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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]

In this work we are concerned with the analysis on a simultaneous finite element reconstruction of the convection velocity and source strength in a time-dependent convection–diffusion equation. The ill-posed problem is formulated into an output least-squares nonlinear minimization by an appropriately selected Tikhonov regularization. The regularizing effect and mathematical properties of the regularized system are justified and demonstrated. The nonlinear optimization problem is approximated by a fully discrete finite element method, whose convergence is rigorously established. [ABSTRACT FROM AUTHOR]

This paper aims to solve an inverse heat conduction problem in two-dimensional space under transient regime, which consists of the estimation of multiple time-dependent heat sources placed at the boundaries. Robin boundary condition (third type boundary condition) is considered at the working domain boundary. The simultaneous identification problem is formulated as a constrained minimization problem using the output least squares method with Tikhonov regularization. The properties of the continuous and discrete optimization problem are studied. Differentiability results and the adjoint problems are established. The numerical estimation is investigated using a modified conjugate gradient method. Furthermore, to verify the performance of the proposed algorithm, obtained results are compared with results obtained from the well-known finite-element software COMSOL Multiphysics under the same conditions. The numerical results show that the proposed algorithm is accurate, robust and capable of simultaneously representing the time effects on reconstructing the time-dependent Robin coefficient and heat flux. [ABSTRACT FROM AUTHOR]

In this paper, we shall derive and propose an efficient algorithm for simultaneously reconstructing the Robin coefficient and heat flux in an elliptic system from part of the boundary measurements. The uniqueness of the simultaneous identification is demonstrated. The ill-posed inverse problem is formulated into an output least-squares nonlinear and non-convex minimization with Tikhonov regularization, while the regularizing effects of the regularized system are justified. The Levenberg–Marquardt method is applied to change the non-convex minimization into convex minimization, which will be solved by surrogate functional method so as to get the explicit expression of the minimizer. Numerical experiments are provided to show the accuracy and efficiency of the algorithm. [ABSTRACT FROM PUBLISHER]

In this paper, we investigate the convergence rates of the widely used output least squares method with Tikhonov regularization for the identification of the magnetic permeability in a Maxwell system. By introducing some new techniques, we derive a simple and easily interpretable source condition and guarantee the usual convergence rate O() for the regularized solutions, whereis assumed to be the noise level of the data. More importantly, we finally construct the source functions successfully in two-dimensional spaces. [ABSTRACT FROM PUBLISHER]

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]