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 website:
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, Random matrix models and nonparametric method for uncertainty quantificationHandbook of Uncertainty Quantification, pp. 219-287, 2017. https://doi.org/10.1007/978-3-319-12385-1_5
              • C. Soize, Random Vectors and Random Fields in High Dimension: Parametric Model-Based Representation, Identification from Data, and Inverse ProblemsHandbook of Uncertainty Quantification, pp. 883-935, 2017. 
              • 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
              • C. Farhat, A. Bos, P. Avery, C. Soize, Modeling and quantification of model-form uncertainties in eigenvalue computations using a stochastic reduced model, AIAA Journal, vol. 56, pp. 1198-1210, 2017. https://doi.org/10.2514/1.J056314
              • 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
              Monte Carlo methods:
              • A. Cunha Jr, R. Nasser, R. Sampaio, H. Lopes and K. Breitman, Uncertainty quantification through Monte Carlo method in a cloud computing setting. Computer Physics Communications, vol. 185, pp. 1355-1363, 2014.
              Polynomial chaos:
              • PD Spanos, R Ghanem, Stochastic finite element expansion for random media, Journal of Engineering Mechanics, vol. 115, pp. 1035-1053, 1989. https://doi.org/10.1061/(ASCE)0733-9399(1989)115:5(1035)
              • 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
              • 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
              • 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
              • B. J. Debusschere, H. N. Najm, P. P. Pébay, O. M.  Knio, R. G. Ghanem, O. Le Maitre, Numerical challenges in the use of polynomial chaos representations for stochastic processesSIAM Journal on Scientific Computing, vol. 26, pp. 698-719, 2004. https://doi.org/10.1137/S1064827503427741
              • O. P. Le Maıtre, O. M. Knio, H. N. Najm, R. G. Ghanem, Uncertainty propagation using Wiener–Haar expansions, Journal of Computational Physics,vol. 197, pp. 28-57, 2004. https://doi.org/10.1016/j.jcp.2003.11.033
              • 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
              • A. Nouy, A generalized spectral decomposition technique to solve a class of linear stochastic partial differential equationsComputer Methods in Applied Mechanics and Engineering, vol. 196, pp. 4521-4537, 2007. 
              • G. Blatman, B. Sudret, Sparse polynomial chaos expansions and adaptive stochastic finite elements using a regression approachComptes Rendus Mécaniquevol.  336, pp. 518-523, 2008. https://doi.org/10.1016/j.crme.2008.02.013
              • 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
              • A. Nouy, O. Le Maitre, Generalized spectral decomposition for stochastic nonlinear problemsJournal of Computational Physics, vol. 228, pp. 202-235, 2009. https://doi.org/10.1016/j.jcp.2008.09.010
              • 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, An adaptive algorithm to build up sparse polynomial chaos expansions for stochastic finite element analysis, Probabilistic Engineering Mechanics, vol. 25, pp. 183-197,2010. 
              • 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
              • A. Kundu, S. Adhikari, M.I. Friswell, Stochastic finite elements of discretely parameterized random systems on domains with boundary uncertaintyInternational Journal for Numerical Methods in Engineering, vol. 100, pp. 183-221, 2014.
              • N. Fajraoui, S. Marelli B. Sudret, Sequential design of experiment for sparse polynomial chaos expansionsSIAM/ASA Journal on Uncertainty Quantification, vol. 5, pp. 1061-1085, 2017. https://doi.org/10.1137/16m1103488
              • V. Yaghoubi, S. Marelli, B. Sudret, T. Abrahamsson, Sparse polynomial chaos expansions of frequency response functions using stochastic frequency transformationProbabilistic Engineering Mechanics, vol. 48, pp. 39-58, 2017. 
              • Chu V. Mai, Polynomial chaos expansions for uncertain dynamical systems. Applications in earthquake engineeringIBK Bericht, vol. 502, 2018. https://doi.org/10.3929/ethz-b-000287120
              • S.E. Pryse, S. Adhikari, A. Kundu, Sample-based and sample-aggregated based Galerkin projection schemes for structural dynamicsProbabilistic Engineering Mechanics, 54, 118-130, 2018. https://doi.org/10.1016/j.probengmech.2017.09.002
              • P. Tsilifis, R. G. Ghanem, Bayesian adaptation of chaos representations using variational inference and sampling on geodesicsProceedings of the Royal Society A: Mathematical, Physical and Engineering, vol. 474 pp. 20180285, 2018.
              • P. Tsilifis, X. Huan, C. Safta, K. Sargsyan, G. Lacaze, J. C. Oefelein, H. N. Najm, R.H. Ghanem, Compressive sensing adaptation for polynomial chaos expansionsJournal of Computational Physics, vol. 380, pp. 29-47, 2019. 
              Sensitivity analysis:
              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 PlausibilityICES 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
              • J. B. Nagel B. Sudret, Bayesian multilevel model calibration for inverse problems under uncertainty with perfect dataJournal of Aerospace Information Systems, vol. 12, pp. 97-113, 2015. https://doi.org/10.2514/1.I010264
              • 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
              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 exampleSIAM Review, vol. 55, pp. 149–167, 2013. https://doi.org/10.1137/100788604
              • Y. M. Marzouk, H. N. Najm, L. A. Rahn, Stochastic spectral methods for efficient Bayesian solution of inverse problemsJournal of Computational Physics, vol. 224, pp. 560-586, 2007. https://doi.org/10.1016/j.jcp.2006.10.010
              • H. Owhadi, C. Scovel, and T. Sullivan, On the brittleness of Bayesian inferenceSIAM Review, vol. 57, pp. 566–582, 2015. https://doi.org/10.1137/130938633
              • J. B. Nagel, B. Sudret, A unified framework for multilevel uncertainty quantification in Bayesian inverse problems, Probabilistic Engineering Mechanics, vol. 43, pp. 68-84, 2016. 
              • J. B. Nagel. B. Sudret, Spectral likelihood expansions for Bayesian inferenceJournal of Computational Physics, vol. 309, pp. 267-294, 2016. https://doi.org/10.1016/j.jcp.2015.12.047
              Surrogate/Reduced Order Modeling:
              • V. Dubourg, B. Sudret, F. Deheeger, Metamodel-based importance sampling for structural reliability analysis, Probabilistic Engineering Mechanics, vol. 33, pp. 47-57,2013. https://doi.org/10.1016/j.probengmech.2013.02.002
              • V. Dubourg, B. Sudret, Meta-model-based importance sampling for reliability sensitivity analysis, Structural Safety, vol. 49, pp. 27-36, 2014. https://doi.org/10.1016/j.strusafe.2013.08.010
              • 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
              • R. Schöbi, B. Sudret, Joe Wiart, Polynomial chaos-based Kriging, International Journal for Uncertainty Quantification, vol. 5, pp. 171-193, 2015. https://doi.org/10.1615/Int.J.UncertaintyQuantification.2015012467
              • P. Kersaudy, B. Sudret, N. Varsier, O. Picon, J. Wiart, A new surrogate modeling technique combining Kriging and polynomial chaos expansions: Application to uncertainty analysis in computational dosimetryJournal of Computational Physics, vol. 286, pp. 103-117, 2015. https://doi.org/10.1016/j.jcp.2015.01.034
              • R. Schöbi, B. Sudret, S. Marelli, Rare event estimation using Polynomial-Chaos Kriging ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering, pp. D4016002, 2016.  https://doi.org/10.1061/AJRUA6.0000870
              • C. V. Mai, M. D. Spiridonakos, E. N. Chatzi, B. Sudret, Surrogate modeling for stochastic dynamical systems by combining nonlinear autoregressive with exogenous input models and polynomial chaos expansions, International Journal for Uncertainty Quantification, vol. 6: no. 4, pp. 313-339, 2016. 
              • K, Konakli, B. Sudret, Polynomial meta-models with canonical low-rank approximations: Numerical insights and comparison to sparse polynomial chaos expansionsJournal of Computational Physics, vol. 321, pp. 1144-1169, 2016. 
              • K. Konakli, B. Sudret, Reliability analysis of high-dimensional models using low-rank tensor approximations Probabilistic Engineering Mechanics, vol. 46, pp. 18-36, 2016. https://doi.org/10.1016/j.probengmech.2016.08.002
              • M. Moustapha, B. Sudret, JM Bourinet, B. Guillaume, Quantile-based optimization under uncertainties using adaptive Kriging surrogate modelsStructural and Multidisciplinary Optimization, vol. 54: no. 6, pp. 1403-1421, 2016.
              • C. V. Mai, B. Sudret, Surrogate models for oscillatory systems using sparse polynomial chaos expansions and stochastic time warping SIAM/ASA Journal on Uncertainty Quantification, vol. 5, pp. 540-571, 2017. 
              • M. Billaud-Friess, A. Nouy, Dynamical model reduction method for solving parameter-dependent dynamical systemsSIAM Journal on Scientific Computing, vol. 39, pp. A1766-A1792, 2017. https://doi.org/10.1137/16M1071493
              • O. Zahm, M. Billaud-Friess, A. Nouy, Projection-based model order reduction methods for the estimation of vector-valued variables of interestSIAM Journal on Scientific Computing, vol. 39, pp. A1647-A1674, 2017. 
              • A. Nouy, Low-rank tensor methods for model order reductionHandbook of Uncertainty Quantification, 857-882, 2017.
              • C. Lataniotis, S. Marelli B. Sudret, The Gaussian process modelling module in UQLabSoft Computing in Civil Engineering, vol. 2, pp. 91-116, 2018. https://doi.org/10.22115/scce.2018.129323.1062
              • M. Moustapha, JM Bourinet, B. Guillaume, B. Sudret, Comparative study of kriging and support vector regression for structural engineering applications, ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering, vol. 4, pp. 04018005, 2018. https://doi.org/10.1061/AJRUA6.0000950
              • V. Yaghoubi, S. Rahrovani, H. Nahvi, S. Marelli, Reduced order surrogate modeling technique for linear dynamic systemsMechanical Systems and Signal Processing, vol. 111, pp. 172-193, 2018. 
              • 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
                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
                • A. Kundu, H. G. Matthies, M.I. Friswell, Probabilistic optimization of engineering system with prescribed target design in a reduced parameter spaceComputer Methods in Applied Mechanics and Engineering, 337, 281-304, 2018.
                • C. Soize, Design optimization under uncertainties of a mesoscale implant in biological tissues using a probabilistic learning algorithmComputational Mechanics, vol. 62, pp. 477-497, 2018. https://doi.org/10.1007/s00466-017-1509-x
                • R. Ghanem, C. Soize, Probabilistic nonconvex constrained optimization with fixed number of function evaluations, International Journal for Numerical Methods in Engineering, vol. 113, pp. 719-741, 2018. 
                • M. Moustapha, B. Sudret, Surrogate-assisted reliability-based design optimization: a survey and a unified modular frameworkStructural and Multidisciplinary Optimization, pp. 1-21, 2019. https://doi.org/10.1007/s00158-019-02290-y
                Multi-fidelity methods
                • T. D. Robinson, M. S. Eldred, K. E. Willcox, R. Haimes, Surrogate-based optimization using multifidelity models with variable parameterization and corrected space mappingAIAA Journal, vol. 46, pp. 2814-2822, 2008. https://doi.org/10.2514/1.36043
                • A. March, K. Willcox, Constrained multifidelity optimization using model calibrationStructural and Multidisciplinary Optimization, vol. 46, pp. 93-109, 2012. https://doi.org/10.1007/s00158-011-0749-1
                • L. W. T. Ng, K. E. Willcox, Multifidelity approaches for optimization under uncertaintyInternational Journal for Numerical Methods in Engineering, vol. 100, pp. 746-772, 2014. https://doi.org/10.1002/nme.4761
                • D. Allaire, K. Willcox, A mathematical and computational framework for multifidelity design and analysis with computer modelsInternational Journal for Uncertainty Quantification, vol. 4, pp. 1-20, 2014. 
                • B. Peherstorfer, K. Willcox, M. Gunzburger, Optimal model management for multifidelity Monte Carlo estimationSIAM Journal on Scientific Computing, vol. 38, pp. A3163-A3194, 2016. https://doi.org/10.1137/15M1046472
                • B. Peherstorfer, T. Cui, Y. Marzouk, K. Willcox, Multifidelity importance samplingComputer Methods in Applied Mechanics and Engineering, vol. 300, 490-509, 2016. https://doi.org/10.1016/j.cma.2015.12.002
                • A. Chaudhuri, R. Lam, K. Willcox, Multifidelity Uncertainty propagation via adaptive surrogates in coupled multidisciplinary systemsAIAA Journal, 2017. https://doi.org/10.2514/1.J055678
                • B. Peherstorfer, B. Kramer, K. Willcox, Multifidelity preconditioning of the cross-entropy method for rare event simulation and failure probability estimation, SIAM/ASA Journal on Uncertainty Quantification, vol. 6, 737-761, 2018.
                • E. Qian, B. Peherstorfer, D. O’Malley, V.V.  Vesselinov, K. Willcox, Multifidelity Monte Carlo estimation of variance and sensitivity indicesSIAM/ASA Journal for Uncertainty Quantification, vol. 6, pp. 683–706, 2018. 
                • B. Peherstorfer, M. Gunzburger, K. Willcox, Convergence analysis of multifidelity Monte Carlo estimationNumerische Mathematik, vol. 139, pp. 683-707, 2018. https://doi.org/10.1007/s00211-018-0945-7
                • R. Lam, O. Zahm, Y. Marzouk, K. Willcox,  Multifidelity Dimension Reduction via Active Subspaces, 2018 (preprint). 
                • B. Peherstorfer, K. Willcox, and M. Gunzburger, Survey of multifidelity methods in uncertainty propagation, inference, and optimizationSIAM Review, vol. 60, pp. 550–591, 2018. https://doi.org/10.1137/16M1082469
                • B. Kramer, A. N. Marques, B. Peherstorfer, U. Villa, K. Willcox, Multifidelity probability estimation via fusion of estimatorsJournal of Computational Physics, 392, 385-402, 2019. https://doi.org/10.1016/j.jcp.2019.04.071
                • A. Chaudhuri, A. N.  Marques, K. E. Willcox, mfEGRA: Multifidelity Efficient Global Reliability Analysis, 2019 (preprint) 
                Data-driven, machine learning methods, 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
                • 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
                • S. Marelli B. Sudret, An active-learning algorithm that combines sparse polynomial chaos expansions and bootstrap for structural reliability analysisStructural Safety, vol. 75, pp. 67-74, 2018. https://doi.org/10.1016/j.strusafe.2018.06.003
                • E. Torre, S. Marelli, P. Embrechts B. Sudret, A general framework for data-driven uncertainty quantification under complex input dependencies using vine copulasProbabilistic Engineering Mechanics, vol. 55, pp. 1-16, 2019.  
                • E. Torre, S. Marelli, P. Embrechts, B. Sudret, Data-driven polynomial chaos expansion for machine learning regression, Journal of Computational Physics, vol. 388, pp. 601-623, 2019. https://doi.org/10.1016/j.jcp.2019.03.039
                • C Soize, R Ghanem, Physics‐constrained non‐Gaussian probabilistic learning on manifoldsInternational Journal for Numerical Methods in Engineering, 2019. https://doi.org/10.1002/nme.6202
                • C. Farhat, R. Tezaur, T. Chapman, P. Avery, C. Soize, Feasible probabilistic learning method for model-form uncertainty quantification in vibration analysisAIAA Journal, pp. 1-14, 2019. https://doi.org/10.2514/1.J057797
                • C. Soize, C. Farhat, Probabilistic learning for modeling and quantifying model‐form uncertainties in nonlinear computational mechanicsInternational Journal for Numerical Methods in Engineering, vol. 117, pp. 819-843, 2019. 
                Applications of UQ tools:
                • L. G. G. Villani, S. da Silva, A. Cunha Jr and M. D. Todd,  

                  On the detection of a nonlinear damage in an uncertain nonlinear beam using stochastic Volterra series. Structural Health Monitoring An International Journal, 2019 (in press).

                • D. A. Santos and A. Cunha Jr,  

                  Flight control of a hexa-rotor airship: Uncertainty quantification for a range of temperature and pressure conditions. ISA Transactions,  vol. 93, pp. 268-279, 2019.

                • J. P. Dias, S. Ekwaro-Osire, A. Cunha Jr, S. Dabetwar, A. Nispel,  F. M. Alemayehu and H. B. Endeshaw,  

                   Parametric probabilistic approach for cumulative fatigue damage using double linear damage rule considering limited data. International Journal of Fatigue

                  vol. 127, pp. 246-258, 2019


                • P. Wolszczak, P. Lonkwic, A. Cunha Jr, G. Litak and S. Molski,  

                  Robust optimization and uncertainty quantification in the nonlinear mechanics of an elevator brake system. Meccanica,  vol. 54, pp. 1057-1069, 2019.

                • L. G. G. Villani, S. da Silva, A. Cunha Jr and M. D. Todd,  

                  Damage detection in an uncertain nonlinear beam based on stochastic Volterra series: an experimental application. Mechanical Systems and Signal Processing,  vol. 128, pp. 463-478, 2019.

                • L. G. G. Villani, S. da Silva and A. Cunha Jr,  

                  Damage detection in uncertain nonlinear systems based on stochastic Volterra series. Mechanical Systems and Signal Processing,  vol. 125, pp. 288-310, 2019.


                • N. Cuellar, A. Pereira, I. F. M. Menezes, and A. Cunha Jr, 
                • Non-intrusive polynomial chaos expansion for topology optimization using polygonal meshes. 
                • Journal of the Brazilian Society of Mechanical Sciences and Engineering, vol. 40, pp. 561, 2018.

                • https://doi.org/10.1007/s40430-018-1464-2
                • E. Dantas, M. Tosin and A. Cunha Jr, Calibration of a SEIR–SEI epidemic model to describe the Zika virus outbreak in Brazil. Applied Mathematics and Computation, vol. 338, pp. 249-259, 2018.
                • A. Cunha Jr, J. L. P. Felix and J. M. Balthazar,  

                  Quantification of parametric uncertainties induced by irregular soil loading in orchard tower sprayer nonlinear dynamics. Journal of Sound and Vibration, vol. 408, pp. 252-269, 2017.

                • D. Colón, A. Cunha Jr, S. Kaczmarczyk and J. M. Balthazar,  

                  On dynamic analysis and control of an elevator system using polynomial chaos and Karhunen-Loève approachesProcedia Engineering, vol. 199, pp. 1629-1634, 2017.

                • L.G.G. Villani, S. da Silva and A. Cunha Jr, 
                • Damage detection in an uncertain nonlinear beam 

                  Procedia Engineering, vol. 199, pp. 2090-2095, 2017.


                • http://dx.doi.org/10.1016/j.proeng.2017.09.480
                • A. Cunha Jr, C. Soize and R. Sampaio, Computational modeling of the nonlinear stochastic dynamics of horizontal drillstrings. Computational Mechanics, vol. 55, pp. 849-878, 2015.
                • A. Cunha Jr and R. Sampaio, On the nonlinear stochastic dynamics of a continuous system with discrete attached elements. Applied Mathematical Modelling, vol. 39, pp. 809-819, 2015.
                • A. Cunha Jr and R. Sampaio, Study of the nonlinear longitudinal dynamics of a stochastic system. MATEC Web of Conferences, vol. 16, pp. 05004, 2014.
                • M. G. Sandoval, A. Cunha Jr and R. Sampaio, Identification of parameters in the torsional dynamics of a drilling process through Bayesian statistics. Mecánica Computacional, vol. 32, pp. 763-773, 2013.
                • http://www.cimec.org.ar/ojs/index.php/mc/article/view/4388/4318
                • A. Cunha Jr and R. Sampaio,  Effect of an attached end mass in the dynamics of uncertainty nonlinear continuous random system. Mecánica Computacional, vol. 31, pp. 2673-2683, 2012. 
                • http://www.cimec.org.ar/ojs/index.php/mc/article/view/4214/4140

                Short-courses related to this thematic were offered by the instructor in following institutions/events:
                • Georgia Tech School of Physics / CCIS 2019
                • Universidade Federal de Uberlândia
                • Texas Tech University
                • Universidade NOVA de Lisboa
                • ICVRAM-ISUMA UNCERTAINTIES 2018
                • 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