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Multiresolution Analysis for Uncertainty Quantification

  • Olivier P. Le Maı̂treEmail author
  • Omar M. Knio
Reference work entry

Abstract

We survey the application of multiresolution analysis (MRA) methods in uncertainty propagation and quantification problems. The methods are based on the representation of uncertain quantities in terms of a series of orthogonal multiwavelet basis functions. The unknown coefficients in this expansion are then determined through a Galerkin formalism. This is achieved by injecting the multiwavelet representations into the governing system of equations and exploiting the orthogonality of the basis in order to derive suitable evolution equations for the coefficients. Solution of this system of equations yields the evolution of the uncertain solution, expressed in a format that readily affords the extraction of various properties. One of the main features in using multiresolution representations is their natural ability to accommodate steep or discontinuous dependence of the solution on the random inputs, combined with the ability to dynamically adapt the resolution, including basis enrichment and reduction, namely, following the evolution of the surfaces of steep variation or discontinuity. These capabilities are illustrated in light of simulations of simple dynamical system exhibiting a bifurcation and more complex applications to a traffic problem and wave propagation in gas dynamics.

Keywords

Multiresolution analysis Multiwavelet basis Stochastic refinement Stochastic bifurcation 

Notes

Acknowledgements

The authors are thankful to Dr. Alexandre Ern and Dr. Julie Tryoen for their helpful discussions and for their contributions to the work presented in this chapter.

References

  1. 1.
    Chorin, A.J.: Gaussian fields and random flow. J. Fluid Mech. 63, 21–32 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Meecham, W.C., Jeng, D.T.: Use of the Wiener-Hermite expansion for nearly normal turbulence. J. Fluid Mech. 32, 225 (1968)CrossRefzbMATHGoogle Scholar
  3. 3.
    Le Maître, O., Knio, O., Najm, H., Ghanem, R.: Uncertainty propagation using Wiener-Haar expansions. J. Comput. Phys. 197(1), 28–57 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Le Maître, O.P., Najm, H.N., Ghanem, R.G., Knio, O.M.: Multi-resolution analysis of Wiener-type uncertainty propagation schemes. J. Comput. Phys. 197(2), 502–531 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Le Maître, O.P., Najm, H.N., Pébay, P.P., Ghanem, R.G., Knio, O.M.: Multi-resolution-analysis scheme for uncertainty quantification in chemical systems. SIAM J. Sci. Comput. 29(2), 864–889 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Le Maître, O., Knio, O.: Spectral Methods for Uncertainty Quantification. Scientific Computation. Springer, Dordrecht/New York (2010)CrossRefzbMATHGoogle Scholar
  7. 7.
    Gorodetsky, A., Marzouk, Y.: Efficient localization of discontinuities in complex computational simulations. SIAM J. Sci. Comput. 36, A2584–A2610 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Beran, P.S., Pettit, C.L., Millman, D.R.: Uncertainty quantification of limit-cycle oscillations. J. Comput. Phys. 217, 217–247 (2006)CrossRefzbMATHGoogle Scholar
  9. 9.
    Pettit, C.L., Beran, P.S.: Spectral and multiresolution Wiener expansions of oscillatory stochastic processes. J. Sound Vib. 294, 752–779 (2006)CrossRefGoogle Scholar
  10. 10.
    Tryoen, J., Le Maître, O., Ndjinga, M., Ern, A.: Multi-resolution analysis and upwinding for uncertain nonlinear hyperbolic systems. J. Comput. Phys. 228, 6485–6511 (2010)CrossRefzbMATHGoogle Scholar
  11. 11.
    Tryoen, J., Le Maître, O., Ern, A.: Adaptive anisotropic spectral stochastic methods for uncertain scalar conservation laws. SIAM J. Sci. Comput. 34, 2459–2481 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Ren, X., Wu, W., Xanthis, L.S.: A dynamically adaptive wavelet approach to stochastic computations based on polynomial chaos – capturing all scales of random modes on independent grids. J. Comput. Phys. 230, 7332–7346 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Sahai, T., Pasini, J.M.: Uncertainty quantification in hybrid dynamical systems. J. Comput. Phys. 237, 411–427 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Pettersson, P., Iaccarino, G., Nordström, J.: A stochastic galerkin method for the Euler equations with roe variable transformation. J. Comput. Phys. 257, 481–500 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ghanem, R., Spanos, P.: Stochastic Finite Elements: A Spectral Approach. Dover, Minneola (2003)zbMATHGoogle Scholar
  16. 16.
    Alpert, B.K.: A class of bases in L 2 for the sparse representation of integral operators. J. Math. Anal. 24, 246–262 (1993)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Strang, G.: Introduction to Applied Mathematics. Wellesley-Cambridge Press, Wellesley (1986)zbMATHGoogle Scholar
  18. 18.
    Cohen, A., Müller, S., Postel, M., Kaber, S.: Fully adaptive multiresolution schemes for conservation laws. Math. Comput. 72, 183–225 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Cohen, A., Dahmen, W., DeVore, R.: Adaptive wavelet techniques in numerical simulation. In: Stein, E., de Borst, R., Hughes, T.J.R. (eds.) Encyclopedia of Computational Mechanics, vol. 1, pp. 157–197. Wiley, Chichester (2004)Google Scholar
  20. 20.
    Harten, A.: Multiresolution algorithms for the numerical solution of hyperbolic conservation laws. Commun. Pure Appl. Math. 48(12), 1305–1342 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Le Maître, O.P., Knio, O.M.: Spectral Methods for Uncertainty Quantification. Springer, Dordrecht/New York (2010)CrossRefzbMATHGoogle Scholar
  22. 22.
    Tryoen, J., Le Maître, O., Ndjinga M., Ern, A.: Intrusive projection methods with upwinding for uncertain nonlinear hyperbolic systems. J. Comput. Phys. 228(18), 6485–6511 (2010)CrossRefzbMATHGoogle Scholar
  23. 23.
    Tryoen, J., Le Maître, O., Ndjinga, M., Ern, A.: Roe solver with entropy corrector for uncertain nonlinear hyperbolic systems. J. Comput. Appl. Math. 235(2), 491–506 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Crestaux, T., Le Maître, O.P., Martinez, J.M.: Polynomial chaos expansion for sensitivity analysis. Reliab. Eng. Syst. Saf. 94(7), 1161–1172 (2009)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2017

Authors and Affiliations

  1. 1.LIMSI-CNRSOrsayFrance
  2. 2.Pratt School of Engineering, Mechanical Engineering and Materials ScienceDuke UniversityDurhamUSA

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