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First order second moment analysis for stochastic interfaceproblems based on low-rank approximation∗

Published online by Cambridge University Press:  14 August 2013

Helmut Harbrecht  and
Jingzhi Li
Helmut Harbrecht
Affiliation:
Helmut Harbrecht, Mathematisches Institut, Universität Basel, Rheinsprung 21, 4051 Basel, Switzerland.. helmut.harbrecht@unibas.ch
Jingzhi Li
Affiliation:
Faculty of Science, South University of Science and Technology of China, Shenzhen 518055, P. R. China.; li.jz@sustc.edu.cn
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Abstract

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

Keywords

Elliptic interface problemstochastic interfacelow-rank approximationpivoted Cholesky decomposition
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 5 , September 2013 , pp. 1533 - 1552
Copyright
© EDP Sciences, SMAI, 2013

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References

Babuška, I. and Chatzipantelidis, P., On solving elliptic stochastic partial differential equations. Comput. Methods Appl. Mech. Engrg. 191 (2002) 4093–4122. Google Scholar
Babuška, I., Nobile, F. and Tempone, R., Worst case scenario analysis for elliptic problems with uncertainty. Numer. Math. 101 (2005) 185–219. Google Scholar
Babuška, I., Nobile, F. and Tempone, R., A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Num. Anal. 45 (2007) 1005–1034. Google Scholar
Babuška, I., Tempone, R. and Zouraris, G.E., Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J. Numer. Anal. 42 (2004) 800–825. Google Scholar
Barth, A., Schwab, C. and Zollinger, N., Multi-Level Monte Carlo Finite Element method for elliptic PDE’s with stochastic coefficients. Numer. Math. 119 (2011) 123–161. Google Scholar
V.I. Bogachev, Gaussian Measures, Mathematical Surveys and Monographs in vol. 62. AMS, Providence, RI (1998).
Bramble, J.H. and King, J.T., A finite element method for interface problems with smooth boundaries and interfaces. Adv. Comput. Math. 6 (1996) 109–138. Google Scholar
Barrett, J.W. and Elliott, C.M., Fitted and unfitted finite-element methods for elliptic equations with interfaces. IMA J. Numer. Anal. 7 (1987) 283–300. Google Scholar
Canuto, C. and Kozubek, T., A fictitious domain approach to the numerical solution of pdes in stochastic domains. Numer. Math. 107 (2007) 257–293. Google Scholar
Chen, Z. and Zou, J., Finite element methods and their convergence for elliptic and parabolic interface problems. Numer. Math. 79 (1998) 175–202. Google Scholar
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).
Deb, M.K., Babuska, I.M. and Oden, J.T., Solution of stochastic partial differential equations using Galerkin finite element techniques. Comput. Methods Appl. Mech. Engrg. 190 (2001) 6359–6372. Google Scholar
Debusschere, B.J., Najm, H.N., Pébay, P.P., Knio, O.M., Ghanem, R.G. and Maître, O.P.L., Numerical challenges in the use of polynomial chaos representations for stochastic processes. SIAM J. Sci. Comput. 26 (2004) 698–719. Google Scholar
M.C. Delfour and J.-P. Zolesio, Shapes and Geometries — Analysis, Differential Calculus, and Optimization. SIAM, Society for Industrial and Appl. Math., Philadelphia (2001).
Desaint, F.R. and Zolésio, J.-P., Manifold derivative in the Laplace-Beltrami equation. J. Functional Anal. 151 (1997) 234–269. Google Scholar
Frauenfelder, P., Schwab, C. and Todor, R.A., Finite elements for elliptic problems with stochastic coefficients. Comput. Methods Appl. Mech. Engrg. 194 (2005) 205–228. Google Scholar
R.G. Ghanem and P.D. Spanos, Stochastic finite elements: a spectral approach. Springer-Verlag (1991).
M. Griebel and H. Harbrecht, Approximation of bivariate functions: singular value decomposition versus sparse grids. IMA J. Numer. Anal. To appear (2013).
Harbrecht, H., A finite element method for elliptic problems with stochastic input data. Appl. Numer. Math. 60 (2010) 227–244. Google Scholar
Harbrecht, H., On output functionals of boundary value problems on stochastic domains. Math. Meth. Appl. Sci. 33 (2010) 91–102. Google Scholar
Harbrecht, H., Peters, M. and Schneider, R., On the low-rank approximation by the pivoted Cholesky decomposition. Appl. Numer. Math. 62 (2012) 428–440. Google Scholar
Harbrecht, H., Schneider, R. and Schwab, C., Multilevel frames for sparse tensor product spaces. Numer. Math. 110 (2008) 199–220. Google Scholar
Harbrecht, H., Schneider, R. and Schwab, C., Sparse second moment analysis for elliptic problems in stochastic domains. Numer. Math. 109 (2008) 385–414. Google Scholar
Hettlich, F. and Rundell, W., The determination of a discontinuity in a conductivity from a single boundary measurement. Inverse Problems 14 (1998) 67–82. Google Scholar
Hettlich, F. and Rundell, W., Identification of a discontinuous source in the heat equation. Inverse Problems 17 (2001) 1465–1482. Google Scholar
Ito, K., Kunisch, K. and Li, Z., Level-set function approach to an inverse interface problem. Inverse Problems 17 (2001) 1225–1242. Google Scholar
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.
M. Kleiber and T.D. Hien, The stochastic finite element method: basic perturbation technique and computer implementation. Wiley, Chichester (1992).
P.E. Kloeden and E. Platen, Numerical solution of stochastic differential equations. Springer, Berlin 3rd ed. (1999).
M. Ledoux and M. Talagrand, Probability in Banach Spaces. Isoperimetry and Processes. Springer, Berlin (1991).
Li, J., Melenk, J.M., Wohlmuth, B. and Zou, J., Optimal a priori estimates for higher order finite elements for elliptic interface problems. Appl. Numer. Math. 60 (2010) 19–37. Google Scholar
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).
Matthies, H.G. and Keese, A., Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations. Comput. Methods Appl. Mech. Engrg. 194 (2005) 1295–1331. Google Scholar
O. Pironneau, Optimal Shape Design for Elliptic Systems. Springer, New York (1984).
P. Protter, Stochastic Integration and Differential Equations: A New Approach. Springer, Berlin, 3rd ed. (1995).
Schwab, C. and Todor, R.A., Sparse finite elements for elliptic problems with stochastic loading. Numer. Math. 95 (2003) 707–734. Google Scholar
Schwab, C. and Todor, R.A., Sparse finite elements for stochastic elliptic problems – higher order moments. Comput. 71 (2003) 43–63. Google Scholar
Schwab, C. and Todor, R.A., Karhunen-Loéve approximation of random fields by generalized fast multipole methods. J. Comput. Phys. 217 (2006) 100–122. Google Scholar
J. Sokolowski and J.-P. Zolesio, Introduction to Shape Optimization: Shape Sensitivity Analysis. Springer-Verlag (1992).
von Petersdorff, T. and Schwab, C., Sparse finite element methods for operator equations with stochastic data. Appl. Math. 51 (2006) 145–180. Google Scholar
Wan, X., Rozovskii, B. and Karniadakis, G. E., A stochastic modeling method based on weighted Wiener chaos and Malliavan calculus. PNAS 106 (2009) 14189–14194. Google ScholarPubMed
J. Wloka, Partial Differential Equations. Cambridge University Press, Cambridge (1987).
Xiu, D. and Karniadakis, G.E., Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos. Comput. Methods Appl. Mech. Engrg. 191 (2002) 4927–4948. Google Scholar
Xiu, D. and Tartakovsky, D.M., Numerical methods for differential equations in random domains. SIAM J. Scientific Comput. 28 (2006) 1167–1185.Google Scholar