A new approach to convergence rate analysis of Tikhonov regularization for parameter identification in heat conduction

In this paper we investigate the stability and convergence rates of the widely used output least-squares method with Tikhonov regularization for the identification of the conductivity distribution in a heat conduction system. Due to the rather restrictive source conditions and regularity assumptions on the nonlinear parameter-to-solution operator concerned, the existing Tikhonov regularization theory for nonlinear inverse problems is difficult to apply for the convergence rate analysis here. By introducing some new techniques, we are able to relax these regularity requirements and derive a much simpler and easily interpretable source condition but still achieve the same convergence rates as the standard Tikhonov regularization theory does.

[1]  Zhiming Chen,et al.  An Augmented Lagrangian Method for Identifying Discontinuous Parameters in Elliptic Systems , 1999 .

[2]  Karl Kunisch,et al.  Regularization of nonlinear illposed problems with closed operators , 1993 .

[3]  William Rundell,et al.  Surveys on solution methods for inverse problems , 2000 .

[4]  H. Engl,et al.  Convergence rates for Tikhonov regularisation of non-linear ill-posed problems , 1989 .

[5]  H. Engl,et al.  Identification of a temperature dependent heat conductivity by Tikhonov regularization , 2002 .

[6]  Andreas Neubauer,et al.  Tikhonov regularisation for non-linear ill-posed problems: optimal convergence rates and finite-dimensional approximation , 1989 .

[7]  Otmar Scherzer,et al.  Convergence Rates Results for Iterative Methods for Solving Nonlinear Ill-Posed Problems , 2000 .

[8]  William Rundell,et al.  A Regularization Scheme for an Inverse Problem in Age-Structured Populations , 1994 .

[9]  Jun Zou,et al.  An Efficient Linear Solver for Nonlinear Parameter Identification Problems , 2000, SIAM J. Sci. Comput..

[10]  Otmar Scherzer,et al.  Parameter identification from boundary measurements in a parabolic equation arising from geophysics , 1993 .

[11]  Curtis R. Vogel,et al.  Well posedness and convergence of some regularisation methods for non-linear ill posed problems , 1989 .

[12]  Otmar Scherzer,et al.  Finite-dimensional approximation of tikhonov regularized solutions of non-linear ill-posed problems , 1990 .

[13]  Q. Jin,et al.  Tikhonov regularization for nonlinear ill-posed problems , 1997 .

[14]  C. Kravaris,et al.  Identification of parameters in distributed parameter systems by regularization , 1983, The 22nd IEEE Conference on Decision and Control.

[15]  Jun Zou,et al.  Numerical identifications of parameters in parabolic systems , 1998 .