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.
@article{M2AN_2013__47_5_1533_0,
author = {Harbrecht, Helmut and Li, Jingzhi},
title = {First order second moment analysis for stochastic interface problems based on low-rank approximation},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {47},
year = {2013},
pages = {1533-1552},
doi = {10.1051/m2an/2013079},
mrnumber = {3100774},
zbl = {1297.65009},
language = {en},
url = {http://dml.mathdoc.fr/item/M2AN_2013__47_5_1533_0}
}
Harbrecht, Helmut; Li, Jingzhi. First order second moment analysis for stochastic interface problems based on low-rank approximation. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 47 (2013) pp. 1533-1552. doi : 10.1051/m2an/2013079. http://gdmltest.u-ga.fr/item/M2AN_2013__47_5_1533_0/
[1] and , On solving elliptic stochastic partial differential equations. Comput. Methods Appl. Mech. Engrg. 191 (2002) 4093-4122. | MR 1919790 | Zbl 1019.65010
[2] , and , Worst case scenario analysis for elliptic problems with uncertainty. Numer. Math. 101 (2005) 185-219. | MR 2195342 | Zbl 1082.65115
[3] , and , A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Num. Anal. 45 (2007) 1005-1034. | MR 2318799 | Zbl 1151.65008
[4] , and , Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J. Numer. Anal. 42 (2004) 800-825. | MR 2084236 | Zbl 1080.65003
[5] , and , Multi-Level Monte Carlo Finite Element method for elliptic PDE's with stochastic coefficients. Numer. Math. 119 (2011) 123-161. | MR 2824857 | Zbl 1230.65006
[6] , Gaussian Measures, Mathematical Surveys and Monographs in vol. 62. AMS, Providence, RI (1998). | MR 1642391 | Zbl 0913.60035
[7] and , A finite element method for interface problems with smooth boundaries and interfaces. Adv. Comput. Math. 6 (1996) 109-138. | MR 1431789 | Zbl 0868.65081
[8] and , Fitted and unfitted finite-element methods for elliptic equations with interfaces. IMA J. Numer. Anal. 7 (1987) 283-300. | MR 968524 | Zbl 0629.65118
[9] and , A fictitious domain approach to the numerical solution of pdes in stochastic domains. Numer. Math. 107 (2007) 257-293. | MR 2328847 | Zbl 1126.65004
[10] and , Finite element methods and their convergence for elliptic and parabolic interface problems. Numer. Math. 79 (1998) 175-202. | MR 1622502 | Zbl 0909.65085
[11] and , First order k-th moment finite element analysis of nonlinear operator equations with stochastic data. Math. Comput. To appear (2012). | MR 3073184 | Zbl 1281.65008
[12] , and , Solution of stochastic partial differential equations using Galerkin finite element techniques. Comput. Methods Appl. Mech. Engrg. 190 (2001) 6359-6372. | MR 1870425 | Zbl 1075.65006
[13] , , , , and , Numerical challenges in the use of polynomial chaos representations for stochastic processes. SIAM J. Sci. Comput. 26 (2004) 698-719. | MR 2116369 | Zbl 1072.60042
[14] and , Shapes and Geometries - Analysis, Differential Calculus, and Optimization. SIAM, Society for Industrial and Appl. Math., Philadelphia (2001). | MR 1855817 | Zbl 1251.49001
[15] and , Manifold derivative in the Laplace-Beltrami equation. J. Functional Anal. 151 (1997) 234-269. | MR 1487777 | Zbl 0903.58059
[16] , and , Finite elements for elliptic problems with stochastic coefficients. Comput. Methods Appl. Mech. Engrg. 194 (2005) 205-228. | MR 2105161 | Zbl 1143.65392
[17] and , Stochastic finite elements: a spectral approach. Springer-Verlag (1991). | MR 1083354 | Zbl 0722.73080
[18] and , Approximation of bivariate functions: singular value decomposition versus sparse grids. IMA J. Numer. Anal. To appear (2013). | MR 3168277 | Zbl 1287.65009
[19] , A finite element method for elliptic problems with stochastic input data. Appl. Numer. Math. 60 (2010) 227-244. | MR 2602675 | Zbl 1237.65009
[20] , On output functionals of boundary value problems on stochastic domains. Math. Meth. Appl. Sci. 33 (2010) 91-102. | MR 2591227 | Zbl 1181.35354
[21] , and , On the low-rank approximation by the pivoted Cholesky decomposition. Appl. Numer. Math. 62 (2012) 428-440. | MR 2899254 | Zbl 1244.65042
[22] , and , Multilevel frames for sparse tensor product spaces. Numer. Math. 110 (2008) 199-220. | MR 2425155 | Zbl 1151.65010
[23] , and , Sparse second moment analysis for elliptic problems in stochastic domains. Numer. Math. 109 (2008) 385-414. | MR 2399150 | Zbl 1146.65007
[24] and , The determination of a discontinuity in a conductivity from a single boundary measurement. Inverse Problems 14 (1998) 67-82. | MR 1607628 | Zbl 0894.35126
[25] and , Identification of a discontinuous source in the heat equation. Inverse Problems 17 (2001) 1465-1482. | MR 1862202 | Zbl 0986.35129
[26] , and , Level-set function approach to an inverse interface problem. Inverse Problems 17 (2001) 1225-1242. | MR 1862188 | Zbl 0986.35130
[27] , Stochastic equations and wave propagation in random media. Proc. Symp. Appl. Math. in vol. 16. AMS, Providence, R.I. (1964) 145-170. | MR 178638 | Zbl 0202.46703
[28] and , The stochastic finite element method: basic perturbation technique and computer implementation. Wiley, Chichester (1992). | MR 1198887 | Zbl 0902.73004
[29] and , Numerical solution of stochastic differential equations. Springer, Berlin 3rd ed. (1999). | MR 1214374 | Zbl 0752.60043
[30] and , Probability in Banach Spaces. Isoperimetry and Processes. Springer, Berlin (1991). | MR 1102015 | Zbl 1226.60003
[31] , , and , Optimal a priori estimates for higher order finite elements for elliptic interface problems. Appl. Numer. Math. 60 (2010) 19-37. | MR 2566075 | Zbl 1208.65168
[32] and , The immersed interface method: numerical solutions of PDEs involving interfaces and irregular domains. SIAM, Society for Industrial and Appl. Math., Philadelphia (2006). | MR 2242805 | Zbl 1122.65096
[33] and , Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations. Comput. Methods Appl. Mech. Engrg. 194 (2005) 1295-1331. | MR 2121216 | Zbl 1088.65002
[34] , Optimal Shape Design for Elliptic Systems. Springer, New York (1984). | MR 725856 | Zbl 0496.93029
[35] , Stochastic Integration and Differential Equations: A New Approach. Springer, Berlin, 3rd ed. (1995). | MR 1037262 | Zbl 0694.60047
[36] and , Sparse finite elements for elliptic problems with stochastic loading. Numer. Math. 95 (2003) 707-734. | MR 2013125 | Zbl 1044.65085
[37] and , Sparse finite elements for stochastic elliptic problems - higher order moments. Comput. 71 (2003) 43-63. | MR 2009650 | Zbl 1044.65006
[38] and , Karhunen-Loéve approximation of random fields by generalized fast multipole methods. J. Comput. Phys. 217 (2006) 100-122. | MR 2250527 | Zbl 1104.65008
[39] and , Introduction to Shape Optimization: Shape Sensitivity Analysis. Springer-Verlag (1992). | MR 1215733 | Zbl 0761.73003
[40] and , Sparse finite element methods for operator equations with stochastic data. Appl. Math. 51 (2006) 145-180. | MR 2212311 | Zbl 1164.65300
[41] , and , A stochastic modeling method based on weighted Wiener chaos and Malliavan calculus. PNAS 106 (2009) 14189-14194. | MR 2539729 | Zbl 1203.60066
[42] , Partial Differential Equations. Cambridge University Press, Cambridge (1987). | MR 895589 | Zbl 0623.35006
[43] and , Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos. Comput. Methods Appl. Mech. Engrg. 191 (2002) 4927-4948. | MR 1932024 | Zbl 1016.65001
[44] and , Numerical methods for differential equations in random domains. SIAM J. Scientific Comput. 28 (2006) 1167-1185. | MR 2240809 | Zbl 1114.60056