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Modeling and Quantification of Physical Systems Uncertainties in a Probabilistic Framework

  • Americo CunhaJr.
Chapter

Abstract

Uncertainty quantification (UQ) is a multidisciplinary area, that deals with quantitative characterization and reduction of uncertainties in applications. It is essential to certify the quality of numerical and experimental analyses of physical systems. The present manuscript aims to provide the reader with an introductory view about modeling and quantification of uncertainties in physical systems. In this sense, the text presents some fundamental concepts in UQ, a brief review of probability basics notions, discusses, through a simplistic example, the fundamental aspects of probabilistic modeling of uncertainties in a physical system, and explains what is the uncertainty propagation problem.

Keywords

Uncertainty quantification Stochastic modeling of uncertainties Probabilistic approach 

Notes

Acknowledgements

The author’s research is supported by the Brazilian agencies CNPq (National Council for Scientific and Technological Development), CAPES (Coordination for the Improvement of Higher Education Personne) and FAPERJ (Research Support Foundation of the State of Rio de Janeiro).

References

  1. 1.
    L. Biegler, G. Biros, O. Ghattas, M. Heinkenschloss, D. Keye, B. Mallick, Y. Marzouk, L. Tenorio, B.B. Waanders, K. Willcox, Large-Scale Inverse Problems and Quantification of Uncertainty (Wiley, 2010)Google Scholar
  2. 2.
    O.P. Le Maître, O.M. Knio, Spectral Methods for Uncertainty Quantification: With Applications to Computational Fluid Dynamics (Springer, 2010)Google Scholar
  3. 3.
    D. Xiu, Numerical Methods for Stochastic Computations: A Spectral Method Approach (Princeton University Press, 2010)Google Scholar
  4. 4.
    C. Soize, Stochastic Models of Uncertainties in Computational Mechanics (American Society of Civil Engineers, 2012)Google Scholar
  5. 5.
    M. Grigoriu, Stochastic Systems: Uncertainty Quantification and Propagation (Springer, 2012)Google Scholar
  6. 6.
    R.C. Smith, Uncertainty Quantification: Theory, Implementation, and Applications (SIAM, 2013)Google Scholar
  7. 7.
    H. Bijl, D. Lucor, S. Mishra, C. Schwab, Uncertainty Quantification in Computational Fluid Dynamics (Springer, 2013)Google Scholar
  8. 8.
    M.P. Pettersson, G. Iaccarino, J. Nordström, Polynomial Chaos Methods for Hyperbolic Partial Differential Equations: Numerical Techniques for Fluid Dynamics Problems in the Presence of Uncertainties (Springer, 2015)Google Scholar
  9. 9.
    R. Ohayon, C. Soize, Advanced Computational Vibroacoustics: Reduced-Order Models and Uncertainty Quantification (Cambridge University Press, 2015)Google Scholar
  10. 10.
    T.J. Sullivan, Introduction to Uncertainty Quantification (Springer, 2015)Google Scholar
  11. 11.
    S. Sarkar, J.A.S. Witteveen, Uncertainty Quantification in Computational Science (World Scientific Publishing Company, 2016)Google Scholar
  12. 12.
    R. Ghanem, D. Higdon, H. Owhadi, Handbook of Uncertainty Quantification (Springer, 2017)Google Scholar
  13. 13.
    C. Soize, Uncertainties and Stochastic Modeling (Short Course at PUC-Rio, Aug 2008)Google Scholar
  14. 14.
    C. Soize, Stochastic Models in Computational Mechanics (Short Course at PUC-Rio, Aug 2010)Google Scholar
  15. 15.
    C. Soize, Probabilité et Modélisation des Incertitudes: Eléments de base et concepts fondamentaux (Course Notes, Université Paris-Est Marne-la-Vallée, Paris, 2013)Google Scholar
  16. 16.
    G. Iaccarino, A. Doostan, M.S. Eldred, O. Ghattas, Introduction to uncertainty quantification techniques, in Minitutorial at SIAM CSE Conference, 2009Google Scholar
  17. 17.
    G. Iaccarino, Introduction to Uncertainty Quantification (Lecture at KAUST, 2012)Google Scholar
  18. 18.
    A. Doostan, P. Constantine, Numerical Methods for Uncertainty Propagation (Short Course at USNCCM13, 2015)Google Scholar
  19. 19.
    C. Soize, A comprehensive overview of a non-parametric probabilistic approach of model uncertainties for predictive models in structural dynamics. J. Sound Vib. 288, 623–652 (2005)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Guide for the verification and validation of computational fluid dynamics simulations. Technical Report AIAA G-077-1998 (American Institute of Aeronautics and Astronautics, Reston, 1998)Google Scholar
  21. 21.
    W.L. Oberkampf, T.G. Trucano, Verification and validation in computational fluid dynamics. Technical Report SAND 2002-0529 (Sandia National Laboratories, Livermore, 2002)Google Scholar
  22. 22.
    W. Oberkampf, T. Trucano, C. Hirsch, Verification, validation, and predictive capability in computational engineering and physics. Appl. Mech. Rev. 57, 345–384 (2004)CrossRefGoogle Scholar
  23. 23.
    ASME Guide for Verification and Validation in Computational Solid Mechanics. Technical Report ASME Standard V&V 10-2006 (American Society of Mechanical Engineers, New York, 2006)Google Scholar
  24. 24.
    W.L. Oberkampf, C.J. Roy, Verification and Validation in Scientific Computing (Cambridge University Press, 2010)Google Scholar
  25. 25.
    U.M. Ascher, C. Greif, A First Course in Numerical Methods (SIAM, 2011)Google Scholar
  26. 26.
    P.J. Roache, Code verification by the method of manufactured solutions. J. Fluids Eng. 124, 4–10 (2001)CrossRefGoogle Scholar
  27. 27.
    C.J. Roy, Review of code and solution verification procedures for computational simulation. J. Comput. Phys. 205, 131–156 (2005)CrossRefMATHGoogle Scholar
  28. 28.
    L.A. Petri, P. Sartori, J.K. Rogenski, L.F. de Souza, Verification and validation of a direct numerical simulation code. Comput. Methods Appl. Mech. Eng. 291, 266–279 (2015)MathSciNetCrossRefGoogle Scholar
  29. 29.
    G.I. Schuëller, A state-of-the-art report on computational stochastic mechanics. Probabilistic Eng. Mech. 12, 197–321 (1997)CrossRefGoogle Scholar
  30. 30.
    G.I. Schuëller, Computational stochastic mechanics recent advances. Comput. Struct. 79, 2225–2234 (2001)CrossRefGoogle Scholar
  31. 31.
    C. Soize, Stochastic modeling of uncertainties in computational structural dynamics—recent theoretical advances. J. Sound Vib. 332, 2379–2395 (2013)CrossRefGoogle Scholar
  32. 32.
    D. Moens, D. Vandepitte, A survey of non-probabilistic uncertainty treatment in finite element analysis. Comput. Methods Appl. Mech. Eng. 194, 1527–1555 (2005)CrossRefMATHGoogle Scholar
  33. 33.
    D. Moens, M. Hanss, Non-probabilistic finite element analysis for parametric uncertainty treatment in applied mechanics: recent advances. Finite Elem. Anal. Des. 47, 4–16 (2011)CrossRefGoogle Scholar
  34. 34.
    M. Beer, S. Ferson, V. Kreinovich, Imprecise probabilities in engineering analyses. Mech. Syst. Signal Process. 37, 4–29 (2013)CrossRefGoogle Scholar
  35. 35.
    C. Soize, A nonparametric model of random uncertainties for reduced matrix models in structural dynamics. Probabilistic Eng. Mech. 15, 277–294 (2000)CrossRefGoogle Scholar
  36. 36.
    C. Soize, Generalized probabilistic approach of uncertainties in computational dynamics using random matrices and polynomial chaos decompositions. Int. J. Numer. Methods Eng. 81, 939–970 (2010)MathSciNetMATHGoogle Scholar
  37. 37.
    A. Batou, C. Soize, M. Corus, Experimental identification of an uncertain computational dynamical model representing a family of structures. Comput. Struct. 89, 1440–1448 (2011)CrossRefGoogle Scholar
  38. 38.
    G. Grimmett, D. Welsh, Probability: An Introduction, 2nd edn. (Oxford University Press, 2014)Google Scholar
  39. 39.
    J. Jacod, P. Protter, Probability Essentials, 2nd edn. (Springer, 2004)Google Scholar
  40. 40.
    A. Klenke, Probability Theory: A Comprehensive Course, 2nd edn. (Springer, 2014)Google Scholar
  41. 41.
    A. Papoulis, S.U. Pillai, Probability, Random Variables and Stochastic Processes, 4th edn. (McGraw-Hill, 2002)Google Scholar
  42. 42.
    C.E. Shannon, A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423 (1948)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    L. Wasserman, All of Nonparametric Statistics (Springer, 2007)Google Scholar
  44. 44.
    L. Wasserman, All of Statistics: A Concise Course in Statistical Inference (Springer, 2004)Google Scholar
  45. 45.
    E.T. Jaynes, Information theory and statistical mechanics. Phys. Rev. Ser. II(106), 620–630 (1957)MathSciNetMATHGoogle Scholar
  46. 46.
    E.T. Jaynes, Information theory and statistical mechanics II. Phys. Rev. Ser. II(108), 171–190 (1957)MathSciNetMATHGoogle Scholar
  47. 47.
    J.N. Kapur, H.K. Kesavan, Entropy Optimization Principles with Applications (Academic Press, 1992)Google Scholar
  48. 48.
    J.N. Kapur, Maximum Entropy Models in Science and Engineering (New Age, 2009)Google Scholar
  49. 49.
    F.E. Udwadia, Response of uncertain dynamic systems. I. Appl. Math. Comput. 22, 115–150 (1987)CrossRefMATHGoogle Scholar
  50. 50.
    F.E. Udwadia, Response of uncertain dynamic systems. II. Appl. Math. Comput. 22, 151–187 (1987)CrossRefMATHGoogle Scholar
  51. 51.
    F.E. Udwadia, Some results on maximum entropy distributions for parameters known to lie in finite intervals. SIAM Rev. 31, 103–109 (1989)MathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    K. Sobezyk, J. Trçbicki, Maximum entropy principle in stochastic dynamics. Probabilistic Eng. Mech. 5, 102–110 (1990)CrossRefGoogle Scholar
  53. 53.
    K. Sobezyk, J. Trȩbicki, Maximum entropy principle and nonlinear stochastic oscillators. Phys. A: Stat. Mech. Appl. 193, 448–468 (1993)CrossRefMATHGoogle Scholar
  54. 54.
    J. Trȩbicki, K. Sobezyk, Maximum entropy principle and non-stationary distributions of stochastic systems. Probabilistic Eng. Mech. 11, 169–178 (1996)CrossRefGoogle Scholar
  55. 55.
    A. Cunha Jr., R. Nasser, R. Sampaio, H. Lopes, K. Breitman, Uncertainty quantification through Monte Carlo method in a cloud computing setting. Comput. Phys. Commun. 185, 1355–1363 (2014)CrossRefGoogle Scholar
  56. 56.
    N. Metropolis, S. Ulam, The Monte Carlo method. J. Am. Stat. Assoc. 44, 335–341 (1949)CrossRefMATHGoogle Scholar
  57. 57.
    C. Lemieux, Monte Carlo and Quasi-Monte Carlo Sampling (Springer, 2009)Google Scholar
  58. 58.
    D.P. Kroese, T. Taimre, Z.I. Botev, Handbook of Monte Carlo Methods (Wiley, 2011)Google Scholar
  59. 59.
    J.S. Liu, Monte Carlo Strategies in Scientific Computing (Springer, 2001)Google Scholar
  60. 60.
    G. Fishman, Monte Carlo: Concepts, Algorithms, and Applications, corrected edn. (Springer, 2003)Google Scholar
  61. 61.
    R.Y. Rubinstein, D.P. Kroese, Simulation and the Monte Carlo Method, 2nd edn. (Wiley, 2007)Google Scholar
  62. 62.
    S. Asmussen, P.W. Glynn, Stochastic Simulation: Algorithms and Analysis (Springer, 2007)Google Scholar
  63. 63.
    R.W. Shonkwiler, F. Mendivil, Explorations in Monte Carlo Methods (Springer, 2009)Google Scholar
  64. 64.
    C.P. Robert, G. Casella, Monte Carlo Statistical Methods (Springer, 2010)Google Scholar
  65. 65.
    R. Ghanem, P.D. Spanos, Polynomial chaos in stochastic finite elements. J. Appl. Mech. 57, 197–202 (1990)CrossRefMATHGoogle Scholar
  66. 66.
    R. Ghanem, P.D. Spanos, Stochastic Finite Elements: A Spectral Approach, 2nd edn. (Dover Publications, 2003)Google Scholar
  67. 67.
    D. Xiu, G.E. Karniadakis, The Wiener-Askey Polynomial Chaos for stochastic differential equations. SIAM J. Sci. Comput. 24, 619–644 (2002)MathSciNetCrossRefMATHGoogle Scholar
  68. 68.
    P. Vos, Time-dependent polynomial chaos. Master Thesis, Delft University of Technology, Delft, 2006Google Scholar
  69. 69.
    P. Constantine, in A Primer on Stochastic Galerkin Methods. Lecture Notes, 2007Google Scholar
  70. 70.
    A. O’Hagan, in Polynomial Chaos: A Tutorial and Critique from a Statistician’s Perspective, (submitted to publication, 2013)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.NUMERICO - Nucleus of Modeling and Experimentation with ComputersUniversidade do Estado do Rio de JaneiroRio de JaneiroBrazil

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