ライブラリ登録: Guest
Begell Digital Portal Begellデジタルライブラリー 電子書籍 ジャーナル 参考文献と会報 リサーチ集
International Journal for Uncertainty Quantification
インパクトファクター: 3.259 5年インパクトファクター: 2.547 SJR: 0.531 SNIP: 0.8 CiteScore™: 1.52

ISSN 印刷: 2152-5080
ISSN オンライン: 2152-5099

Open Access

International Journal for Uncertainty Quantification

DOI: 10.1615/Int.J.UncertaintyQuantification.2019029046
pages 161-186

NUMERICAL APPROXIMATION OF ELLIPTIC PROBLEMS WITH LOG-NORMAL RANDOM COEFFICIENTS

Xiaoliang Wan
Department of Mathematics and Center of Computation and Technology, Louisiana State University, Baton Rouge, LA, 70803
Haijun Yu
NCMIS & LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Beijing 100190; School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

要約

In this work, we consider a non-standard preconditioning strategy for the numerical approximation of the classical elliptic equations with log-normal random coefficients. In earlier work, a Wick-type elliptic model was proposed by modeling the random flux through the Wick product. Due to the lower-triangular structure of the uncertainty propagator, this model can be approximated efficiently using the Wiener chaos expansion in the probability space. Such a Wick-type model provides, in general, a second-order approximation of the classical one in terms of the standard deviation of the underlying Gaussian process. Furthermore, when the correlation length of the underlying Gaussian process goes to infinity, the Wick-type model yields the same solution as the classical one. These observations imply that the Wick-type elliptic equation can provide an effective preconditioner for the classical random elliptic equation under appropriate conditions. We use the Wick-type elliptic model to accelerate the Monte Carlo method and the stochastic Galerkin finite element method. Numerical results are presented and discussed.

参考

  1. Lototsky, S., Rozovskii, B., and Wan, X., Elliptic Equations of Higher Stochastic Order, ESAIM: Math. Model. Numer. Anal., 5(4):1135–1153, 2010.

  2. Galvis, J. and Sarkis, M., Aproximating Infinity-Dimensional Stochastic Darcy’s Equations without Uniform Ellipticity, SIAM J. Numer. Anal., 47(5):3624–3651, 2009.

  3. Gittelson, C.J., Stochastic Galerkin Discretization of the Log-Normal Isotropic Diffusion Problems, Math. Models Methods Appl. Sci., 20(2):237–263, 2010.

  4. Mugler, A. and Starkloff, H.-J., On Elliptic Partial Differential Equations with Random Coefficients, Stud. Univ. Babes-Bolyai Math., 56(2):473–487, 2011.

  5. Da Prato, G. and Zabczyk, J., Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and Its Applications, vol. 44, Cambridge, UK: Cambridge University Press, 1992.

  6. Charrier, J., Strong and Weak Error Estimates for Elliptic Partial Differential Equations with Random Coefficients, SIAM J. Numer. Anal., 50(1):216–246, 2012.

  7. Ghanem, R. and Spanos, P., Stochastic Finite Element: A Spectral Approach, New York: Springer-Verlag, 1991.

  8. Babuska, I., Tempone, R., and Zouraris, G., Galerkin Finite Element Approximations of Stochastic Elliptic Differential Equations, SIAM J. Numer. Anal., 42:800–825, 2004.

  9. Frauenfelder, P., Schwab, C., and Todor, R., Finite Elements for Elliptic Problems with Stochastic Coefficients, Comput. Methods Appl. Mech. Eng., 194:205–228, 2005.

  10. Todor, R. and Schwab, C., Convergence Rates for Sparse Chaos Approximations of Elliptic Problems with Stochastic Coefficients, IMA J. Numer. Anal., 27(2):232–261, 2007.

  11. Bonizzoni, F. and Nobile, F., Perturbation Analysis for the Darcy Problem with Log-Normal Permeability, SIAM/ASA J. Uncertainty Quantif., 2:223–244, 2014.

  12. Bonizzoni, F., Nobile, F., and Kressner, D., Tensor Train Approximation of Moment Equations for Elliptic Equations with Lognormal Coefficient, Comput. Meth. Appl. Mech. Eng., 308:349–376, 2016.

  13. Chkifa, A., Cohen, A., DeVore, R., and Schwab, C., Sparse Adaptive Taylor Approximation Algorithms for Parametric and Stochastic Elliptic PDEs, ESAIM: Math. Model. Numer. Anal., 47(1):253–280, 2013.

  14. Nobile, F. and Tesei, F., A Multi Level Monte Carlo Method with Control Variate for Elliptic PDEs with Log-Normal Coefficients, Stochastic PDE: Anal. Comput., 3(3):398–444, 2015.

  15. Nobile, F., Tamellini, L., Tesei, F., and Tempone, R., An Adaptive Sparse Grid Algorithm for Elliptic PDEs with Lognormal Diffusion Coefficient, in Sparse Grids and Applications—Stuttgart 2014, J. Garcke and D. Pflger, Eds., Cham, Switzerland: Springer International Publishing, vol. 109, pp. 191–220, 2016.

  16. Powell, C.E. and Elman, H.C., Block-Diagonal Preconditioning for Spectral Stochastic Finite-Element Systems, IMA J. Numer. Anal., 29(2):350–375, 2009.

  17. Powell, C. and Ullmann, E., Preconditioning Stochastic Galerkin Saddle Point Systems, SIAM J. Matrix Anal. Appl., 31(5):2813–2840, 2010.

  18. Hampton, J., Fairbanks, H., Narayan, A., and Doostan, A., Parametric/Stochastic Model Reduction: Low-Rank Representation, Non-Intrusive Bi-Fidelity Approximation, and Convergence Analysis, arXiv:1709.03661 [math], Sep. 2017.

  19. Holden, H., Oksendal, B., and Zhang, T., Stochastic Partial Differential Equations: A Modeling, White Noise Functional Approach, Boston: Birkhauser, 1996.

  20. Theting, T., Solving Wick-Stochastic Boundary Value Problems using a Finite Element Method, Stochastics: Int. J. Prob. Stochastic Proc., 70:241–270, 2000.

  21. Wan, X., Rozovskii, B., and Karniadakis, G., A Stochastic Modeling Methodology based on Weighted Wiener Chaos and Malliavin Calculus, Proc. Natl. Acad. Sci., 106:14189–14194, 2009.

  22. Wan, X., A Note on Stochastic Elliptic Models, Comput. Methods Appl. Mech. Eng., 199(45-48):2987–2995, 2010.

  23. Wan, X., A Discussion on Two Stochastic Modeling Strategies for Elliptic Problems, Commun. Comput. Phys., 11:775–796, 2012.

  24. Mikulevicius, R. and Rozovskii, B.L., On Unbiased Stochastic Navier-Stokes Equations, Probab. Theory Relat. Fields, 154(3- 4):787–834, 2012.

  25. Nualart, D., Malliavin Calculus and Related Topics, 2nd ed., New York: Springer, 2006.

  26. Venturi, D., Wan, X., Mikulevicius, R., Rozovskii, B., and Karniadakis, G., Wick-Malliavin Approximation to Nonlinear Stochastic PDEs: Analysis and Simulations, Proc. R. Soc. A, 469:20130001, 2013.

  27. Wan, X. and Rozovskii, B.L., The Wick-Malliavin Approximation of Elliptic Problems with Log-Normal Random Coefficients, SIAM J. Sci. Comput., 35(5):A2370–A2392, 2013.

  28. Hu, Y. and Yan, J., Wick Calculus for Nonlinear Gaussian Functionals, Acta Math. Appl. Sin. Eng. Ser., 25:399–414, 2009.

  29. Cameron, R. and Martin,W., The Orthogonal Development of Nonlinear Functionals in Series of Fourier-Hermite Functionals, Ann. Math., 48:385, 1947.

  30. Lototsky, S. and Rozovskii, B., Stochastic Differential Equations Driven by Purely Spatial Noise, SIAM J Math. Anal., 41(4):1295–1322, 2009.

  31. Shen, J. and Yu, H., Efficient Spectral Sparse Grid Methods and Applications to High-Dimensional Elliptic Problems, SIAM J. Sci. Comput., 32:3228–3250, 2010.

  32. Shen, J. and Yu, H., Efficient Spectral Sparse Grid Methods and Applications to High-Dimensional Elliptic Equations II: Unbounded Domains, SIAM J. Sci. Comput., 34:1141–1164, 2012.

  33. Shen, J., Wang, L.-L., and Yu, H., Approximations by Orthonormal Mapped Chebyshev Functions for Higher-Dimensional Problems in Unbounded Domains, J. Comput. Appl. Math., 265:264–275, 2014.

  34. Riesz, F. and Nagy, B.S., Functional Analysis, New York: Dover, 1990.

  35. Landau, L.D. and Lifshitz, E.M., Electrodynamics of Continuous Media, Oxford: Pergamon Press, 1960.

  36. Matheron, G., Elements pour une Theorie des Milieux Poreux, Paris: Masson, 1967.

  37. Ciarlet, P., The Finite Element Method for Elliptic Problems, Philadelphia: SIAM, 2002.

  38. Karniadakis, G. and Sherwin, S., Spectral/hp Element Methods for CFD, 2nd ed., Oxford: Oxford University Press, 2005.

  39. Schwab, C., p- and hp- Finite Element Methods, Oxford: Oxford University Press, 1998.

  40. Saad, Y., Iterative Methods for Sparse Linear Systems, 2nd ed., Philadelphia: SIAM, 2003.


Articles with similar content:

MODELING SCALAR MIXING PROCESS IN TURBULENT FLOW
Turbulence and Shear Flow Phenomena -1 First International Symposium, Vol.0, 1999, issue
Jacek Pozorski, Jean-Pierre Minier
ADAPTIVE SELECTION OF SAMPLING POINTS FOR UNCERTAINTY QUANTIFICATION
International Journal for Uncertainty Quantification, Vol.7, 2017, issue 4
Casper Rutjes, Enrico Camporeale, Ashutosh Agnihotri
COMPUTING GREEN'S FUNCTIONS FOR FLOW IN HETEROGENEOUS COMPOSITE MEDIA
International Journal for Uncertainty Quantification, Vol.3, 2013, issue 1
David A. Barajas-Solano, Daniel M. Tartakovsky
MODELING SCALAR MIXING PROCESS IN TURBULENT FLOW
TSFP DIGITAL LIBRARY ONLINE, Vol.1, 1999, issue
Jacek Pozorski, Jean-Pierre Minier
BLOCK AND MULTILEVEL PRECONDITIONING FOR STOCHASTIC GALERKIN PROBLEMS WITH LOGNORMALLY DISTRIBUTED PARAMETERS AND TENSOR PRODUCT POLYNOMIALS
International Journal for Uncertainty Quantification, Vol.7, 2017, issue 5
Ivana Pultarová