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.
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References
Chorin, A.J.: Gaussian fields and random flow. J. Fluid Mech. 63, 21–32 (1974)
Meecham, W.C., Jeng, D.T.: Use of the Wiener-Hermite expansion for nearly normal turbulence. J. Fluid Mech. 32, 225 (1968)
Le Maître, O., Knio, O., Najm, H., Ghanem, R.: Uncertainty propagation using Wiener-Haar expansions. J. Comput. Phys. 197(1), 28–57 (2004)
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)
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)
Le Maître, O., Knio, O.: Spectral Methods for Uncertainty Quantification. Scientific Computation. Springer, Dordrecht/New York (2010)
Gorodetsky, A., Marzouk, Y.: Efficient localization of discontinuities in complex computational simulations. SIAM J. Sci. Comput. 36, A2584–A2610 (2014)
Beran, P.S., Pettit, C.L., Millman, D.R.: Uncertainty quantification of limit-cycle oscillations. J. Comput. Phys. 217, 217–247 (2006)
Pettit, C.L., Beran, P.S.: Spectral and multiresolution Wiener expansions of oscillatory stochastic processes. J. Sound Vib. 294, 752–779 (2006)
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)
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)
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)
Sahai, T., Pasini, J.M.: Uncertainty quantification in hybrid dynamical systems. J. Comput. Phys. 237, 411–427 (2013)
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)
Ghanem, R., Spanos, P.: Stochastic Finite Elements: A Spectral Approach. Dover, Minneola (2003)
Alpert, B.K.: A class of bases in L 2 for the sparse representation of integral operators. J. Math. Anal. 24, 246–262 (1993)
Strang, G.: Introduction to Applied Mathematics. Wellesley-Cambridge Press, Wellesley (1986)
Cohen, A., Müller, S., Postel, M., Kaber, S.: Fully adaptive multiresolution schemes for conservation laws. Math. Comput. 72, 183–225 (2002)
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)
Harten, A.: Multiresolution algorithms for the numerical solution of hyperbolic conservation laws. Commun. Pure Appl. Math. 48(12), 1305–1342 (1995)
Le Maître, O.P., Knio, O.M.: Spectral Methods for Uncertainty Quantification. Springer, Dordrecht/New York (2010)
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)
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)
Crestaux, T., Le Maître, O.P., Martinez, J.M.: Polynomial chaos expansion for sensitivity analysis. Reliab. Eng. Syst. Saf. 94(7), 1161–1172 (2009)
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.
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Le Maı̂tre, O.P., Knio, O.M. (2017). Multiresolution Analysis for Uncertainty Quantification. In: Ghanem, R., Higdon, D., Owhadi, H. (eds) Handbook of Uncertainty Quantification. Springer, Cham. https://doi.org/10.1007/978-3-319-12385-1_18
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DOI: https://doi.org/10.1007/978-3-319-12385-1_18
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