Topological Derivatives of Shape Functionals. Part III: Second-Order Method and Applications

  • Antonio André Novotny
  • Jan SokołowskiEmail author
  • Antoni Żochowski
Invited Paper


The framework of asymptotic analysis in singularly perturbed geometrical domains presented in the first part of this series of review papers can be employed to produce two-term asymptotic expansions for a class of shape functionals. In Part II (Novotny et al. in J Optim Theory Appl 180(3):1–30, 2019), one-term expansions of functionals are required for algorithms of shape-topological optimization. Such an approach corresponds to the simple gradient method in shape optimization. The Newton method of shape optimization can be replaced, for shape-topology optimization, by two-term expansions of shape functionals. Thus, the resulting approximations are more precise and the associated numerical methods are much more complex compared to one-term expansion topological derivative algorithms. In particular, numerical algorithms associated with first-order topological derivatives of shape functionals have been presented in Part II (Novotny et al. 2019), together with an account of their applications currently found in the literature, with emphasis on shape and topology optimization. In this last part of the review, second-order topological derivatives are introduced. Second-order algorithms of shape-topological optimization are used for numerical solution of representative examples of inverse reconstruction problems. The main feature of these algorithms is that the method is non-iterative and thus very robust with respect to noisy data as well as independent of initial guesses.


Topological derivatives Second-order method Applications in inverse problems 

Mathematics Subject Classification

35J15 35Q74 49J20 49M15 49N45 



This research was partly supported by CNPq (Brazilian Research Council), CAPES (Brazilian Higher Education Staff Training Agency) and FAPERJ (Research Foundation of the State of Rio de Janeiro). The support is gratefully acknowledged. We also thank Habib Ammari, Alfredo Canelas, Michael Hintermüller, Hyeonbae Kang, Antoine Laurain, Jairo Faria, Ravi Prakash and the former students Lucas Fernandez, Andrey Ferreira, Thiago Machado and Suelen Rocha.


  1. 1.
    Novotny, A.A., Sokołowski, J., Żochowski, A.: Topological derivatives of shape functionals. Part II: first order method and applications. J. Optim. Theory Appl. 180(3), 1–28 (2019)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Novotny, A.A., Sokołowski, J., Żochowski, A.: Topological derivatives of shape functionals. Part I: theory in singularly perturbed geometrical domains. J. Optim. Theory Appl. 180(2), 1–33 (2019)MathSciNetzbMATHGoogle Scholar
  3. 3.
    de Faria, J.R., Novotny, A.A.: On the second order topologial asymptotic expansion. Struct. Multidiscipl. Optim. 39(6), 547–555 (2009)zbMATHGoogle Scholar
  4. 4.
    Novotny, A.A., Sokołowski, J.: Topological Derivatives in Shape Optimization. Interaction of Mechanics and Mathematics. Springer, Berlin (2013)zbMATHGoogle Scholar
  5. 5.
    Bonnet, M., Cornaggia, R.: Higher order topological derivatives for three-dimensional anisotropic elasticity. ESAIM Control Optim. Calc. Var. 51(6), 2069–2092 (2017)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Hintermüller, M., Laurain, A., Novotny, A.A.: Second-order topological expansion for electrical impedance tomography. Adv. Comput. Math. 36(2), 235–265 (2012)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Ferreira, A., Novotny, A.A.: A new non-iterative reconstruction method for the electrical impedance tomography problem. Inverse Probl. 33(3), 035005 (2017)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Canelas, A., Laurain, A., Novotny, A.A.: A new reconstruction method for the inverse source problem from partial boundary measurements. Inverse Probl. 31(7), 075009 (2015)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Isakov, V.: Inverse Problems for Partial Differential Equations. Applied Mathematical Sciences, vol. 127. Springer, New York (2006)zbMATHGoogle Scholar
  10. 10.
    Fernandez, L., Novotny, A.A., Prakash, R.: A non-iterative reconstruction method for an inverse potential problem modeled by a modified Helmholtz equation. Numer. Funct. Anal. Optim. 39(9), 937–966 (2018)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Canelas, A., Laurain, A., Novotny, A.A.: A new reconstruction method for the inverse potential problem. J. Comput. Phys. 268, 417–431 (2014)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Machado, T.J., Angelo, J.S., Novotny, A.A.: A new one-shot pointwise source reconstruction method. Math. Methods Appl. Sci. 40(15), 1367–1381 (2017)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Rocha, S.S., Novotny, A.A.: Obstacles reconstruction from partial boundary measurements based on the topological derivative concept. Struct. Multidiscip. Optim. 55(6), 2131–2141 (2017)MathSciNetGoogle Scholar
  14. 14.
    Burger, M.: A level set method for inverse problems. Inverse Probl. 17, 1327–1356 (2001)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Hintermüller, M., Laurain, A.: Electrical impedance tomography: from topology to shape. Control Cybern. 37(4), 913–933 (2008)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Isakov, V., Leung, S., Qian, J.: A fast local level set method for inverse gravimetry. Commun. Comput. Phys. 10(4), 1044–1070 (2011)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Leitão, A., Baumeister, J.: Topics in Inverse Problems. IMPA Mathematical Publications, Rio de Janeiro (2005)zbMATHGoogle Scholar
  18. 18.
    Tricarico, P.: Global gravity inversion of bodies with arbitrary shape. Geophys. J. Int. 195(1), 260–275 (2013)Google Scholar
  19. 19.
    Calderón, A.P.: On an inverse boundary value problem. Comput. Appl. Math. 25(2–3), 133–138 (2006). (Reprinted from the Seminar on Numerical Analysis and its Applications to Continuum Physics, Sociedade Brasileira de Matemática, Rio de Janeiro, 1980)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Amstutz, S., Horchani, I., Masmoudi, M.: Crack detection by the topological gradient method. Control Cybern. 34(1), 81–101 (2005)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Carpio, A., Rapún, M.L.: Solving inhomogeneous inverse problems by topological derivative methods. Inverse Probl. 24(4), 045,014 (2008)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Caubet, F., Conca, C., Godoy, M.: On the detection of several obstacles in 2D Stokes flow: topological sensitivity and combination with shape derivatives. Inverse Probl. Imaging 10(2), 327–367 (2016)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Guzina, B.B., Bonnet, M.: Small-inclusion asymptotic of misfit functionals for inverse problems in acoustics. Inverse Probl. 22(5), 1761–1785 (2006)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Jackowska-Strumiłło, L., Sokołowski, J., Żochowski, A., Henrot, A.: On numerical solution of shape inverse problems. Comput. Optim. Appl. 23(2), 231–255 (2002)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Laurain, A., Hintermüller, M., Freiberger, M., Scharfetter, H.: Topological sensitivity analysis in fluorescence optical tomography. Inverse Probl. 29(2), 025,003,30 (2013)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Masmoudi, M., Pommier, J., Samet, B.: The topological asymptotic expansion for the Maxwell equations and some applications. Inverse Probl. 21(2), 547–564 (2005)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Ammari, H., Kang, H.: High-order terms in the asymptotic expansions of the steady-state voltage potentials in the presence of inhomogeneities of small diameter. SIAM J. Math. Anal. 34(5), 1152–1166 (2003)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Brühl, M., Hanke, M., Vogelius, M.S.: A direct impedance tomography algorithm for locating small inhomogeneities. Numer. Math. 93(4), 635–654 (2003)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Capdeboscq, Y., Vogelius, M.S.: A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction. Math. Model. Numer. Anal. 37(1), 159–173 (2003)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Capdeboscq, Y., Vogelius, M.S.: Optimal asymptotic estimates for the volume of internal inhomogeneities in terms of multiple boundary measurements. Math. Model. Numer. Anal. 37(2), 227–240 (2003)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Cedio-Fengya, D.J., Moskow, S., Vogelius, M.S.: Identification of conductivity imperfections of small diameter by boundary measurements. Continuous dependence and computational reconstruction. Inverse Probl. 14(3), 553–595 (1998)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Friedman, A., Vogelius, M.: Identification of small inhomogeneities of extreme conductivity by boundary measurements: a theorem on continuous dependence. Arch. Ration. Mech. Anal. 105(4), 299–326 (1989)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Ammari, H., Bretin, E., Garnier, J., Jing, W., Kang, H., Wahab, A.: Localization, stability, and resolution of topological derivative based imaging functionals in elasticity. SIAM J. Imaging Sci. 6(4), 2174–2212 (2013)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Ammari, H., Calmon, P., Iakovleva, E.: Direct elastic imaging of a small inclusion. SIAM J. Imaging Sci. 1, 169–187 (2008)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Ammari, H., Kang, H.: Reconstruction of Small Inhomogeneities from Boundary Measurements. Lectures Notes in Mathematics, vol. 1846. Springer, Berlin (2004)zbMATHGoogle Scholar
  36. 36.
    Bonnet, M.: Higher-order topological sensitivity for 2-D potential problems. Int. J. Solids Struct. 46(11–12), 2275–2292 (2009)zbMATHGoogle Scholar
  37. 37.
    Silva, M., Matalon, M., Tortorelli, D.A.: Higher order topological derivatives in elasticity. Int. J. Solids Struct. 47(22–23), 3053–3066 (2010)zbMATHGoogle Scholar
  38. 38.
    Isakov, V.: Inverse Problems for Partial Diferential Equations. Springer, New York (1998)zbMATHGoogle Scholar
  39. 39.
    Isakov, V.: Inverse Source Problems. American Mathematical Society, Providence, RI (1990)zbMATHGoogle Scholar
  40. 40.
    Kohn, R., Vogelius, M.: Determining conductivity by boundary measurements. Commun. Pure Appl. Math. 37(3), 289–298 (1984)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Canelas, A., Novotny, A.A., Roche, J.R.: A new method for inverse electromagnetic casting problems based on the topological derivative. J. Comput. Phys. 230, 3570–3588 (2011)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Canelas, A., Novotny, A.A., Roche, J.R.: Topology design of inductors in electromagnetic casting using level-sets and second order topological derivatives. Struct. Multidiscip. Optim. 50(6), 1151–1163 (2014)MathSciNetGoogle Scholar
  43. 43.
    Fernandez, L., Novotny, A.A., Prakash, R.: Topological asymptotic analysis of an optimal control problem modeled by a coupled system. Asympt. Anal. 109(1–2), 1–26 (2018)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Laboratório Nacional de Computação Científica LNCC/MCTCoordenação de Matemática Aplicada e ComputacionalPetrópolisBrazil
  2. 2.UMR 7502 Laboratoire de Mathématiques, Institut Élie CartanUniversité de LorraineVandoeuvre Lès Nancy CedexFrance
  3. 3.Systems Research InstitutePolish Academy of SciencesWarsawPoland

Personalised recommendations