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Abstract: In this paper, we consider the projection algorithm studied by Iiduka and Takahashi (2008) for finding a solution of the variational inequality problem for an inverse strongly monotone operator in a Banach space. We first remark that, under the assumptions imposed on the operator in their paper, the iterative sequence converges weakly to a zero of the operator, not just a solution of the variational inequality problem. In our proof, slightly modified from the original, we do not assume the uniform smoothness of a space as was the case there. Finally, using Halpern’s type method, we modify this algorithm to obtain the strong convergence to a zero of an inverse strongly monotone operator which is nearest to the initial element of the algorithm in the sense of the Bergman distance associated with the function . [Copyright &y& Elsevier]

Abstract: Electrical capacitance tomography (ECT) is considered as a promising tomography technology, and exactly reconstructing the original objects is highly desirable in real applications. In this paper, a generalized image reconstruction model that simultaneously considers the inaccurate property in the measured capacitance data and the linearization approximation error is presented. A generalized objective function, which has been developed using a combinational M-estimation and an extended stabilizing item, is proposed. The objective function unifies six estimation methods into a concise formula, where different estimation methods can be easily obtained by selecting different parameters. The homotopy method that integrates the beneficial advantages of the alternant iteration scheme is employed to solve the proposed objective function. Numerical simulations are implemented to evaluate the numerical performances and effectiveness of the proposed algorithm, and the numerical results reveal that the proposed algorithm is efficient and overcomes the numerical instability in the process of ECT image reconstruction. For the reconstructed objects in this paper, a dramatic improvement in accuracy and spatial resolution can be achieved, which indicates that the proposed algorithm is a promising candidate for solving ECT inverse problems. [Copyright &y& Elsevier]

Abstract: In this paper we propose the Multipower Defect-Correction method, a generalization of the double iteration, to compute a cluster of eigenvalues and the associated invariant subspace from discretized integral operators. It consists of an inner/outer iteration where, inside a defect-correction iteration, p power iteration steps are performed. The approximate inverse used in the defect correction is built with an approximation to the reduced resolvent operator of a coarse discretization of the integral operator. The proposed method computes eigenpairs approximations by refining initial approximations obtained from a coarser dimensional problem. It is therefore meant for large dimensional problems. Furthermore, the kernel of the integral operator may be weakly singular. We provide a proof for the convergence of this Multipower Defect-Correction method. A numerical example illustrating the theory and the behavior of the method is also presented. [Copyright &y& Elsevier]

Abstract: Recently, a Newton’s iterative method is attracting more and more attention from various fields of science and engineering. This method is generally quadratically convergent. In this paper, some Chebyshev-type methods with the third order convergence are analyzed in detail and used to compute approximate inverse preconditioners for solving the linear system Ax = b. Theoretic analysis and numerical experiments show that Chebyshev’s method is more effective than Newton’s one in the case of constructing approximate inverse preconditioners. [Copyright &y& Elsevier]

Abstract: In this paper we propose an iterative algorithm to solve large size linear inverse ill posed problems. The regularization problem is formulated as a constrained optimization problem. The dual Lagrangian problem is iteratively solved to compute an approximate solution. Before starting the iterations, the algorithm computes the necessary smoothing parameters and the error tolerances from the data. The numerical experiments performed on test problems show that the algorithm gives good results both in terms of precision and computational efficiency. [Copyright &y& Elsevier]

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: It is well known that the determination of a general potential function employs two sets of information. One or both may be a sequence of eigenvalues. In this work a partially known potential and one finite sequence of eigenvalues are given in order to approximate a non-symmetric potential. We utilize not only the finite difference method but also Numerov’s method to approximate the necessary eigenvalues in each iteration step. A complete set of eigenvalues is the main information to approximate a potential. In this paper a method recovering a missing eigenvalue in the sequence is given. [Copyright &y& Elsevier]