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In this paper we consider a class of regularized Gauss-Newton methods for solving nonlinear inverse problems for which an a posteriori stopping rule is proposed to terminate the iteration. Such methods have the frozen feature that they require only the computation of the Fréchet derivative at the initial approximation. Thus the computational work is considerably reduced. Under certain mild conditions, we give the convergence analysis and derive various estimates, including the order optimality, on these methods. [ABSTRACT FROM AUTHOR]

Abstract: When the damped Morozov discrepancy principle is used to determine the Tikhonov regularization parameter, one should theoretically solve a nonlinear equation by some iteration process, which is generally of local convergence with large amount of computations. This paper considers an approximation of the regularization parameter under the model function framework, which solves an approximate Morozov equation with an explicit expression iteratively. For this approximation, three kinds of new model functions are proposed. The corresponding new algorithms for determining the regularization parameters are also established, with the rigorous proof of global convergence under a unified framework. Our work is a generalization and improvement of the earlier model function method [J.L. Xie, J. Zou, Inverse Problems 18 (5) (2002) 631–643]. Numerical implementations for some ill-posed problems are presented to illustrate the validity of the proposed algorithms. [Copyright &y& Elsevier]

Abstract: This paper is devoted to the class of inverse problems for coefficient identification with an adjoint problem approach related to nonlinear parabolic partial differential equations. The unknown coefficient depends on the gradient of the solution and belongs to a set of admissible coefficients. First, we present a coefficient-to-data mapping. Then, based on maximum principle and the adjoint problem of the direct problem, the integral identities will be obtained. Using these identities, we can show that the coefficient-to-data mapping is continuous and strictly monotone. Furthermore, an approximate solution to the inverse problem is constructed, and the error is analyzed. Finally, the applicability of the method is demonstrated in numerical examples. [Copyright &y& Elsevier]