Uncertainty Quantification

Prof. Americo Barbosa da Cunha Junior
Tel.: (21) 2334-0323 ramal: 208
Sala 6032, Bloco B


1 - Basic Notions on Uncertainty Quantification
2 - Elements of Probability Theory
3 - Elements on Statistical Inference
4 - Sampling Methods
5 - Probabilistic Modeling of Uncertainties
6 - Notions of Polynomial Chaos

Complementary material:
GNU Octave:

In this course we will develop some computational activities. 
To do this, students can use the programming environment  GNU Octave
UQ Software:
Reference for this course:

R. Smith, 
Uncertainty Quantification: Theory, Implementation, and Applications,
SIAM: Society for Industrial and Applied Mathematics, 2013
Available at SIAM
C. Soize,
Uncertainty Quantification: An Accelerated Course with Advanced Applications in Computational Engineering, Springer, 2017
Available at Amazon

References for supplementary studies:

Fundamentals of UQ:
    R. Ghanem, D. Higdon, H. Owhadi (Editors),
    Handbook of Uncertainty Quantification,
    Springer, 2017
    Available at Amazon
    T. J. Sulivan,
    Introduction to Uncertainty Quantification,
    Springer, 2015
    Available at Amazon
    C. Soize,
    Stochastic Models of Uncertainties in Computational Mechanics,
    American Society of Civil Engineers, 2012.
    Available at Amazon
    J. E. S. de Cursi and R. Sampaio,
    Modelagem Estocástica e Quantificação de Incertezas,
    Sociedade Brasileira de Matemática Aplicada e Computacional, 2012
    D. A. Castello and T. G. Ritto,
    Quantificação de Incertezas e Estimação de Parâmetros em Dinâmica Estrutural: uma introdução a partir de exemplos computacionais, Sociedade Brasileira de Matemática Aplicada e Computacional, 2015

    Probability Theory:
      G. Grimmett and D. Welsh, 
      Probability: An Introduction,
      Oxford University Press, 2 edition, 2014
      Available at Amazon
      A. Papoulis and S. U. Pillai,
      Probability, Random Variables and Stochastic Processes,
      McGraw-Hill Europe; 4th edition, 2002
      Available at Amazon
      E. T. Jaynes,
      Probability Theory: The Logic of Science,
      Cambridge University Press, 2003
      E. Cataldo,
      Introdução aos Processos Estocásticos,
      Sociedade Brasileira de Matemática Aplicada e Computacional, 2012

      Statistical Inference:
        L. Wasserman,,
        All of Statistics: A Concise Course in Statistical Inference,
        Springer, 2004
        Available at Amazon

        D. P. Kroese, T. Taimre, and Z. I. Botev,

        Handbook of Monte Carlo Methods,

        Wiley, 2011

        Available at Amazon
        S. Asmussen and P. W. Glynn,
        Stochastic Simulation: Algorithms and Analysis,
        Springer, 2007
        Available at Amazon
        F. B. Ribeiro, E. C. Molina,
        Uma Introdução ao Método de Monte Carlo,
        Sociedade Brasileira de Matemática Aplicada e Computacional, 2017

        Inverse Problems:
           L. Biegler, G. Biros, O. Ghattas et al. (Editors),
          Large-Scale Inverse Problems and Quantification of Uncertainty,
          Wiley, 2010
          Available at Amazon
          J. Kaipio and E. Somersalo,
          Statistical and Computational Inverse Problems,
          Springer, 2005
          Available at Amazon
          A. Tarantola,
          Inverse Problem Theory and Methods for Model Parameter Estimation,
          SIAM, 2004
          L. Tenorio,
          An Introduction to Data Analysis and Uncertainty Quantification for Inverse Problems,
          SIAM, 2017
          Available at SIAM
          J. M. Bardsley,
          Computational Uncertainty Quantification for Inverse Problems,
          SIAM, 2018
          Available at SIAM

          Spectral Methods:
            R. G. Ghanem and P. D. Spanos,
            Stochastic Finite Elements: A Spectral Approach,
            Dover Publications, Revised Edition, 2012
            Available at Amazon
            D. Xiu,
            Numerical Methods for Stochastic Computations: A Spectral Method Approach,
            Princeton University Press, 2010
            Available at Amazon
            O. Le Maitre and O. Knio,
            Spectral Methods for Uncertainty Quantification: With Applications to Computational Fluid Dynamics,
            Springer, 2010
            Available at Amazon
            M. P. Pettersson, G. Iaccarino and J. Nordström,
            Polynomial Chaos Methods for Hyperbolic Partial Differential Equations: Numerical Techniques for Fluid Dynamics Problems in the Presence of Uncertainties, Springer, 2015
            Available at Amazon

            Verification and Validation:
              W. L. Oberkampf and C. J. Roy,
              Verification and Validation in Scientific Computing,
              Cambridge University Press, 2010
              Available at Amazon

              Journals with UQ content:

              SIAM/ASA Journal on Uncertainty Quantification

              International Journal for Uncertainty Quantification

              Reliability Engineering & System Safety

              Probabilistic Engineering Mechanics

              ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems

              Journal of Verification, Validation and Uncertainty Quantification

              Computer Methods in Applied Mechanics and Engineering

              Journal of Computational Physics

              Interesting articles:

              Fundamental ideas in UQ:
              Probabilistic approaches for UQ:
              • G. I. Schueller, A state-of-the-art report on computational stochastic mechanicsProbabilistic Engineering Mechanics, vol. 12, pp. 197-321, 1997. https://doi.org/10.1016/S0266-8920(97)00003-9
              • C. Soize,  A comprehensive overview of a non-parametric probabilistic approach of model uncertainties for predictive models in structural dynamicsJournal of Sound and Vibration,  vol. 288, pp.623-65, 2005. 
              • R. Ghanem and A Doostan, On the construction and analysis of stochastic models: characterization and propagation of the errors associated with limited dataJournal of Computational Physics, vol 217, pp. 63-81, 2008. 
              • J. Jakeman, M. Eldred and D. Xiu, Numerical approach for quantification of epistemic uncertainty, Journal of Computational Physics, vol. 229, pp. 4648-4663, 2010. https://doi.org/10.1016/j.jcp.2010.03.003
              • X. Chen, E.-J. Park and D. Xiu, A flexible numerical approach for quantification of epistemic uncertaintyJournal of Computational Physics, vol. 240, pp. 211-224, 2013. https://doi.org/10.1016/j.jcp.2013.01.018
              • C. Soize, Stochastic modeling of uncertainties in computational structural dynamics - Recent theoretical advancesJournal of Sound and Vibration, vol. 332, pp. 2379-2395, 2013. https://dx.doi.org/10.1016/j.jsv.2011.10.010
              • C. Soize and C. Farhat, A nonparametric probabilistic approach for quantifying uncertainties in low-dimensional and high-dimensional nonlinear modelsInternational Journal for Numerical Methods in Engineering, vol. 109, pp. 837-888, 2017. https://dx.doi.org/10.1002/nme.5312
              • J.T. Oden, Foundations of Predictive Computational Science, ICES Report 17-01, The University of Texas at Austin, 2017.
              • J. T. Oden, Adaptive multiscale predictive modelling, Acta Numerica, vol. 27, pp. 353-450, 2018. 
              Maximum entropy principle:
              • C. E. Shannon, A mathematical theory of communication, Bell System Technology Journal, vol. 27,pp. 379-423, 1948. https://doi.org/10.1002/j.1538-7305.1948.tb01338.x
              • E.T. Jaynes, Information Theory and Statistical Mechanics, Physical Review, vol. 106, pp. 620-630, 1957. https://doi.org/10.1103/PhysRev.106.620
              • S. Guiasu and A. Shenitzer, The principle of maximum entropy, The Mathematical Intelligencer, vol. 7, pp. 42–48, 1985. https://doi.org/10.1007/BF03023004
              • F. E. Udwadia, Some results on maximum entropy distributions for parameters known to lie in finite intervalsSIAM Review, vol. 31, pp. 103-109, 1989. https://doi.org/10.1137/1031004
              • K. Sobczyk, J. Trebicki, Maximum entropy principle in stochastic dynamics, Probabilistic Engineering Mechanics, vol. 5, pp. 102-110, 1990. https://doi.org/10.1016/0266-8920(90)90001-Z
              • J. Trebicki, K.J. Sobczyk, Maximum entropy principle and non-stationary distributions of stochastic systems,
              • Probabilistic Engineering Mechanics, vol. 11, pp. 169-178, 1996. https://doi.org/10.1016/0266-8920(96)00008-2
              • C. Soize, Maximum entropy approach for modeling random uncertainties in transient elastodynamicsThe Journal of the Acoustical Society of America, vol. 109, pp. 1979-1996, 2001. https://doi.org/10.1121/1.1360716
              • C. Soize, Construction of probability distributions in high dimension using the maximum entropy principle: Applications to stochastic processes, random fields and random matrices, International Journal for Numerical Methods in Engineering, vol. 76, pp. 1583-1611, 2008. https://dx.doi.org/10.1002/nme.2385
              Polynomial chaos:
              • D. Xiu and G.E. Karniadakis, The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM Journal of Scientific Computing, vol. 24, pp. 619-644, 2002.  https://doi.org/10.1137/S1064827501387826
              • R. Ghanem, Ingredients for a general purpose stochastic finite elements implementationComputer Methods in Applied Mechanics and Engineering, vol. 168, pp. 19-34, 1999. https://doi.org/10.1016/S0045-7825(98)00106-6
              • C. Soize and R. Ghanem, Physical systems with random uncertainties: Chaos representations with arbitrary probability measure, SIAM Journal of Scientific Computing,  vol. 26, pp. 395–410, 2004. https://doi.org/10.1137/S1064827503424505
              • A. Doostan, R. Ghanem and J. Red-Horse, Stochastic model reduction for chaos representationsComputer Methods in Applied Mechanics and Engineering, vol. 196, pp. 3951-3966, 2007. https://doi.org/10.1016/j.cma.2006.10.047
              • D. Xiu, Fast numerical methods for stochastic computations: a reviewCommunications in Computational Physics, vol. 5, pp. 242-272, 2009. http://sci.utah.edu/publications/xiu09/Xiu_CiCP2009.pdf
              • I. Babuška, F. Nobile, and R. Tempone, A stochastic collocation method for elliptic partial differential equations with random input data, SIAM Review, vol. 52, pp. 317–355, 2010. https://doi.org/10.1137/100786356
              • G. Blatman, B. Sudret, Adaptive sparse polynomial chaos expansion based on least angle regression, Journal of Computational Physics, vol. 230, pp. 2345-2367, 2011. https://doi.org/10.1016/j.jcp.2010.12.021
              Calibration, verification and validation:
              • W. L. OberkampfT. G. Trucano and C. Hirsch, Verification, validation, and predictive capability in computational engineering and physics,  Applied Mechanics Review, vol. 57, pp. 345-384, 2004. https://doi.org/10.1115/1.1767847
              • R. Ghanem, A. Doostan and J. Red-Horse, A probabilistic construction of model validationComputer Methods in Applied Mechanics and Engineering, vol. 197, pp. 2585-2595, 2008. https://doi.org/10.1016/j.cma.2007.08.029
              • C. J.Roy and W. L. Oberkampf, A comprehensive framework for verification, validation, and uncertainty quantification in scientific computingComputer Methods in Applied Mechanics and Engineering vol 200 pp. 2131-2144, 2011.
              • K. Farell and J.T. Oden, Model Misspecification and Plausibility, ICES Report 17-01, The University of Texas at Austin, 2014. http://www.ices.utexas.edu/media/reports/2014/1421.pdf
              • J. T. Oden et al., Toward predictive multiscale modeling of vascular tumor growth: computational and experimental oncology for tumor predictionICES Report 15-10, The University of Texas at Austin, 2015.
              • K. Sargsyan, H. N. Najm and R. Ghanem, On the statistical calibration of physical modelsInternational Journal of Chemical Kinetics, vol. 47, pp. 246-276, 2015. https://doi.org/10.1002/kin.20906
              • Y. He and D. Xiu, Numerical strategy for model correction using physical constraintsJournal of Computational Physics, vol. 313, pp. 617-634, 2016. https://doi.org/10.1016/j.jcp.2016.02.054
              • R. E.  Morrison, T. A. Oliver, R. D.  Moser, Representing model inadequacy: A stochastic operator approachSIAM/ASA Journal on Uncertainty Quantification, vol. 6, pp. 457-496, 2018. https://doi.org/10.1137/16M1106419
              Sensitivity analysis:
              Inverse problems:
              • M. Allmaras, W. Bangerth, J. M. Linhart, J. Polanco, F. Wang, K. Wang, J. Webster, and S. Zedler, Estimating parameters in physical models through Bayesian inversion: A complete example, SIAM Review, vol. 55, pp. 149–167, 2013. https://doi.org/10.1137/100788604
              • H. Owhadi, C. Scovel, and T. Sullivan, On the brittleness of Bayesian inference, SIAM Review, vol. 57, pp. 566–582, 2015. https://doi.org/10.1137/130938633
              Optimization under uncertainty:
              • D. Bertsimas, D. B. Brown, and C. Caramanis, Theory and applications of robust optimization, SIAM Review, vol. 53, pp. 464–501, 2011. https://doi.org/10.1137/080734510
              • H. Owhadi, C. Scovel, T. J. Sullivan, M. McKerns, and M. Ortiz, Optimal uncertainty quantification, SIAM Review, vol. 55, pp. 271–345, 2013. https://doi.org/10.1137/10080782X
              Data-driven, machine learning, multi-fidelity, etc:
              • A. Narayan, Y. Marzouk and D. Xiu, Sequential data assimilation with multiple modelsJournal of Computational Physics, vol. 231, pp. 6401-6418, 2012. https://doi.org/10.1016/j.jcp.2012.06.002
              • P. G. Constantine, E. Dow, Q. Wang, Active subspace methods in theory and practice: applications to kriging surfacesSIAM Journal on Scientific Computing, vol. 36, pp. A1500-A1524, 2014. https://doi.org/10.1137/130916138
              • C. Soize and R. Ghanem, Data-driven probability concentration and sampling on manifoldJournal of Computational Physics, vol. 321, pp. 242-258, 2016. https://doi.org/10.1016/j.jcp.2016.05.044
              • C. Soize and R. Ghanem, Polynomial chaos representation of databases on manifoldsJournal of Computational Physics, vol. 335, pp. 201-221, 2017. https://doi.org/10.1016/j.jcp.2017.01.031
              • M. Raissi, P. Perdikaris, G. E. Karniadakis, Physics informed deep learning (Part I): Data-driven solutions of nonlinear partial differential equations, 2017 (preprint). https://arxiv.org/abs/1711.10561
              • M. Raissi, P. Perdikaris, G. E. Karniadakis, Physics informed deep learning (Part II): Data-driven solutions of nonlinear partial differential equations, 2017 (preprint). https://arxiv.org/abs/1711.10566
              • K. Wu and D. Xiu, Numerical aspects for approximating governing equations using data, 2018 (preprint). 
              • D. Zhang, L. Lu, L. Guo and G. E. Karniadakis, Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems, 2018 (preprint). https://arxiv.org/abs/1809.08327
              • A. J. Majda and D. Qi, Strategies for reduced-order models for predicting the statistical responses and uncertainty quantification in complex turbulent dynamical systems, SIAM Review, vol. 60, pp. 491–549, 2018. https://doi.org/10.1137/16M1104664
              • B. Peherstorfer, K. Willcox, and M. Gunzburger, Survey of multifidelity methods in uncertainty propagation, inference, and optimization, SIAM Review, vol. 60, pp. 550–591, 2018. https://doi.org/10.1137/16M1082469
              Short-courses related to this thematic were offered by the instructor in following institutions or events:
              • Georgia Tech School of Physics / CCIS 2019
              • Universidade Federal de Uberlândia
              • Texas Tech University
              • Universidade NOVA de Lisboa
              • Universidade de Brasília
              • UNESP Ilha Solteira
              • Programa de Verão LNCC 2017
              If you have interest in this short-course for your event, company or university, please contact the instructor: americo@ime.uerj.br