Numerische Mathematik

, Volume 120, Issue 1, pp 45–77 | Cite as

Convergence rates for Tikhonov regularization of a two-coefficient identification problem in an elliptic boundary value problem

  • Dinh Nho HàoEmail author
  • Tran Nhan Tam Quyen


We investigate the convergence rates for Tikhonov regularization of the problem of simultaneously estimating the coefficients q and a in the Neumann problem for the elliptic equation \({{-{\rm div}(q \nabla u) + au = f \;{\rm in}\; \Omega, q{\partial u}/{\partial n} = g}}\) on the boundary \({{\partial\Omega, \Omega \subset \mathbb{R}^d, d \geq 1}}\) , when u is imprecisely given by \({{{z^\delta} \in {H^1}(\Omega), \|u-z^\delta\|_{H^1(\Omega)}\le\delta, \delta > 0}}\). We regularize this problem by minimizing the strictly convex functional of (q, a)
$$\begin{array}{lll}\int\limits_{\Omega}\left(q| \nabla (U(q,a)-z^{\delta})|^2 + a(U(q,a)-z^{\delta})^2\right)dx\\\quad+\rho\left(\|q-q^*\|^2_{L^2(\Omega)} + \|a-a^*\|^2_{L^2(\Omega)}\right)\end{array}$$
over the admissible set K, where ρ > 0 is the regularization parameter and (q*, a*) is an a priori estimate of the true pair (q, a) which is identified, and consider the unique solution of these minimization problem as the regularized one to that of the inverse problem. We obtain the convergence rate \({{{\mathcal {O}}(\sqrt{\delta})}}\), as δ → 0 and ρ ~ δ, for the regularized solutions under the simple and weak source condition
$${\rm there\;is\;a\;function}\;w^* \in V^*\;{\rm such\;that}\;{U^\prime (q^ \dagger, a^\dagger)}^*w^* = (q^\dagger - q^*, a^\dagger - a^*)$$
with \({{(q^\dagger, a^\dagger)}}\) being the (q*, a*)-minimum norm solution of the coefficient identification problem, U′(·, ·) the Fréchet derivative of U(·, ·), V the Sobolev space on which the boundary value problem is considered. Our source condition is without the smallness requirement on the source function which is popularized in the theory of regularization of nonlinear ill-posed problems. Furthermore, some concrete cases of our source condition are proved to be simply the requirement that the sought coefficients belong to certain smooth function spaces.

Mathematics Subject Classification (2000)

35R30 35R25 65J20 65J22 


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  1. 1.
    Acar R.: Identification of the coefficient in elliptic equations. SIAM J. Control Optim. 31(5), 1221–1244 (1993)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Alessandrini G.: An identification problem for an elliptic equation in two variables. Ann. Mat. Pura Appl. 145, 265–296 (1986)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Anderssen R.S., Hegland M.: For numerical differentiation, dimensionality can be a blessing!. Math. Comput. 68(227), 1121–1141 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Banks, H.T., Kunisch, K.: Estimation Techniques for Distributed Parameter Systems. Systems & Control: Foundations & Applications, vol. 1, Birkhäuser Boston Inc., Boston (1989)Google Scholar
  5. 5.
    Baumeister J., Kunisch K.: Identifiability and stability of a two-parameter estimation problem. Appl. Anal. 40(4), 263–279 (1991)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Beirão da Veiga H.: On a stationary transport equation. Ann. Univ. Ferrara Sez. VII, Sci. Math. 32, 79–91 (1986)zbMATHGoogle Scholar
  7. 7.
    Chan T.F., Tai X.C.: Identification of discontinuous coefficients in elliptic problems using total variation regularization. SIAM J. Sci. Comput. 25(3), 881–904 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Chan T.F., Tai X.C.: Level set and total variation regularization for elliptic inverse problems with discontinuous coefficients. J. Comput. Phys. 193, 40–66 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Chavent G.: Nonlinear Least Squares for Inverse Problems. Theoretical Foundations and Step-by-Step Guide for Applications. Scientific Computation. Springer, New York (2009)Google Scholar
  10. 10.
    Chavent G., Kunisch K.: The output least squares identifiability of the diffusion coefficient from an H 1-observation in a 2-D elliptic equation. ESAIM Control Optim. Calc. Var. 8, 423–440 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Chen Z., Zou J.: An augmented Lagrangian method for identifying discontinuous parameters in elliptic systems. SIAM J. Control Optim. 37(3), 892–910 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Cherlenyak, I.: Numerische Lösungen inverser Probleme bei elliptischen Differentialgleichungen. Dr. rer. nat. Dissertation, Universität Siegen, 2009, Verlag Dr. Hut, München (2010)Google Scholar
  13. 13.
    Colonius F., Kunisch K.: Output least squares stability in elliptic systems. Appl. Math. Optim. 19, 33–63 (1989)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Engl H.W., Hanke M., Neubauer A.: Regularization of Inverse Problems. Mathematics and its Applications. 375. Kluwer Academic Publishers, Dordrecht (1996)Google Scholar
  15. 15.
    Engl H.W., Kunisch K., Neubauer A.: Convergence rates for Tikhonov regularization of nonlinear ill-posed problems. Inverse Probl. 5, 523–540 (1989)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Falk R.: Error estimates for the numerical identification of a variable coefficient. Math. Comput. 40, 537–546 (1983)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Hanke M.: A regularizing Levenberg-Marquardt scheme, with applications to inverse groundwater filtration problems. Inverse Probl. 13, 79–95 (1997)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Hào, D.N., Quyen, N.T.T.: Convergence rates for Tikhonov regularization of coefficient identification problems in Laplace-type equation, Inverse Problems 26, (2010) 125014 (23pp)Google Scholar
  19. 19.
    Hein T., Meyer M.: Simultaneous identification of independent parameters in elliptic equations—numerical studies. J. Inv. Ill Posed Probl. 16, 417–433 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Ito K., Kunisch K.: The augmented Lagrangian method for parameter estimation in elliptic systems. SIAM J. Control Optim. 28(1), 113–136 (1990)zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Ito K., Kunisch K.: On the injectivity and linearization of the coefficient-to-solution mapping for elliptic boundary value problems. J. Math. Anal. Appl. 188, 1040–1066 (1994)zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Ito, K., Kunisch, K.: Lagrange Multiplier Approach to Variational Problems and Applications. Advances in Design and Control, vol. 15, Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2008)Google Scholar
  23. 23.
    Ito K., Kroller M., Kunisch K.: A numerical study of an augmented Lagrangian method for the estimation of parameters in elliptic systems. SIAM J. Sci. Stat. Comput. 12(4), 884–910 (1991)zbMATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Kaltenbacher B., Schöberl J.: A saddle point variational formulation for projection-regularized parameter identification. Numer. Math. 91(4), 675–697 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Keung Y.L., Zou J.: An efficient linear solver for nonlinear parameter identification problems. SIAM J. Sci. Comput 22, 1511–1526 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Knowles I.: Uniqueness for an elliptic inverse problem. SIAM J. Appl. Math. 59, 1356–1370 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Knowles, I.: Coefficient identification in elliptic differential equations. In: Direct and Inverse Problems of Mathematical Physics (Newark, DE, 1997), Int. Soc. Anal. Appl. Comput., vol. 5, pp. 149–160. Kluwer Academic Publishers, Dordrecht (2000)Google Scholar
  28. 28.
    Knowles I.: Parameter identification for elliptic problems. J. Comput. Appl. Math. 131, 19–175 (2001)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Knowles I., Le T., Yan A.: On the recovery of multiple flow parameters from transient head data. J. Comput. Appl. Math. 169(1), 1–15 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Knowles I., Wallace R.: A variational method for numerical differentiation. Numer. Math. 70(1), 91–110 (1995)zbMATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    Kohn R.V., Lowe B.D.: A variational method for parameter identification. RAIRO Modél. Math. Anal. Numér. 22(1), 119–158 (1988)zbMATHMathSciNetGoogle Scholar
  32. 32.
    Ladyzhenskaya O.A.: The Boundary Value Problems of Mathematical Physics. Springer, New York (1984)Google Scholar
  33. 33.
    Neubauer A.: Tikhonov regularization of nonlinear ill-posed problems in Hilbert scales. Appl. Anal. 46(1-2), 59–72 (1992)zbMATHMathSciNetCrossRefGoogle Scholar
  34. 34.
    Richter G.R.: An inverse problem for the steady state diffusion equation. SIAM J. Appl. Math. 41, 210–221 (1981)zbMATHMathSciNetCrossRefGoogle Scholar
  35. 35.
    Sun N.-Z.: Inverse Problems in Groundwater Modeling. Kluwer Academic Publishers, Dordrecht (1994)Google Scholar
  36. 36.
    Troianiello G.M.: Elliptic Differential Equations and Obstacle Problems. Plenum, New York (1987)zbMATHGoogle Scholar
  37. 37.
    Vainikko, G.: Identification of filtration coefficient. In: Ill-Posed Problems in Natural Sciences, (Moscow, 1991), pp. 202–213. VSP, Utrecht (1992)Google Scholar
  38. 38.
    Vainikko G.: On the discretization and regularization of ill-posed problems with noncompact operators. Numer. Funct. Anal. Optim. 13, 381–396 (1992)zbMATHMathSciNetCrossRefGoogle Scholar
  39. 39.
    Vainikko G., Kunisch K.: Identifiabilty of the transmissivity coefficient in an elliptic boundary value problem. Zeischrift für Analysis und ihre Anwendungen 12, 327–341 (1993)zbMATHMathSciNetGoogle Scholar
  40. 40.
    Yeh W.W.G.: Review of parameter identification procedures in ground water hydrology: the inverse problem. Water Resour. Res. 22, 95–108 (1986)CrossRefGoogle Scholar
  41. 41.
    Zou J.: Numerical methods for elliptic inverse problems. Int. J. Comput. Math. 70, 211–232 (1998)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Hanoi Institute of MathematicsHanoiVietnam
  2. 2.Da Nang University of EducationDa NangVietnam

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