First order second moment analysis for stochastic interface problems based on low-rank approximation

Helmut Harbrecht; Jingzhi Li

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2013)

  • Volume: 47, Issue: 5, page 1533-1552
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Harbrecht, Helmut, and 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 47.5 (2013): 1533-1552. <http://eudml.org/doc/273178>.

@article{Harbrecht2013,
abstract = {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.},
author = {Harbrecht, Helmut, Li, Jingzhi},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {elliptic interface problem; stochastic interface; low-rank approximation; pivoted Cholesky decomposition; perturbation approach; stochastic elliptic interface problem; error estimates; Monte Carlo method},
language = {eng},
number = {5},
pages = {1533-1552},
publisher = {EDP-Sciences},
title = {First order second moment analysis for stochastic interface problems based on low-rank approximation},
url = {http://eudml.org/doc/273178},
volume = {47},
year = {2013},
}

TY - JOUR
AU - Harbrecht, Helmut
AU - Li, Jingzhi
TI - First order second moment analysis for stochastic interface problems based on low-rank approximation
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2013
PB - EDP-Sciences
VL - 47
IS - 5
SP - 1533
EP - 1552
AB - 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.
LA - eng
KW - elliptic interface problem; stochastic interface; low-rank approximation; pivoted Cholesky decomposition; perturbation approach; stochastic elliptic interface problem; error estimates; Monte Carlo method
UR - http://eudml.org/doc/273178
ER -

References

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  1. [1] I. Babuška and P. Chatzipantelidis, On solving elliptic stochastic partial differential equations. Comput. Methods Appl. Mech. Engrg.191 (2002) 4093–4122. Zbl1019.65010MR1919790
  2. [2] I. Babuška, F. Nobile and R. Tempone, Worst case scenario analysis for elliptic problems with uncertainty. Numer. Math.101 (2005) 185–219. Zbl1082.65115MR2195342
  3. [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. Zbl1151.65008MR2318799
  4. [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. Zbl1080.65003MR2084236
  5. [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. Zbl1230.65006MR2824857
  6. [6] V.I. Bogachev, Gaussian Measures, Mathematical Surveys and Monographs in vol. 62. AMS, Providence, RI (1998). Zbl0913.60035MR1642391
  7. [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. Zbl0868.65081MR1431789
  8. [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. Zbl0629.65118MR968524
  9. [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. Zbl1126.65004MR2328847
  10. [10] Z. Chen and J. Zou, Finite element methods and their convergence for elliptic and parabolic interface problems. Numer. Math.79 (1998) 175–202. Zbl0909.65085MR1622502
  11. [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). Zbl1281.65008MR3073184
  12. [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. Zbl1075.65006MR1870425
  13. [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. Zbl1072.60042MR2116369
  14. [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). Zbl1251.49001MR1855817
  15. [15] F.R. Desaint and J.-P. Zolésio, Manifold derivative in the Laplace-Beltrami equation. J. Functional Anal.151 (1997) 234–269. Zbl0903.58059MR1487777
  16. [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. Zbl1143.65392MR2105161
  17. [17] R.G. Ghanem and P.D. Spanos, Stochastic finite elements: a spectral approach. Springer-Verlag (1991). Zbl0722.73080MR1083354
  18. [18] M. Griebel and H. Harbrecht, Approximation of bivariate functions: singular value decomposition versus sparse grids. IMA J. Numer. Anal. To appear (2013). Zbl1287.65009MR3168277
  19. [19] H. Harbrecht, A finite element method for elliptic problems with stochastic input data. Appl. Numer. Math.60 (2010) 227–244. Zbl1237.65009MR2602675
  20. [20] H. Harbrecht, On output functionals of boundary value problems on stochastic domains. Math. Meth. Appl. Sci.33 (2010) 91–102. Zbl1181.35354MR2591227
  21. [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. Zbl1244.65042MR2899254
  22. [22] H. Harbrecht, R. Schneider and C. Schwab, Multilevel frames for sparse tensor product spaces. Numer. Math.110 (2008) 199–220. Zbl1151.65010MR2425155
  23. [23] H. Harbrecht, R. Schneider and C. Schwab, Sparse second moment analysis for elliptic problems in stochastic domains. Numer. Math.109 (2008) 385–414. Zbl1146.65007MR2399150
  24. [24] F. Hettlich and W. Rundell, The determination of a discontinuity in a conductivity from a single boundary measurement. Inverse Problems14 (1998) 67–82. Zbl0894.35126MR1607628
  25. [25] F. Hettlich and W. Rundell, Identification of a discontinuous source in the heat equation. Inverse Problems17 (2001) 1465–1482. Zbl0986.35129MR1862202
  26. [26] K. Ito, K. Kunisch and Z. Li, Level-set function approach to an inverse interface problem. Inverse Problems17 (2001) 1225–1242. Zbl0986.35130MR1862188
  27. [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. Zbl0202.46703MR178638
  28. [28] M. Kleiber and T.D. Hien, The stochastic finite element method: basic perturbation technique and computer implementation. Wiley, Chichester (1992). Zbl0902.73004MR1198887
  29. [29] P.E. Kloeden and E. Platen, Numerical solution of stochastic differential equations. Springer, Berlin 3rd ed. (1999). Zbl0752.60043MR1214374
  30. [30] M. Ledoux and M. Talagrand, Probability in Banach Spaces. Isoperimetry and Processes. Springer, Berlin (1991). Zbl1226.60003MR1102015
  31. [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. Zbl1208.65168MR2566075
  32. [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). Zbl1122.65096MR2242805
  33. [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. Zbl1088.65002MR2121216
  34. [34] O. Pironneau, Optimal Shape Design for Elliptic Systems. Springer, New York (1984). Zbl0496.93029MR725856
  35. [35] P. Protter, Stochastic Integration and Differential Equations: A New Approach. Springer, Berlin, 3rd ed. (1995). Zbl0694.60047MR1037262
  36. [36] C. Schwab and R.A. Todor, Sparse finite elements for elliptic problems with stochastic loading. Numer. Math.95 (2003) 707–734. Zbl1044.65085MR2013125
  37. [37] C. Schwab and R.A. Todor, Sparse finite elements for stochastic elliptic problems – higher order moments. Comput.71 (2003) 43–63. Zbl1044.65006MR2009650
  38. [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. Zbl1104.65008MR2250527
  39. [39] J. Sokolowski and J.-P. Zolesio, Introduction to Shape Optimization: Shape Sensitivity Analysis. Springer-Verlag (1992). Zbl0761.73003MR1215733
  40. [40] T. von Petersdorff and C. Schwab, Sparse finite element methods for operator equations with stochastic data. Appl. Math.51 (2006) 145–180. Zbl1164.65300MR2212311
  41. [41] X. Wan, B. Rozovskii and G. E. Karniadakis, A stochastic modeling method based on weighted Wiener chaos and Malliavan calculus. PNAS106 (2009) 14189–14194. Zbl1203.60066MR2539729
  42. [42] J. Wloka, Partial Differential Equations. Cambridge University Press, Cambridge (1987). Zbl0623.35006MR895589
  43. [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. Zbl1016.65001MR1932024
  44. [44] D. Xiu and D.M. Tartakovsky, Numerical methods for differential equations in random domains. SIAM J. Scientific Comput.28 (2006) 1167–1185. Zbl1114.60056MR2240809

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