seismic response of nuclear reactors, J. Nucl. Eng. Design, 58 (1980), pp. 425--435.
E. B. Dynkin, Sufficient statistics and extreme points, Ann. Probab., 6 (1978), pp. 705--730.
B. Efron and C. Morris, Stein's paradox in statistics, Sci. Amer., 236 (1977), pp. 119--127.
I. H. Eldred, C. G. Webster, and P. G. Constantine, Design under Uncertainty Employing Stochastic Expansion Methods, Paper 2008--6001, American Institute of Aeronautics and Astronautics, 2008.
I. Elishakoff and M. Ohsaki, Optimization and Anti-Optimization of Structures under Uncertainty, World Scientific, London, 2010.
L. Esteva, Seismic risk and seismic design, in Seismic Design for Nuclear Power Plants, The MIT Press, Cambridge, MA, 1970, pp. 142--182.
R. Ghanem, Ingredients for a general purpose stochastic finite elements implementation, Comput. Methods Appl. Mech. Engrg., 168 (1999), pp. 19--34.
R. Ghanem and S. Dham, Stochastic finite element analysis for multiphase flow in heterogeneous porous media, Transp. Porous Media, 32 (1998), pp. 239--262.
W. D. Gillford, Risk analysis and the acceptable probability of failure, Structural Engineer, 83 (2005), pp. 25--26.
H. J. Godwin, On generalizations of Tchebychef's inequality, J. Amer. Statist. Assoc., 50 (1955), pp. 923--945.
W. Hoeffding, On the distribution of the number of successes in independent trials, Ann. Math. Statist., 27 (1956), pp. 713--721.
W. Hoeffding, The role of assumptions in statistical decisions, in Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954--1955, Vol. I, University of California Press, Berkeley, Los Angeles, 1956, pp. 105--114.
W. Hoeffding, Probability inequalities for sums of bounded random variables, J. Amer. Statist. Assoc., 58 (1963), pp. 13--30.
W. A. Hustrulid, M. McCarter, and D. J. A. Van Zyl, eds., Slope Stability in Surface Mining, Society for Mining Metallurgy & Exploration, 2001.
K. Isii, On a method for generalizations of Tchebycheff's inequality, Ann. Inst. Statist. Math. Tokyo, 10 (1959), pp. 65--88.
K. Isii, The extrema of probability determined by generalized moments. I. Bounded random variables, Ann. Inst. Statist. Math., 12 (1960), pp. 119--134; errata, p. 280.
K. Isii, On sharpness of Tchebycheff-type inequalities, Ann. Inst. Statist. Math., 14 (1962/1963), pp. 185--197.
H. Joe, Majorization, randomness and dependence for multivariate distributions, Ann. Probab., 15 (1987), pp. 1217--1225.
R. A. Johnson, Products of two Borel measures, Trans. Amer. Math. Soc., 269 (1982), pp. 611--625.
S. Karlin and W. J. Studden, Tchebycheff Systems: With Applications in Analysis and Statistics, Pure Appl. Math. 15, Interscience Publishers, John Wiley & Sons, New York, London, Sydney, 1966.
A. F. Karr, Extreme points of certain sets of probability measures, with applications, Math. Oper. Res., 8 (1983), pp. 74--85.
K. E. Kelly, The myth of $10^{-6}$ as a definition of acceptable risk, in Proceedings of the International Congress on the Health Effects of Hazardous Waste (Atlanta, 1993), Agency for Toxic Substances and Disease Registry, 1993.
D. G. Kendall, Simplexes and vector lattices, J. London Math. Soc., 37 (1962), pp. 365--371.
A. A. Kidane, A. Lashgari, B. Li, M. McKerns, M. Ortiz, H. Owhadi, G. Ravichandran, M. Stalzer, and T. J. Sullivan, Rigorous model-based uncertainty quantification with application to terminal ballistics. Part I: Systems with controllable inputs and small scatter, J. Mech. Phys. Solids, 60 (2012), pp. 983--–1001.
N. Lama, J. Wilsona, and G. Hutchinsona, Generation of synthetic earthquake accelograms using seismological modeling: A review, J. Earthquake Engrg., 4 (2000), pp. 321--354.
T. Leonard and J. S. J. Hsu, Bayesian Methods: An Analysis for Statisticians and Interdisciplinary Researchers, Camb. Ser. Stat. Prob. Math. 5, Cambridge University Press, Cambridge, UK, 1999.
J. Li and Z. An, Random function model research on strong ground motion, in Proceedings of the 14th World Conference on Earthquake Engineering, Beijing, China, 2008; available online from http://www.iitk.ac.in/nicee/wcee/article/14_02-0049.PDF.
J. S. Liu, Monte Carlo Strategies in Scientific Computing, Springer Ser. Statist., Springer, New York, 2008.
A. L. Lopez, L. E. P. Rocha, D. L. Escobedo, and J. S. Sesma, Reliability and vulnerability analysis of electrical substations and transmission towers for definition of wind and seismic damage maps for Mexico, in Proceedings of the 11th Americas Conference on Wind Engineering, San Juan, Puerto Rico, 2009; available online from http://www.iawe.org/Proceedings/11ACWE/11ACWE-Lopez.pdf.
L. J. Lucas, H. Owhadi, and M. Ortiz, Rigorous verification, validation, uncertainty quantification and certification through concentration-of-measure inequalities, Comput. Methods Appl. Mech. Engrg., 197 (2008), pp. 4591--4609.
N. Mantel and W. R. Bryan, ``Safety” testing of carcinogenic agents, J. Natl. Cancer Inst., 27 (1961), pp. 455--470.
A. W. Marshall and I. Olkin, Multivariate Chebyshev inequalities, Ann. Math. Statist., 31 (1960), pp. 1001--1014.
A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Applications, Math. Sci. Engrg. 143, Academic Press, Harcourt Brace Jovanovich Publishers, New York, 1979.
MATruss 1.3, MA Software, 2008; http://www.soft3k.com/MATruss-p4424.htm.
S. Matsuda and K. Asano, Analytical expression of earthquake input energy spectrum based on spectral density model of ground acceleration, in Proceeding of the 8th U.S. National Conference on Earthquake Engineering, San Francisco, CA, 2006.
C. McDiarmid, On the method of bounded differences, in Surveys in Combinatorics, 1989 (Norwich, 1989), London Math. Soc. Lecture Note Ser. 141, J. Siemons, ed., Cambridge University Press, Cambridge, UK, 1989, pp. 148--188.
C. McDiarmid, Concentration, in Probabilistic Methods for Algorithmic Discrete Mathematics, Algorithms Combin. 16, M. Habib, C. McDiarmid, J. Ramirez-Alfonsin, and B. Reed, eds., Springer, Berlin, 1998, pp. 195--248.
M. McKerns, P. Hung, and M. Aivazis, Mystic: A Simple Model-Independent Inversion Framework, 2009; http://dev.danse.us/trac/mystic.
M. McKerns, H. Owhadi, C. Scovel, T. J. Sullivan, and M. Ortiz, The Optimal Uncertainty Algorithm in the Mystic Framework, Tech. Report, CACR Technical Publication, Caltech, 2011; available online from http://arxiv.org/pdf/1202.1055v1.pdf.
M. M. McKerns, L. Strand, T. J. Sullivan, A. Fang, and M. A. G. Aivazis, Building a framework for predictive science, in Proceedings of the 10th Python in Science Conference (SciPy 2011), 2011; available online from http://arxiv.org/pdf/1202.1056v1.pdf.
H. P. Mulholland and C. A. Rogers, Representation theorems for distribution functions, Proc. London Math. Soc. (3), 8 (1958), pp. 177--223.
N. M. Newmark and E. Rosenblueth, Fundamentals of Earthquake Engineering, Prentice-Hall, Englewood Cliffs, NJ, 1972.
H. Niederreiter, Random Number Generation and Quasi-Monte Carlo Methods, CBMS-NSF Regional Conf. Ser. in Appl. Math. 63, SIAM, Philadelphia, 1992.
I. Olkin and J. W. Pratt, A multivariate Tchebycheff inequality, Ann. Math. Statist, 29 (1958), pp. 226--234.
H. Owhadi, Anomalous slow diffusion from perpetual homogenization, Ann. Probab., 31 (2003), pp. 1935--1969.
P. M. Pardalos and S. A. Vavasis, Quadratic programming with one negative eigenvalue is NP-hard, J. Global Optim., 1 (1991), pp. 15--22.
R. R. Phelps, Lectures on Choquet's Theorem, 2nd ed., Lecture Notes in Math. 1757, Springer, Berlin, 2001.
I. Pinelis, Exact inequalities for sums of asymmetric random variables, with applications, Probab. Theory Related Fields, 139 (2007), pp. 605--635.
I. Pinelis, On inequalities for sums of bounded random variables, J. Math. Inequal., 2 (2008), pp. 1--7.
M. J. D. Powell, A direct search optimization method that models the objective and constraint functions by linear interpolation, in Advances in Optimization and Numerical Analysis, S. Gomez and J. P. Hennart, eds., Kluwer Academic, Dordrecht, 1994, pp. 51--67.
K. V. Price, R. M. Storn, and J. A. Lampinen, Differential Evolution: A Practical Approach to Global Optimization, Natural Computing Series, Springer, Berlin, 2005.
A. D. Rikun, A convex envelope formula for multilinear functions, J. Global Optim., 10 (1997), pp. 425--437.
R. T. Rockafellar, Augmented Lagrange multiplier functions and duality in nonconvex programming, SIAM J. Control, 12 (1974), pp. 268--285.
A. Saltelli, K. Chan, and E. M. Scott, eds., Sensitivity Analysis, Wiley Series in Probability and Statistics, John Wiley & Sons, Chichester, UK, 2000.
A. Saltelli, M. Ratto, T. Andres, F. Campolongo, J. Cariboni, D. Gatelli, M. Saisana, and S. Tarantola, Global Sensitivity Snalysis. The Primer, John Wiley & Sons, Chichester, UK, 2008.
S. M. Samuels, On a Chebyshev-type inequality for sums of independent random variables, Ann. Math. Statist., 37 (1966), pp. 248--259.
C. Scovel and I. Steinwart, Hypothesis testing for validation and certification, J. Complexity, submitted.
X. Shen and L. Wasserman, Rates of convergence of posterior distributions, Ann. Statist., 29 (2001), pp. 687--714.
H. D. Sherali, Convex envelopes of multilinear functions over a unit hypercube and over special discrete sets, Acta Math. Vietnam., 22 (1997), pp. 245--270.
I. H. Sloan, Sparse Sampling Techniques, presented at 2010 ICMS Uncertainty Quantification workshop, 2010; available online from http://icms.org.uk/downloads/uq/Sloan.pdf.
I. H. Sloan and S. Joe, Lattice Methods for Multiple Integration, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1994.
H. M. Soekkha, ed., Aviation Safety: Human Factors, System Engineering, Flight Operations, Economics, Strageties, Management, CRC Press, 1997.
A. Srivastav and P. Stangier, Algorithmic Chernoff--Hoeffding inequalities in integer programming, Random Structures Algorithms, 8 (1996), pp. 27--58.
P. J. Stafford, J. B. Berrill, and J. R. Pettinga, New predictive equations for Arias intensity from crustal earthquakes in New Zealand, J. Seismology, 13 (2009), pp. 31--52.
S. Stein and M. Wysession, An Introduction to Seismology, Earthquakes, and Earth Structure, Wiley-Blackwell, Oxford, 2002.
R. M. Storn and K. V. Price, Differential evolution---a simple and efficient heuristic for global optimization over continuous spaces, J. Global Optim., 11 (1997), pp. 341--359.
A. M. Stuart, Inverse problems: A Bayesian perspective, Acta Numer., 19 (2010), pp. 451--559.
T. J. Sullivan, M. McKerns, D. Meyer, F. Theil, H. Owhadi, and M. Ortiz, Optimal uncertainty quantification for legacy data observations of Lipschitz functions, M2AN Math. Model. Numer. Anal., to appear.
T. J. Sullivan, U. Topcu, M. McKerns, and H. Owhadi, Uncertainty quantification via codimension-one partitioning, Internat. J. Numer. Methods Engrg., 85 (2011), pp. 1499--1521.
I. Takewaki, Seismic critical excitation method for robust design: A review, J. Struct. Engrg., 128 (2002), pp. 665--672.
I. Takewaki, Critical Excitation Methods in Earthquake Engineering, Elsevier Science, New York, 2007.
R. A. Todor and C. Schwab, Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients, IMA J. Numer. Anal., 27 (2007), pp. 232--261.
F. Topsøe, Topology and Measure, Lecture Notes in Math. 133, Springer, Berlin, 1970.
H. Trainsson, A. S. Kiremidjian, and S. Winterstein, Modeling of Earthquake Ground Motion in the Frequency Domain, Tech. Report 134, Department of Civil and Environmental Engineering, Stanford University, 2000.
J. W. van de Lindt and G.-H. Goh, Earthquake duration effect on structural reliability, J. Struct. Engrg., 130 (2004), pp. 821--826.
L. Vandenberghe, S. Boyd, and K. Comanor, Generalized Chebyshev bounds via semidefinite programming, SIAM Rev., 49 (2007), pp. 52--64.
H. von Weizsäcker and G. Winkler, Integral representation in the set of solutions of a generalized moment problem, Math. Ann., 246 (1979/1980), pp. 23--32.
G. Winkler, Extreme points of moment sets, Math. Oper. Res., 13 (1988), pp. 581--587.
D. Xiu, Fast numerical methods for stochastic computations: A review, Commun. Comput. Phys., 5 (2009), pp. 242--272.

Information & Authors


Published In

cover image SIAM Review
SIAM Review
Pages: 271 - 345
ISSN (online): 1095-7200


Submitted: 7 September 2010
Accepted: 22 May 2012
Published online: 8 May 2013


  1. uncertainty quantification
  2. concentration inequalities
  3. sensitivity analysis
  4. Markov--Krein-type reduction theorems for generalized Chebyshev optimization problems

MSC codes

  1. 60E15
  2. 62G99
  3. 65C50
  4. 90C26
  5. 60-08
  6. 28E99



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