Computing quantities of interest for random domains with second order shape sensitivity analysis
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 5, pp. 1285-1302.

We consider random perturbations of a given domain. The characteristic amplitude of these perturbations is assumed to be small. We are interested in quantities of interest which depend on the random domain through a boundary value problem. We derive asymptotic expansions of the first moments of the distribution of this output function. A simple and efficient method is proposed to compute the coefficients of these expansions provided that the random perturbation admits a low-rank spectral representation. By numerical experiments, we compare our expansions with Monte–Carlo simulations.

Reçu le :
DOI : 10.1051/m2an/2015012
Classification : 60G35, 65N75, 65N99
Mots clés : Random domain, second order shape sensitivity, low-rank approximation
Dambrine, Marc 1 ; Harbrecht, Helmut 2 ; Puig, Bénédicte 1

1 Universitéde Pau et des Pays de l’Adour, 64000 Pau, France.
2 Universität Basel. 
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Dambrine, Marc; Harbrecht, Helmut; Puig, Bénédicte. Computing quantities of interest for random domains with second order shape sensitivity analysis. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 5, pp. 1285-1302. doi : 10.1051/m2an/2015012. http://www.numdam.org/articles/10.1051/m2an/2015012/

L. Afraites, M. Dambrine and D. Kateb, On second order shape optimization methods for electrical impedance tomography. SIAM J. Control Optim. 47 (2008) 1556–1590. | DOI | Zbl

G. Allaire, F. Jouve and A.-M. Toader, Structural optimization using sensitivity analysis and a level-set method. J. Comput. Phys. 194 (2004) 363–393. | DOI | Zbl

F. Caubet, Instability of an inverse problem for the stationary Navier-Stokes equations. SIAM J. Control Optim. 51 (2013) 2949–2975. | DOI | Zbl

F. Caubet and M. Dambrine, Stability of critical shapes for the drag minimization problem in Stokes flow. J. Math. Pures Appl. 100 (2013) 327–346. | DOI | Zbl

A. Chernov and C. Schwab, First order k–th moment finite element analysis of nonlinear operator equations with stochastic data. Math. Comput. 82 (2013) 1859–1888. | DOI | Zbl

M. Dambrine, On variations of the shape Hessian and sufficient conditions for the stability of critical shapes. RACSAM, Rev. R. Acad. Cien. Serie A. Mat. 96 (2002) 95–121. | Zbl

M. Dambrine and D. Kateb, On the shape sensitivity of the first Dirichlet eigenvalue for two-phase problems. Appl. Math. Optim. 63 (2011) 45–74. | DOI | Zbl

M. Dambrine and J. Lamboley, Stability in shape optimization with second variation. Preprint HAL-01073089 (2014).

M.C. Delfour and J.-P. Zolésio, Shapes and geometries. Metrics, analysis, differential calculus, and optimization. Vol. 22 of Advances in Design and Control, 2nd edition. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2011). | Zbl

K. Eppler, Boundary integral representations of second derivatives in shape optimization. Discuss. Math. Differ. Incl. Control Optim. 20 (2000) 63–78. | DOI | Zbl

K. Eppler and H. Harbrecht, A regularized Newton method in electrical impedance tomography using shape Hessian information. Control Cybernet. 34 (2005) 203–225. | Zbl

K. Eppler and H. Harbrecht, Second-order shape optimization using wavelet BEM. Optim. Methods Softw. 21 (2006) 135–153. | DOI | Zbl

K. Eppler, S. Schmidt, V. Schulz and C. Ilic, Preconditioning the pressure tracking in fluid dynamics by shape Hessian information. J. Optim. Theory Appl. 141 (2009) 513–531. | DOI | Zbl

H. Harbrecht, A Newton method for Bernoulli’s free boundary problem in three dimensions. Computing 82 (2008) 11–30. | DOI | Zbl

H. Harbrecht, On output functionals of boundary value problems on stochastic domains. Math. Methods Appl. Sci. 33 (2010) 91–102. | Zbl

A. Henrot and M. Pierre, Variation et optimisation de formes. Une analyse géométrique. Vol. 48 of Math. Appl. Springer, Berlin (2005). | Zbl

H. Harbrecht, Second moment analysis for Robin boundary value problems on random domains, in Singular Phenomena and Scaling in Mathematical Models. Edited by M. Griebel. Springer, Berlin (2013) 361–382.

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

H. Harbrecht, M. Peters and M. Siebenmorgen, Combination technique based kth moment analysis of elliptic problems with random diffusion. J. Comput. Phys. 252 (2013) 128–141. | DOI | Zbl

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

M. Hintermüller and W. Ring, An inexact Newton-CG-type active contour approach for the minimization of the Mumford-Shah functional. J. Math. Imaging Vision 20 (2004) 19–42. | DOI | Zbl

A. Novruzi and M. Pierre, Structure of shape derivatives. J. Evol. Equ. 2 (2002) 365–382. | DOI | Zbl

A. Novruzi and J.R. Roche, Newton’s method in shape optimisation: a three-dimensional case. BIT (2000) 102–120. | Zbl

O. Pironneau, Optimal Shape Design for Elliptic Systems. Springer, New York (1983). | Zbl

S. Schmidt and V. Schulz, Impulse response approximations of discrete shape Hessians with application in CFD. SIAM J. Control Optim. 48 (2009) 2562–2580. | DOI | Zbl

V. Schulz, A Riemannian view on shape optimization. Found. Comput. Math. 14 (2014) 483–501. | DOI | Zbl

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

J. Simon, Second variations for domain optimization problems. In Control and estimation of distributed parameter systems. Edited by F. Kappel et al., Vol. 91 of Int. Ser. Numer. Math. Birkhäuser, Basel (1989) 361–378. | Zbl

J. Sokolowski and J.-P. Zolesio, Introduction to Shape Optimization. Springer, Berlin (1992). | Zbl

J. Wloka, Partial Differential Equations. Cambridge University Press, Cambridge (1987). | Zbl

S. Yang, G. Stadler, R. Moser and Omar Ghattas, A shape Hessian-based boundary roughness analysis of Navier-Stokes flow. SIAM J. Appl. Math. 71 333–355. | DOI | Zbl

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