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On Shape Optimization with Parabolic State Equation

  • Helmut HarbrechtEmail author
  • Johannes Tausch
Chapter
Part of the International Series of Numerical Mathematics book series (ISNM, volume 165)

Abstract

The present paper intends to summarize the main results of Harbrecht and Tausch (Inverse Probl 27:065013, 2011; SIAM J Sci Comput 35:A104–A121, 2013) on the numerical solution of shape optimization problems for the heat equation. This is carried out by means of a specific problem, namely the reconstruction of a heat source which is located inside the computational domain under consideration from measurements of the heat flux through the boundary. We arrive at a shape optimization problem by tracking the mismatch of the heat flux at the boundary. For this shape functional, the Hadamard representation of the shape gradient is derived by use of the adjoint method. The state and its adjoint equation are expressed as parabolic boundary integral equations and solved using a Nyström discretization and a space-time fast multipole method for the rapid evaluation of thermal potentials. To demonstrate the similarities to shape optimization problems for elliptic state equations, we consider also the related stationary shape optimization problem which involves the Poisson equation. Numerical results are given to illustrate the theoretical findings.

Keywords

Shape optimization Heat equation Boundary integral equation Multipole method 

Mathematics Subject Classification (2010)

35K05 58J35 65K10 

References

  1. 1.
    W. Dahmen, H. Harbrecht, R. Schneider, Compression techniques for boundary integral equations. Optimal complexity estimates. SIAM J. Numer. Anal. 43, 2251–2271 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    M. Delfour, J.-P. Zolesio, Shapes and Geometries (SIAM, Philadelphia, 2001)zbMATHGoogle Scholar
  3. 3.
    J.E. Dennis, R.B. Schnabel, Numerical Methods for Nonlinear Equations and Unconstrained Optimization Techniques (Prentice-Hall, Englewood Cliffs, 1983)Google Scholar
  4. 4.
    S. El Yacoubi, J. Sokolowski, Domain optimization problems for parabolic control systems. Appl. Math. Comput. Sci. 6, 277–289 (1996)zbMATHMathSciNetGoogle Scholar
  5. 5.
    K. Eppler, H. Harbrecht, Numerical solution of elliptic shape optimization problems using wavelet-based BEM. Optim. Methods Softw. 18, 105–123 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    K. Eppler, H. Harbrecht, A regularized Newton method in electrical impedance tomography using shape Hessian information. Control Cybern. 34, 203–225 (2005)zbMATHMathSciNetGoogle Scholar
  7. 7.
    K. Eppler, H. Harbrecht, Efficient treatment of stationary free boundary problems. Appl. Numer. Math. 56, 1326–1339 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    K. Eppler, H. Harbrecht, Wavelet based boundary element methods in exterior electromagnetic shaping. Eng. Anal. Bound. Elem. 32, 645–657 (2008)CrossRefzbMATHGoogle Scholar
  9. 9.
    K. Eppler, H. Harbrecht, R. Schneider, On convergence in elliptic shape optimization. SIAM J. Control Optim. 45, 61–83 (2007)CrossRefMathSciNetGoogle Scholar
  10. 10.
    A.V. Fiacco, G.P. McCormick, Nonlinear Programming. Sequential Unconstrained Minimization Techniques (Wiley, New York, 1968)Google Scholar
  11. 11.
    R. Fletcher, Practical Methods for Optimization I&II (Wiley, New York, 1980)Google Scholar
  12. 12.
    C. Grossmann, J. Terno, Numerik der Optimierung (B.G. Teubner, Stuttgart, 1993)CrossRefzbMATHGoogle Scholar
  13. 13.
    H. Harbrecht, Analytical and numerical methods in shape optimization. Math. Methods Appl. Sci. 31, 2095–2114 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    H. Harbrecht, A Newton method for Bernoulli’s free boundary problem in three dimensions. Computing 82, 11–30 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    H. Harbrecht, T. Hohage, Fast methods for three-dimensional inverse obstacle scattering. J. Integral Equ. Appl. 19, 237–260 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    H. Harbrecht, R. Schneider, Wavelet Galerkin schemes for boundary integral equations. Implementation and quadrature. SIAM J. Sci. Comput. 27, 1347–1370 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    H. Harbrecht, J. Tausch, An efficient numerical method for a shape identification problem arising from the heat equation. Inverse Probl. 27, 065013 (2011)CrossRefMathSciNetGoogle Scholar
  18. 18.
    H. Harbrecht, J. Tausch, On the numerical solution of a shape optimization problem for the heat equation. SIAM J. Sci. Comput. 35, A104–A121 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    J. Haslinger, P. Neitaanmäki, Finite Element Approximation for Optimal Shape, Material and Topology Design, 2nd edn. (Wiley, Chichester, 1996)zbMATHGoogle Scholar
  20. 20.
    A. Henrot, J. Sokolowski, A shape optimization problem for the heat equation, in Optimal Control (Gainesville, 1997). Applied Optimization, vol. 15 (Kluwer Academic, Dordrecht, 1998), pp. 204–223Google Scholar
  21. 21.
    F. Hettlich, W. Rundell, The determination of a discontinuity in a conductivity from a single boundary measurement. Inverse Probl. 14, 67–82 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    F. Hettlich, W. Rundell, Identification of a discontinuous source in the heat equation. Inverse Probl. 17, 1465–1482 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    K.-H. Hoffmann, J. Sokolowski, Interface optimization problems for parabolic equations. Shape design and optimization. Control Cybern. 23, 445–451 (1994)zbMATHMathSciNetGoogle Scholar
  24. 24.
    V. Isakov, Inverse Source Problems. AMS Mathematical Surveys and Monographs, vol. 34 (American Mathematical Society, Providence, 1990)Google Scholar
  25. 25.
    K. Ito, K. Kunisch, G. Peichl, Variational approach to shape derivatives for a class of Bernoulli problems. J. Math. Anal. Appl. 314, 126–149 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    A.M. Khludnev, J. Sokołowski, Modeling and Control in Solid Mechanics (Birkhäuser, Basel, 1997)Google Scholar
  27. 27.
    F. Murat, J. Simon, Étude de problèmes d’optimal design, in Optimization Techniques, Modeling and Optimization in the Service of Man, ed. by J. Céa. Lecture Notes in Computer Science, vol. 41 (Springer, Berlin, 1976), pp. 54–62Google Scholar
  28. 28.
    O. Pironneau, Optimal Shape Design for Elliptic Systems (Springer, New York, 1983)Google Scholar
  29. 29.
    J.-R. Roche, J. Sokolowski, Numerical methods for shape identification problems. Control Cybern. 25, 867–894 (1996)zbMATHMathSciNetGoogle Scholar
  30. 30.
    J. Simon, Differentiation with respect to the domain in boundary value problems. Numer. Funct. Anal. Optim. 2, 649–687 (1980)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    J. Sokolowski, Shape sensitivity analysis of boundary optimal control problems for parabolic systems. SIAM J. Control Optim. 26, 763–787 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    J. Sokolowski, J.-P. Zolesio, Introduction to Shape Optimization (Springer, Berlin, 1992)CrossRefzbMATHGoogle Scholar
  33. 33.
    J. Tausch, A fast method for solving the heat equation by layer potentials. J. Comput. Phys. 224, 956–969 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    J. Tausch, Nyström discretization of parabolic boundary integral equations. Appl. Numer. Math. 59, 2843–2856 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    T. Tiihonen, Shape optimization and trial methods for free-boundary problems. RAIRO Model. Math. Anal. Numér. 31, 805–825 (1997)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversität BaselBaselSwitzerland
  2. 2.Department of MathematicsSouthern Methodist UniversityDallasUSA

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