Abstract
We consider the numerical solution of elliptic boundary value problems in domains with random boundary perturbations. Assuming normal perturbations with small amplitude and known mean field and two-point correlation function, we derive, using a second order shape calculus, deterministic equations for the mean field and the two-point correlation function of the random solution for a model Dirichlet problem which are 3rd order accurate in the boundary perturbation size. Using a variational boundary integral equation formulation on the unperturbed, “nominal” boundary and a wavelet discretization, we present and analyze an algorithm to approximate the random solution’s mean and its two-point correlation function at essentially optimal order in essentially \({\mathcal{O}(N)}\) work and memory, where N denotes the number of unknowns required for consistent discretization of the boundary of the nominal domain.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Babuška, I., Nobile, F., Tempone, R.: Worst case scenario analysis for elliptic problems with uncertainty. Numer. Math. 101, 185–219 (2005)
Bungartz, H.J., Griebel, M.: Sparse Grids. Acta Numer. 13, 147–269 (2004)
Bogachev, V.I.: Gaussian Measures. AMS Mathematical Surveys and Monographs, vol. 62 (1998)
Dahmen, W.: Wavelet and multiscale methods for operator equations. Acta Numer. 6, 55–228 (1997)
Dahmen, W., Harbrecht, H., Schneider, R.: Compression techniques for boundary integral equations—optimal complexity estimates. SIAM J. Numer. Anal. 43, 2251–2271 (2006)
Delfour M., Zolesio J.-P. (2001). Shapes Geometries. SIAM, Philadelphia
Eppler, K.: Optimal shape design for elliptic equations via BIE-methods. J. Appl. Math. Comput. Sci. 10, 487–516 (2000)
Eppler, K.: Boundary integral representations of second derivatives in shape optimization. Discussiones Math. (Differential Inclusions, Control and Optimization) 20, 63–78 (2000)
Harbrecht, H., Schneider, R.: Wavelet Galerkin Schemes for Boundary Integral Equations—Implementation and Quadrature. SIAM J. Sci. Comput. 27, 1347–1370 (2006)
Harbrecht, H., Schneider, R.: Biorthogonal wavelet bases for the boundary element method. Math. Nachr. 269–270, 167–188 (2005)
Hettlich, F., Rundell, W.: A second degree method for nonlinear inverse problems. SIAM J. Numer. Anal. 37, 587–620 (2000)
Ledoux, M., Talagrand, M.: Probability in Banach Spaces. Isoperimetry and Processes. Springer, Berlin (1991)
Maz’ya, V.G., Shaposhnikova, T.O.: Theory of multipliers in spaces of differentiable functions. Pitman, Boston, 1985. Monographs and Studies in Mathematics, 23. Pitman Advanced Publishing Program. Boston-London-Melbourne: Pitman Publishing Inc. XIII, 344 p.
Mc Lean, W.: Strongly Elliptic Boundary Integral Equations. Cambridge University Press, Cambridge (2000)
Murat, F., Simon, J.: Étude de problèmes d’optimal design. In: CÉa, J. (eds) Optimization Techniques, Modeling and Optimization in the Service of Man, Lecture Notes Computer Science vol. 41, pp. 54–62. Springer, Berlin (1976)
NÉdÉlec, J.C., Planchard, J.P.: Une méthode variationelle d’élements finis pour la résolution numérique d’un problème extérieur dans \({\mathbb{R}^3}\) . RAIRO Anal. Numér. 7, 105–129 (1973)
von Petersdorff, T., Schwab, C.: Numerical solution of parabolic equations in high dimensions. M2AN Math. Model. Numer. Anal. 38, 93–127 (2004)
von Petersdorff, T., Schwab, C.: Sparse wavelet methods for operator equations with stochastic data. Appl. Math. 51(2), 145–180 (2006)
Pironneau, O.: Optimal Shape Design for Elliptic Systems. Springer, New York (1983)
Potthast, R.: Fréchet-Differenzierbarkeit von Randintegraloperatoren und Randwertproblemen zur Helmholtzgleichung und den zeitharmonischen Maxwellgleichungen. PhD Thesis, Universität Göttingen (1994)
Schmidlin, G., Lage, Ch., Schwab, Ch.: Rapid solution of first kind boundary integral equations in R 3. Eng. Anal. Boundary Elements 27(5), 469–490 (2003)
Schneider, R.: Multiskalen- und Wavelet-Matrixkompression: Analysisbasierte Methoden zur Lösung großer vollbesetzter Gleichungssysteme. B.G. Teubner, Stuttgart (1998)
Sauter, S.A., Schwab, C.: Randelementmethoden. B.G. Teubner, Stuttgart (2004)
Schwab, C., Todor, R.: Sparse finite elements for elliptic problems with stochastic loading. Numer. Math. 95, 707–734 (2003)
Schwab, C., Wendland, W.L.: On the extraction technique in boundary integral equations. Math. Comput. 68, 91–122 (1999)
Simon, J.: Differentiation with respect to the domain in boundary value problems. Numer. Funct. Anal. Optim. 2, 649–687 (1980)
Sokolowski, J., Zolesio, J.-P.: Introduction to Shape Optimization. Springer, Berlin (1992)
Tausch, J., White, J.: Multiscale bases for the sparse representation of boundary integral operators on complex geometry. SIAM J. Sci. Stat. Comput. 24, 1610–1629 (2003)
Triebel, H.: Theory of function spaces 2nd edn. Joh. A. Barth Publ., Leipzig (1995)
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by the EEC Human Potential Programme under contract HPRN-CT-2002-00286, “Breaking Complexity.” Work initiated while HH visited the Seminar for Applied Mathematics at ETH Zürich in the Wintersemester 2005/06 and completed during the summer programme CEMRACS2006 “Modélisation de l’aléatoire et propagation d’incertitudes” in July and August 2006 at the C.I.R.M., Marseille, France.
Rights and permissions
About this article
Cite this article
Harbrecht, H., Schneider, R. & Schwab, C. Sparse second moment analysis for elliptic problems in stochastic domains. Numer. Math. 109, 385–414 (2008). https://doi.org/10.1007/s00211-008-0147-9
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00211-008-0147-9