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In this paper, we shall present the quadratic convergence of Levenberg-Marquardt (L-M) method for solving general ill-posed nonlinear inverse problems including two parameters to be identified. As the Tikhonov regularized minimizations are non-convex due to the nonlinearity of the nonlinear inverse problems, the L-M method transforms them into convex minimizations. The quadratic convergence of the L-M method is rigorously demonstrated under some basic assumptions. An application of the L-M method for the simultaneous identification of the Robin coefficient and heat flux in an elliptic system is presented, and the surrogate functional technique is introduced to solve the convex but still strongly ill-conditioned minimizations, resulting in an explicit minimizer of the minimization at each L-M iteration. Numerical experiments are presented to illustrate the efficiency and accuracy of the proposed algorithm. [ABSTRACT FROM AUTHOR]

We study the Levenberg-Marquardt (L-M) method for solving the highly nonlinear and ill-posed inverse problem of identifying the Robin coefficients in elliptic and parabolic systems. The L-M method transforms the Tikhonov regularized nonlinear non-convex minimizations into convex minimizations. And the quadratic convergence of the L-M method is rigorously established for the nonlinear elliptic and parabolic inverse problems for the first time, under a simple novel adaptive strategy for selecting regularization parameters during the L-M iteration. Then the surrogate functional approach is adopted to solve the strongly ill-conditioned convex minimizations, resulting in an explicit solution of the minimisation at each L-M iteration for both the elliptic and parabolic cases. Numerical experiments are provided to demonstrate the accuracy, efficiency and quadratic convergence of the methods. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the inverse problem on determining the spatial component of the source term in the hyperbolic equation with a time-dependent principal part. Based on a Carleman estimate for general hyperbolic operators, we prove a local stability result of Hölder type in both cases of partial boundary and interior observation data. Numerically, we adopt the classical Tikhonov regularization to reformulate the inverse problem into a related optimization problem, for which we develop an iterative thresholding algorithm by using the corresponding adjoint system. Numerical examples up to three spatial dimensions are presented to demonstrate the accuracy and efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with two efficient algorithms for surface construction. Based on the Gauss equations, a discretized nonlinear equation of the formshould be solved in the process of surface construction. We first consider a regularized fixed-point iterative algorithm for solving the discretized equation, in which we determine the actual regularization parameter by the Morozov discrepancy principle. A two-parameter algorithm is employed for solving the Morozov equation, and the convergence of the regularized fixed-point iterative algorithm is demonstrated. Secondly, we also propose the regularizing Levenberg–Marquardt scheme to solve the discretized equation, in which the regularization parameter is chosen to be a small constant. Numerical experiments are provided to demonstrate the robustness and efficiency of the two algorithms. [ABSTRACT FROM AUTHOR]