no. 5
First order second moment analysis for stochastic interface problems based on low-rank approximation
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 5, pp. 1533-1552.

In this paper, we propose a numerical method to solve stochastic elliptic interface problems with random interfaces. Shape calculus is first employed to derive the shape-Taylor expansion in the framework of the asymptotic perturbation approach. Given the mean field and the two-point correlation function of the random interface, we can thus quantify the mean field and the variance of the random solution in terms of certain orders of the perturbation amplitude by solving a deterministic elliptic interface problem and its tensorized counterpart with respect to the reference interface. Error estimates are derived for the interface-resolved finite element approximation in both, the physical and the stochastic dimension. In particular, a fast finite difference scheme is proposed to compute the variance of random solutions by using a low-rank approximation based on the pivoted Cholesky decomposition. Numerical experiments are presented to validate and quantify the method.

DOI : 10.1051/m2an/2013079
Classification : 60H15, 60H35, 65C20, 65C30
Mots clés : elliptic interface problem, stochastic interface, low-rank approximation, pivoted Cholesky decomposition 
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     title = {First order second moment analysis for stochastic interface problems based on low-rank approximation},
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     pages = {1533--1552},
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Harbrecht, Helmut; Li, Jingzhi. First order second moment analysis for stochastic interface problems based on low-rank approximation. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 5, pp. 1533-1552. doi : 10.1051/m2an/2013079. http://www.numdam.org/articles/10.1051/m2an/2013079/

[1] I. Babuška and P. Chatzipantelidis, On solving elliptic stochastic partial differential equations. Comput. Methods Appl. Mech. Engrg. 191 (2002) 4093-4122. | MR | Zbl

[2] I. Babuška, F. Nobile and R. Tempone, Worst case scenario analysis for elliptic problems with uncertainty. Numer. Math. 101 (2005) 185-219. | MR | Zbl

[3] I. Babuška, F. Nobile and R. Tempone, A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Num. Anal. 45 (2007) 1005-1034. | MR | Zbl

[4] I. Babuška, R. Tempone and G.E. Zouraris, Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J. Numer. Anal. 42 (2004) 800-825. | MR | Zbl

[5] A. Barth, C. Schwab and N. Zollinger, Multi-Level Monte Carlo Finite Element method for elliptic PDE's with stochastic coefficients. Numer. Math. 119 (2011) 123-161. | MR | Zbl

[6] V.I. Bogachev, Gaussian Measures, Mathematical Surveys and Monographs in vol. 62. AMS, Providence, RI (1998). | MR | Zbl

[7] J.H. Bramble and J.T. King, A finite element method for interface problems with smooth boundaries and interfaces. Adv. Comput. Math. 6 (1996) 109-138. | MR | Zbl

[8] J.W. Barrett and C.M. Elliott, Fitted and unfitted finite-element methods for elliptic equations with interfaces. IMA J. Numer. Anal. 7 (1987) 283-300. | MR | Zbl

[9] C. Canuto and T. Kozubek, A fictitious domain approach to the numerical solution of pdes in stochastic domains. Numer. Math. 107 (2007) 257-293. | MR | Zbl

[10] Z. Chen and J. Zou, Finite element methods and their convergence for elliptic and parabolic interface problems. Numer. Math. 79 (1998) 175-202. | MR | Zbl

[11] A. Chernov and C. Schwab, First order k-th moment finite element analysis of nonlinear operator equations with stochastic data. Math. Comput. To appear (2012). | MR | Zbl

[12] M.K. Deb, I.M. Babuska and J.T. Oden, Solution of stochastic partial differential equations using Galerkin finite element techniques. Comput. Methods Appl. Mech. Engrg. 190 (2001) 6359-6372. | MR | Zbl

[13] B.J. Debusschere, H.N. Najm, P.P. Pébay, O.M. Knio, R.G. Ghanem and O.P.L. Maître, Numerical challenges in the use of polynomial chaos representations for stochastic processes. SIAM J. Sci. Comput. 26 (2004) 698-719. | MR | Zbl

[14] M.C. Delfour and J.-P. Zolesio, Shapes and Geometries - Analysis, Differential Calculus, and Optimization. SIAM, Society for Industrial and Appl. Math., Philadelphia (2001). | MR | Zbl

[15] F.R. Desaint and J.-P. Zolésio, Manifold derivative in the Laplace-Beltrami equation. J. Functional Anal. 151 (1997) 234-269. | MR | Zbl

[16] P. Frauenfelder, C. Schwab and R.A. Todor, Finite elements for elliptic problems with stochastic coefficients. Comput. Methods Appl. Mech. Engrg. 194 (2005) 205-228. | MR | Zbl

[17] R.G. Ghanem and P.D. Spanos, Stochastic finite elements: a spectral approach. Springer-Verlag (1991). | MR | Zbl

[18] M. Griebel and H. Harbrecht, Approximation of bivariate functions: singular value decomposition versus sparse grids. IMA J. Numer. Anal. To appear (2013). | MR | Zbl

[19] H. Harbrecht, A finite element method for elliptic problems with stochastic input data. Appl. Numer. Math. 60 (2010) 227-244. | MR | Zbl

[20] H. Harbrecht, On output functionals of boundary value problems on stochastic domains. Math. Meth. Appl. Sci. 33 (2010) 91-102. | MR | Zbl

[21] H. Harbrecht, M. Peters and R. Schneider, On the low-rank approximation by the pivoted Cholesky decomposition. Appl. Numer. Math. 62 (2012) 428-440. | MR | Zbl

[22] H. Harbrecht, R. Schneider and C. Schwab, Multilevel frames for sparse tensor product spaces. Numer. Math. 110 (2008) 199-220. | MR | Zbl

[23] H. Harbrecht, R. Schneider and C. Schwab, Sparse second moment analysis for elliptic problems in stochastic domains. Numer. Math. 109 (2008) 385-414. | MR | Zbl

[24] F. Hettlich and W. Rundell, The determination of a discontinuity in a conductivity from a single boundary measurement. Inverse Problems 14 (1998) 67-82. | MR | Zbl

[25] F. Hettlich and W. Rundell, Identification of a discontinuous source in the heat equation. Inverse Problems 17 (2001) 1465-1482. | MR | Zbl

[26] K. Ito, K. Kunisch and Z. Li, Level-set function approach to an inverse interface problem. Inverse Problems 17 (2001) 1225-1242. | MR | Zbl

[27] J.B. Keller, Stochastic equations and wave propagation in random media. Proc. Symp. Appl. Math. in vol. 16. AMS, Providence, R.I. (1964) 145-170. | MR | Zbl

[28] M. Kleiber and T.D. Hien, The stochastic finite element method: basic perturbation technique and computer implementation. Wiley, Chichester (1992). | MR | Zbl

[29] P.E. Kloeden and E. Platen, Numerical solution of stochastic differential equations. Springer, Berlin 3rd ed. (1999). | MR | Zbl

[30] M. Ledoux and M. Talagrand, Probability in Banach Spaces. Isoperimetry and Processes. Springer, Berlin (1991). | MR | Zbl

[31] J. Li, J.M. Melenk, B. Wohlmuth and J. Zou, Optimal a priori estimates for higher order finite elements for elliptic interface problems. Appl. Numer. Math. 60 (2010) 19-37. | MR | Zbl

[32] Z. Li and K. Ito, The immersed interface method: numerical solutions of PDEs involving interfaces and irregular domains. SIAM, Society for Industrial and Appl. Math., Philadelphia (2006). | MR | Zbl

[33] H.G. Matthies and A. Keese, Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations. Comput. Methods Appl. Mech. Engrg. 194 (2005) 1295-1331. | MR | Zbl

[34] O. Pironneau, Optimal Shape Design for Elliptic Systems. Springer, New York (1984). | MR | Zbl

[35] P. Protter, Stochastic Integration and Differential Equations: A New Approach. Springer, Berlin, 3rd ed. (1995). | MR | Zbl

[36] C. Schwab and R.A. Todor, Sparse finite elements for elliptic problems with stochastic loading. Numer. Math. 95 (2003) 707-734. | MR | Zbl

[37] C. Schwab and R.A. Todor, Sparse finite elements for stochastic elliptic problems - higher order moments. Comput. 71 (2003) 43-63. | MR | Zbl

[38] C. Schwab and R.A. Todor, Karhunen-Loéve approximation of random fields by generalized fast multipole methods. J. Comput. Phys. 217 (2006) 100-122. | MR | Zbl

[39] J. Sokolowski and J.-P. Zolesio, Introduction to Shape Optimization: Shape Sensitivity Analysis. Springer-Verlag (1992). | MR | Zbl

[40] T. Von Petersdorff and C. Schwab, Sparse finite element methods for operator equations with stochastic data. Appl. Math. 51 (2006) 145-180. | EuDML | MR | Zbl

[41] X. Wan, B. Rozovskii and G. E. Karniadakis, A stochastic modeling method based on weighted Wiener chaos and Malliavan calculus. PNAS 106 (2009) 14189-14194. | MR | Zbl

[42] J. Wloka, Partial Differential Equations. Cambridge University Press, Cambridge (1987). | MR | Zbl

[43] D. Xiu and G.E. Karniadakis, Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos. Comput. Methods Appl. Mech. Engrg. 191 (2002) 4927-4948. | MR | Zbl

[44] D. Xiu and D.M. Tartakovsky, Numerical methods for differential equations in random domains. SIAM J. Scientific Comput. 28 (2006) 1167-1185. | MR | Zbl

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