The present article is concerned with the numerical solution of a free boundary problem for an elliptic state equation with random diffusion. The domain under consideration is represented by a level set function which is evolved by the objective’s shape gradient. The state is computed by the finite element method, where the underlying triangulation is constructed by means of a marching cubes algorithm. The high-dimensional integral, which is induced by the random diffusion, is approximated by the quasi-Monte Carlo method. By numerical experiments, we validate the feasibility of the approach.
Keywords: Free boundary problem, random diffusion, shape optimization
@article{COCV_2020__26_1_A56_0,
author = {Br\"ugger, Rahel and Croce, Roberto and Harbrecht, Helmut},
title = {Solving a {Bernoulli} type free boundary problem with random diffusion},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2020},
publisher = {EDP Sciences},
volume = {26},
doi = {10.1051/cocv/2019030},
mrnumber = {4146353},
zbl = {1453.35199},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2019030/}
}
TY - JOUR AU - Brügger, Rahel AU - Croce, Roberto AU - Harbrecht, Helmut TI - Solving a Bernoulli type free boundary problem with random diffusion JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2020 VL - 26 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2019030/ DO - 10.1051/cocv/2019030 LA - en ID - COCV_2020__26_1_A56_0 ER -
%0 Journal Article %A Brügger, Rahel %A Croce, Roberto %A Harbrecht, Helmut %T Solving a Bernoulli type free boundary problem with random diffusion %J ESAIM: Control, Optimisation and Calculus of Variations %D 2020 %V 26 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/cocv/2019030/ %R 10.1051/cocv/2019030 %G en %F COCV_2020__26_1_A56_0
Brügger, Rahel; Croce, Roberto; Harbrecht, Helmut. Solving a Bernoulli type free boundary problem with random diffusion. ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 56. doi: 10.1051/cocv/2019030
[1] , On the geometric form of Bernoulli configurations. Math. Meth. Appl. Sci. 10 (1988) 1–14. | MR | Zbl | DOI
[2] and , Existence and regularity for a minimum problem with free boundary. J. Reine Angew. Math. 325 (1981) 105–144. | MR | Zbl
[3] , , , and , Boundary element based multiresolution shape optimisation in electrostatics. J. Comput. Phys. 297 (2015) 584–598. | MR | DOI | Zbl
[4] , On free boundary problems for the Laplace equation, in Seminars on Analytic functions, I. Institute for Advanced Study, Princeton, NJ (1957) 248–263. | Zbl
[5] and , Nested-dissection orderings for sparse LU with partial pivoting. SIAM J. Matrix Anal. Appl. 23 (2002) 998–1012. | MR | Zbl | DOI
[6] , and , Solving a free boundary problem with non-constant coefficients. Math. Meth. Appl. Sci. 41 (2018) 3653–3671. | MR | Zbl | DOI
[7] , How to prove existence in shape optimization. Control Cybernet. 34 (2005) 103–116. | MR | Zbl
[8] , A framework for the construction of level set methods for shape optimization and reconstruction. Interfaces Free Bound. 5 (2003) 301–329. | MR | Zbl | DOI
[9] and , Numerical approximation of a free boundary problem arising in electromagnetic shaping. SIAM J. Numer. Anal. 31 (1994) 1109–1127. | MR | Zbl | DOI
[10] , and , Shape optimization for quadratic functionals and states with random right-hand sides. SIAM J. Control Optim. 53 (2015) 3081–3103. | MR | Zbl | DOI
[11] , , and , On Bernoulli’s free boundary problem with a random boundary. Int. J. Uncertain. Quantif. 7 (2017) 335–353. | MR | Zbl | DOI
[12] and , Shapes and Geometries. SIAM, Philadelphia (2001). | MR | Zbl
[13] and , Exterior electromagnetic shaping using wavelet BEM. Math. Meth. Appl. Sci. 28 (2005) 387–405. | MR | Zbl | DOI
[14] and , Efficient treatment of stationary free boundary problems. Appl. Numer. Math. 56 (2006) 1326–1339. | MR | Zbl | DOI
[15] and , Bernoulli’s free-boundary problem, qualitative theory and numerical approximation. J. Reine Angew. Math. 486 (1997) 165–204. | MR | Zbl
[16] , Nested dissection of a regular finite element mesh. SIAM J. Numer. Anal. 10 (1973) 345–363. | MR | Zbl | DOI
[17] , A Newton method for Bernoulli’s free boundary problem in three dimensions. Computing 82 (2008) 11–30. | MR | Zbl | DOI
[18] , and , On the low-rank approximation by the pivoted Cholesky decomposition. Appl. Numer. Math. 62 (2012) 428–440. | MR | Zbl | DOI
[19] , and , On the quasi-Monte Carlo quadrature with Halton points for elliptic PDEs with log-normal diffusion. Math. Comput. 86 (2017) 771–797. | MR | Zbl | DOI
[20] , , and , Shape optimization and fictitious domain approach for solving free boundary value problems of Bernoulli type. Comput. Optim. Appl. 26 (2003) 231–251. | MR | Zbl | DOI
[21] and , Improving the run time and quality of nested dissection ordering. SIAM J. Sci. Comput. 20 (1998) 468–489. | MR | Zbl | DOI
[22] , and , Variational approach to shape derivatives for a class of Bernoulli problems. J. Math. Anal. Appl. 314 (2006) 126–149. | MR | Zbl | DOI
[23] , and , Generalized nested dissection. SIAM J. Numer. Anal. 16 (1979) 346–358. | MR | Zbl | DOI
[24] , Probability theory. I+II. Graduate Texts in Mathematics 45. Springer, New York, 4th ed. (1977). | MR | Zbl
[25] and , Marching cubes: a high resolution 3d surface construction algorithm. SIGGRAPH Comput. Graph. 21 (1987) 163–169. | DOI
[26] , The flexible, extensible and efficient toolbox of level set methods. J. Sci. Comput. 35 (2008) 300–329. | Zbl | DOI
[27] , A toolbox of level set methods (version 1.1). Department of Computer Science, University of British Columbia, Vancouver, Canada, Tech. Rep. TR-2007-11. Available from: http://www.cs.ubc.ca/~mitchell/ToolboxLS/toolboxLS.pdf (2007).
[28] and , A toolbox of Hamilton-Jacobi solvers for analysis of nondeterministic continuous and hybrid systems. Hybrid Systems: Computation and Control, edited by and . In Vol. 3414 of Lect. Notes Comput. Sci. Springer, Berlin (2005) 480–494. | Zbl | DOI
[29] and , Étude de problèmes d’optimal design. Optimization Techniques, Modeling and Optimization in the Service of Man, edited by . Vol. 40 of Lect. Notes Comput. Sci. Springer, Berlin (1976) 54–62. | Zbl | MR
[30] , Random Number Generation and Quasi-Monte Carlo Methods. Society for Industrial and Applied Mathematics, Philadelphia, PA (1992). | MR | Zbl | DOI
[31] and , Newton’s method in shape optimisation: a three-dimensional case. BIT 40 (2000) 102–120. | MR | Zbl | DOI
[32] and , Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79 (1988) 12–49. | MR | Zbl | DOI
[33] and , Level set methods: an overview and some recent results. J. Comput. Phys. 169 (2001) 463–502. | MR | Zbl | DOI
[34] and , Level set methods and dynamic implicit surfaces. Springer, New York (2003). | MR | Zbl | DOI
[35] , , , and , A PDE-based fast local level set method. J. Comput. Phys. 155 (1999) 410–438. | MR | Zbl | DOI
[36] and , Numerical simulation of tridimensional electromagnetic shaping of liquid metals. Numer. Math. 65 (1993) 203–217. | MR | Zbl | DOI
[37] , Optimal Shape Design for Elliptic Systems. Springer, New York (1983). | MR | Zbl
[38] , Differentiation with respect to the domain in boundary value problems. Numer. Funct. Anal. Optim. 2 (1980) 649–687. | MR | Zbl | DOI
[39] and , Introduction to Shape Optimization. Springer, Berlin (1992). | MR | Zbl | DOI
[40] , and , A level set approach for computing solutions to incompressible two-phase flow. J. Comput. Phys. 114 (1994) 146–159. | Zbl | DOI
[41] , Shape optimization and trial methods for free-boundary problems. ESAIM: M2AN 31 (1997) 805–825. | MR | Zbl | Numdam | DOI
[42] and , Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients. IMA J. Numer. Anal. 27 (2007) 232–261. | MR | Zbl | DOI
[43] , A constructive approach to strong tractability using quasi-Monte Carlo algorithms. J. Complex. 18 (2002) 683–701. | MR | Zbl | DOI
Cité par Sources :
