Bouligand–Landweber iteration for a non-smooth ill-posed problem

  • Christian ClasonEmail author
  • Vu Huu Nhu


This work is concerned with the iterative regularization of a non-smooth nonlinear ill-posed problem where the forward mapping is merely directionally but not Gâteaux differentiable. Using a Bouligand subderivative of the forward mapping, a modified Landweber method can be applied; however, the standard analysis is not applicable since the Bouligand subderivative mapping is not continuous unless the forward mapping is Gâteaux differentiable. We therefore provide a novel convergence analysis of the modified Landweber method that is based on the concept of asymptotic stability and merely requires a generalized tangential cone condition. These conditions are verified for an inverse source problem for an elliptic PDE with a non-smooth Lipschitz continuous nonlinearity, showing that the corresponding Bouligand–Landweber iteration converges strongly for exact data as well as in the limit of vanishing data if the iteration is stopped according to the discrepancy principle. This is illustrated with a numerical example.

Mathematics Subject Classification

49K20 49K40 90C31 



The authors would like to thank the anonymous reviewers for their detailed and constructive comments that helped to significantly improve the presentation. This work was supported by the DFG under the Grants CL 487/2-1 and RO 2462/6-1, both within the priority programme SPP 1962 “Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization”.


  1. 1.
    Chen, X., Nashed, Z., Qi, L.: Smoothing methods and semismooth methods for nondifferentiable operator equations. SIAM J. Numer. Anal. 38(4), 1200–1216 (2000). MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Chipot, M.: Elliptic Equations: An Introductory Course. Birkhäuser Verlag, Basel (2009). CrossRefzbMATHGoogle Scholar
  3. 3.
    Christof, C., Clason, C., Meyer, C., Walter, S.: Optimal control of a non-smooth semilinear elliptic equation. Math. Control Relat. Fields 8(1), 247–276 (2018). MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Christof, C., De los Reyes, J.C., Meyer, C.: A non-smooth trust-region method for locally Lipschitz functions with application to optimization problems constrained by variational inequalities. (2017). arXiv:1711.03208
  5. 5.
    Clarke, F.H.: Optimization and Nonsmooth Analysis, 2nd edn. SIAM, Philadelphia (1990). CrossRefzbMATHGoogle Scholar
  6. 6.
    Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings. Springer Monographs in Mathematics. Springer, Berlin (2014). zbMATHGoogle Scholar
  7. 7.
    Glowinski, R.: Numerical Methods for Nonlinear Variational Problems. Scientific Computation. Springer, Berlin (1984)CrossRefzbMATHGoogle Scholar
  8. 8.
    Hanke, M., Neubauer, A., Scherzer, O.: A convergence analysis of the Landweber iteration for nonlinear ill-posed problems. Numer. Math. 72(1), 21–37 (1995). MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hofmann, B., Kaltenbacher, B., Pöschl, C., Scherzer, O.: A convergence rates result in Banach spaces with non-smooth operators. Inverse Probl. 23(3), 987–1010 (2007). MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ito, K., Kunisch, K.: Lagrange Multiplier Approach to Variational Problems and Applications. Advances in Design and Control, vol. 15. SIAM, Philadelphia, PA (2008). CrossRefzbMATHGoogle Scholar
  11. 11.
    Kaltenbacher, B., Neubauer, A., Scherzer, O.: Iterative Regularization Methods for Nonlinear Ill-Posed Problems. Radon Series on Computational and Applied Mathematics, vol. 6. De Gruyter, Berlin (2008). CrossRefzbMATHGoogle Scholar
  12. 12.
    Kikuchi, F., Nakazato, K., Ushijima, T.: Finite element approximation of a nonlinear eigenvalue problem related to MHD equilibria. Jpn. J. Appl. Math. 1(2), 369–403 (1984). MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Klatte, D., Kummer, B.: Nonsmooth Equations in Optimization. Nonconvex Optimization and Its Applications, vol. 60. Kluwer, Dordrecht (2002). zbMATHGoogle Scholar
  14. 14.
    Knabner, P., Angermann, L.: Numerical Methods for Elliptic and Parabolic Partial Differential Equations. Texts in Applied Mathematics. Springer, Berlin (2003). zbMATHGoogle Scholar
  15. 15.
    Kügler, P.: A derivative-free Landweber iteration for parameter identification in certain elliptic PDEs. Inverse Probl. 19(6), 1407 (2003). MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kügler, P.: Convergence rate analysis of a derivative free Landweber iteration for parameter identification in certain elliptic PDEs. Numer. Math. 101(1), 165–184 (2005). MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Margotti, F., Rieder, A.: An inexact Newton regularization in Banach spaces based on the nonstationary iterated Tikhonov method. J. Inverse Ill-Posed Probl. 23(4), 373–392 (2015). MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Meyer, C., Susu, L.M.: Optimal control of nonsmooth, semilinear parabolic equations. SIAM J. Control Optim. 55(4), 2206–2234 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Oleĭnk, O.A.: A method of solution of the general Stefan problem. Sov. Math. Dokl. 1, 1350–1354 (1960)MathSciNetGoogle Scholar
  20. 20.
    Outrata, J., Kočvara, M., Zowe, J.: Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. Nonconvex Optimization and Its Applications, vol. 28. Kluwer, Dordrecht (1998). zbMATHGoogle Scholar
  21. 21.
    Ramlau, R., Teschke, G., Zhariy, M.: A compressive Landweber iteration for solving ill-posed inverse problems. Inverse Probl. 24(6), 065013, 26 (2008). MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Rappaz, J.: Approximation of a nondifferentiable nonlinear problem related to MHD equilibria. Numer. Math. 45(1), 117–133 (1984). MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Rauls, A.T., Ulbrich, S.: Subgradient computation for the solution operator of the obstacle problem. Technical Report SPP1962-056, DFG Priority Programme 1962 (2018).
  24. 24.
    Rauls, A.T., Wachsmuth, G.: Generalized derivatives for the solution operator of the obstacle problem. (2018). arXiv:1806.05052
  25. 25.
    Scherzer, O.: Convergence criteria of iterative methods based on Landweber iteration for solving nonlinear problems. J. Math. Anal. Appl. 194(3), 911–933 (1995). MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Scherzer, O., Grasmair, M., Grossauer, H., Haltmeier, M., Lenzen, F.: Variational Methods in Imaging. Springer, Berlin (2009). zbMATHGoogle Scholar
  27. 27.
    Scherzer, O.E.: Handbook of Mathematical Methods in Imaging, 2nd edn. Springer, New York (2015). CrossRefzbMATHGoogle Scholar
  28. 28.
    Scholtes, S.: Introduction to Piecewise Differentiable Equations. Springer Science & Business Media, Berlin (2012). CrossRefzbMATHGoogle Scholar
  29. 29.
    Schuster, T., Kaltenbacher, B., Hofmann, B., Kazimierski, K.S.: Regularization Methods in Banach Spaces. Walter de Gruyter GmbH & Co. KG, Berlin (2012). CrossRefzbMATHGoogle Scholar
  30. 30.
    Temam, R.: A non-linear eigenvalue problem: the shape at equilibrium of a confined plasma. Arch. Ration. Mech. Anal. 60(1), 51–73 (1975). MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Tröltzsch, F.: Optimal Control of Partial Differential Equations. Graduate Studies in Mathematics, vol. 112. American Mathematical Society, Providence, RI (2010). (Theory, methods and applications, Translated from the 2005 German original by Jürgen Sprekels)zbMATHGoogle Scholar
  32. 32.
    Ulbrich, M.: Semismooth Newton Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces. MOS-SIAM Series on Optimization. SIAM, Philadelphia (2011). CrossRefzbMATHGoogle Scholar
  33. 33.
    Visintin, A.: Models of Phase Transitions. Birkhäuser, Boston (1996). CrossRefzbMATHGoogle Scholar
  34. 34.
    Wachsmuth, D., Wachsmuth, G.: Regularization error estimates and discrepancy principle for optimal control problems with inequality constraints. Control Cybern. 40(4), 1125–1158 (2011)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Wachsmuth, G., Wachsmuth, D.: Convergence and regularization results for optimal control problems with sparsity functional. ESAIM Control Optim. Calc. Var. 17(3), 858–886 (2011). MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Faculty of MathematicsUniversity of Duisburg-EssenEssenGermany

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