Abstract
We investigate the evolution of the probability distribution function in time for some wave and Maxwell equations in random media for which the parameters, e.g. permeability, permittivity, fluctuate randomly in space; more precisely, two different media interface randomly in space. We numerically compute the probability distribution and density for output solutions. The underlying numerical and statistical techniques are the so-called polynomial chaos Galerkin projection, which has been extensively used for simulating partial differential equations with uncertainties, and the Monte Carlo simulations.
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Babuška, I., Tempone, R., Zouraris, G.E.: Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J. Numer. Anal. 42, 800–825 (2004)
Billingsley, P.: Probability and Measure. A Wiley-Interscience Publication. Wiley, New York (1995)
Bondeson, A., Rylander, T., Ingelstrom, P.: Computational Eletromagnetics. Springer, New York (2005)
Chauvière, C., Hesthaven, J.S., Lurati, L.: Computational modeling of uncertainity in time-domain electromagnetics. SIAM J. Sci. Comput. 28, 751–775 (2006)
Chen, Q.-Y., Gottlieb, D., Hesthaven, J.S.: Uncertainty analysis for the steady-state flows in a dual throat nozzle. J. Comput. Phys. 204, 378–398 (2005)
Chorin, A.J., Hald, O.H.: Stochastic Tools in Mathematics and Science. Springer, New York (2006)
Cockburn, B., Shu, C.-W.: Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16, 173–261 (2001)
Deb, M.K., Babuška, I., Oden, J.T.: Solution of stochastic partial differential equations using Galerkin finite element techniques. Comput. Methods Appl. Mech. Eng. 190, 6359–6372 (2001)
Debusschere, B.J., Najm, H.N., Pébay, P.P., Knio, O.M., Ghanem, R.G., Le Maître, O.P.: Numerical challenges in the use of polynomial chaos representations for stochastic processes. SIAM J. Sci. Comput. 26, 698–719 (2004)
Evans, L.C.: Partial Differential Equations. American Mathematical Society, Providence (1998)
Ghanem, R.G., Spanos, P.D.: Stochastic Finite Elements: A Spectral Approach. Springer, New York (1991)
Gottlieb, D., Xiu, D.: Galerkin method for wave equations with uncertain coefficients. Commun. Comput. Phys. 3, 505–518 (2008)
Herrick, D.R.M., Nason, G.P., Silverman, B.W.: Some new methods for wavelet density estimation. Special issue on wavelets, Sankhya Ser. A 63, 394–411 (2001)
Isakov, V.: Inverse Problems for Partial Differential Equations. Springer, New York (2006)
Jäckel, P.: Monte Carlo Methods in Finance. Wiley, New York (2002)
Kalbfleisch, J.G.: Probability and Statistical Inference. Springer, New York (1985)
Knio, O.M., Le Maître, O.P.: Uncertainity propagation in CFD using polynomial chaos decomposition. Fluid Dyn. Res. 38, 616–640 (2006)
Le Maître, O.P., Knio, O.M., Najm, H.N., Ghanem, R.G.: Uncertainity propagation using Wiener-Haar expansions. J. Comput. Phys. 197, 28–57 (2004)
Lu, Z., Zhang, D.: A comparative study on uncertainty quantification for flow in randomly heterogeneous media using Monte Carlo simulations and conventional and KL-based moment-equation approaches. SIAM J. Sci. Comput. 26, 558–577 (2004)
Mantegna, R.N.: Time evolution of the probability distribution in stochastic and chaotic systems with enhanced diffusion. J. Stat. Phys. 70, 721–736 (1993)
McOwen, R.C.: Partial Differential Equations: Methods and Applications. Prentice-Hall, New York (2003)
Robert, C.P., Casella, G.: Monte Carlo Statistical Methods. Springer, New York (2004)
Sansone, G.: Orthogonal Functions. Dover, New York (1991). Translated from the Italian by Ainsley H. Diamond. With a foreword by Einar Hille. Reprint of the 1959 edition
Schultz, M.H.: Spline Analysis. Prentice-Hall, New York (1973)
Silverman, B.W.: Density Estimation for Statistics and Data Analysis. Chapman & Hall, London (1986)
Strikwerda, J.C.: Finite Difference Schemes and Partial Differential Equations. Chapman & Hall/CRC, London/Boca Raton (1999)
Temam, R.: Navier-Stokes Equations. AMS/Chelsea, Providence/New York (2001)
Vannucci, M.: Nonparametric density estimation using wavelets: a review. Discussion Paper 95-26, ISDS, Duke University, USA (1995)
Velamur Asokan, B., Zabaras, A.: A stochastic variational multiscale method for diffusion in heterogeneous random media. J. Comput. Phys. 218, 654–676 (2006)
Xiu, D.: Fast numerical methods for stochastic computations: a review. Commun. Comput. Phys. 5, 242–272 (2009)
Xiu, D., Hesthaven, J.S.: High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput. 27, 1118–1139 (2005)
Xiu, D., Karniadakis, G.E.: The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24, 619–644 (2002)
Xiu, D., Karniadakis, G.E.: Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos. Comput. Methods Appl. Mech. Eng. 199, 4927–4948 (2002)
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Jung, CY. Evolution of Probability Distribution in Time for Solutions of Hyperbolic Equations. J Sci Comput 41, 13–48 (2009). https://doi.org/10.1007/s10915-009-9284-2
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DOI: https://doi.org/10.1007/s10915-009-9284-2