Preconditioned Iterative Methods for Algebraic Systems from Multiplicative Half-Quadratic Regularization Image Restorations

<正>Image restoration is often solved by minimizing an energy function consisting of a data-fidelity term and a regularization term.A regularized convex term can usually preserve the image edges well in the restored image.In this paper,we consider a class of convex and edge-preserving regularization functions,i.e.,multiplicative half-quadratic regularizations,and we use the Newton method to solve the correspondingly reduced systems of nonlinear equations.At each Newton iterate,the preconditioned conjugate gradient method,incorporated with a constraint preconditioner,is employed to solve the structured Newton equation that has a symmetric positive definite coefficient matrix. The eigenvalue bounds of the preconditioned matrix are deliberately derived,which can be used to estimate the convergence speed of the preconditioned conjugate gradient method.We use experimental results to demonstrate that this new approach is efficient, and the effect of image restoration is reasonably well.     

On a third‐order Newton‐type method free of bilinear operators

This paper is devoted to the study of a third‐order Newton‐type method. The method is free of bilinear operators, which constitutes the main limitation of the classical third‐order iterative schemes. First, a global convergence theorem in the real case is presented. Second, a semilocal convergence theorem and some examples are analyzed, including quadratic equations and integral equations. Finally, an approximation using divided differences is proposed and used for the approximation of boundary‐value problems. Copyright © 2009 John Wiley & Sons, Ltd.     

关于差商和Newton插值公式教学的一些体会

针对讲授Newton插值多项式之前,如何自然地引入差商概念,介绍了一些心得体会;同时对Newton插值公式给出了一种简便、学生易于理解的证明方法.     

A new inertial-type hybrid projection-proximal algorithm for monotone inclusions

This paper investigates an enhanced proximal algorithm with interesting practical features and convergence properties for solving non-smooth convex minimization problems, or approximating zeroes of maximal monotone operators, in Hilbert spaces. The considered algorithm involves a recent inertial-type extrapolation technique, the use of enlargement of operators and also a recently proposed hybrid strategy, which combines inexact computation of the proximal iteration with a projection. Compared to other existing related methods, the resulting algorithm inherits the good convergence properties of the inertial-type extrapolation and the relaxed projection strategy. It also inherits the relative error tolerance of the hybrid proximal-projection method. As a special result, an update of inexact Newton-proximal method is derived and global convergence results are established.     

ON SMOOTH LU DECOMPOSITIONS WITH APPLICATIONS TO SOLUTIONS OF NONLINEAR EIGENVALUE PROBLEMS

We study the smooth LU decomposition of a given analytic functional A-matrix A（A） and its block-analogue. Sufficient conditions for the existence of such matrix decompositions are given, some differentiability about certain elements arising from them are proved, and several explicit expressions for derivatives of the specified elements are provided. By using these smooth LU decompositions, we propose two numerical methods for computing multiple nonlinear eigenvalues of A（A）, and establish their locally quadratic convergence properties. Several numerical examples are provided to show the feasibility and effectiveness of these new methods.     

Solving nonlinear inverse problems by evolution equations based on Gauss–Newton methods

In this article, we solve nonlinear inverse problems by an evolution equation method which can be viewed as the continuous analogue of the Gauss–Newton method. Under certain conditions we prove the convergence and derive the rate of convergence when the discrepancy principle is coupled.     

Error estimates for the finite element solution of elliptic problems with nonlinear newton boundary conditions

The paper is concerned with the study of an elliptic boundary value problem with a nonlinear Newton boundary condition. The existence and uniqueness of the solution of the continuous pioblem is a consequence of the monotone operator theory. The main attention is paid to the investigation of the finite element approximation using numeriral integration for the evaluation of boundary integrals. The error estimates for the solution of the discrete finite element problem are derived     

A Convergence Analysis of Newton-Like Method for Singular Equations Using Recurrent Functions

We provide new semilocal convergence results for Newton-like method involving outer or generalized inverses in a Banach space setting. Using our new idea of recurrent functions and the same or weaker conditions than before [5-19 A. Ben-Israel and N.E. Greville ( 1974 ). Generalized Inverses: Theory and Applications, Pure and Applied Mathematics . Wiley-Interscience , New York . X. Chen and T. Yamamoto ( 1989 ). Convergence domains of certain iterative methods for solving nonlinear equations . Numer. Funct. Anal. Optimiz. 10 : 37 – 48 . J.E. Dennis , Jr. ( 1968 ). On Newton-like methods . Numer. Math. 11 : 324 – 330 . P. Deuflhard and C. Heindl ( 1979 ). Convergence theorems for Newton's method and extensions to related methods . SIAM J. Numer. Anal. 16 : 1 – 10 . J.M. Gutiérrez ( 1997 ). A new semilocal convergence theorem for Newton's method . J. Comp. Appl. Math. 79 : 131 – 145 . J.M. Gutiérrez , M.A. Hernández , and M.A. Salanova ( 1995 ). Accessibility of solutions by Newton's method . Internat. J. Comput. Math. 57 : 239 – 247 . W.M. Häubler ( 1986 ). A Kantorovich-type convergence analysis for the Gauss–Newton methods . Numer. Math. 48 : 119 – 125 . L.V. Kantorovich and G.P. Akilov ( 1964 ). Functional Analysis . Pergamon Press , Oxford . M.Z. Nashed and X. Chen ( 1993 ). Convergence of Newton-like methods for singular operator equations using outer inverses . Numer. Math. 66 : 235 – 257 . F.A. Potra and V. Ptàk ( 1980 ). Sharp error bounds for Newton's process . Numer. Math. 34 : 67 – 72 . W.C. Rheinboldt ( 1968 ). A unified convergence theory for a class of iterative processes . SIAM J. Numer. Anal. 5 : 42 – 63 . W.C. Rheinboldt ( 1977 ). An adaptive continuation process for solving systems of nonlinear equations . Polish Academy of Sciences, Banach Ctr. Publ. 3 : 129 – 142 . T. Yamamoto ( 1987 ). A method for finding sharp error bounds for Newton's method under the Kantorovich assumptions . Numer. Math. 49 : 203 – 230 . T. Yamamoto ( 1987 ). A convergence theorem for Newton-like methods in Banach spaces . Numer. Math. 51 : 545 – 557 . T. Yamamoto ( 1989 ). Uniqueness of the solution in a Kantorovich-type theorem of Haubler for the Gauss–Newton method . Japan J. Appl. Math. 6 : 77 – 81 . ], we provide more precise information on the location of the solution and finer bounds on the distances involved. Moreover, since our Newton–Kantorovich-type hypothesis is weaker than before, we can now cover cases not previously possible.

Applications and numerical examples involving a nonlinear integral equation of Chandrasekhar-type and a differential equation with Green's function are also provided in this study.     

Regularizing properties of a truncated newton-cg algorithm for nonlinear inverse problems

This paper develops truncated Newton methods as an appropriate tool for nonlinear inverse problems which are ill-posed in the sense of Hadamard. In each Newton step an approximate solution for the linearized problem is computed with the conjugate gradient method as an inner iteration. The conjugate gradient iteration is terminated when the residual has been reduced to a prescribed percentage. Under certain assumptions on the nonlinear operator it is shown that the algorithm converges and is stable if the discrepancy principle is used to terminate the outer iteration. These assumptions are fulfilled, e.g., for the inverse problem of identifying the diffusion coefficient in a parabolic differential equation from distributed data.     

The Bhattacharya Function of Complete Monomial Ideals in Two Variables

We give explicit formulas for the Bhattacharya function of 𝔪-primary complete monomial ideals in two variables in terms of the vertices of the Newton polyhedra or in terms of the decompositions of the ideals as products of simple ideals.