An algorithm for a class of nonlinear complementarity problems with non-Lipschitzian functions

In this paper, we focus on solving a class of nonlinear complementarity problems with non-Lipschitzian functions. We first introduce a generalized class of smoothing functions for the plus function. By combining it with Robinson's normal equation, we reformulate the complementarity problem as a family of parameterized smoothing equations. Then, a smoothing Newton method combined with a new nonmonotone line search scheme is employed to compute a solution of the smoothing equations. The global and local superlinear convergence of the proposed method is proved under mild assumptions. Preliminary numerical results obtained applying the proposed approach to nonlinear complementarity problems arising in free boundary problems are reported. They show that the smoothing function and the nonmonotone line search scheme proposed in this paper are effective.     

Viscosity approximation methods for generalized equilibrium problems and fixed point problems

The purpose of this paper is to investigate the problem of finding a common element of the set of solutions of a generalized equilibrium problem (for short, GEP) and the set of fixed points of a nonexpansive mapping in the setting of Hilbert spaces. By using well-known Fan-KKM lemma, we derive the existence and uniqueness of a solution of the auxiliary problem for GEP. On account of this result and Nadler’s theorem, we propose an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of GEP and the set of fixed points of a nonexpansive mapping. Furthermore, it is proven that the sequences generated by this iterative scheme converge strongly to a common element of the set of solutions of GEP and the set of fixed points of a nonexpansive mapping.     

A CLASS OF GENERALIZED MULTISPLITTING RELAXATION METHODS FOR LINEAR COMPLEMENTARITY PROBLEMS

In this paper,a class of generalized parallel matrix multisplitting relaxation methods for solving linear complementarity problems on the high-speed multiprocessor systems is set up. This class of methods not only includes all the existing relaxation methods for the linear complementarity problems ,but also yields a lot of novel ones in the sense of multisplittlng. We establish the convergence theories of this class of generalized parallel multisplitting relaxation methods under the condition that the system matrix is an H-metrix with positive diagonal elements.     

Modulus‐based matrix splitting iteration methods for a class of implicit complementarity problems

Some modulus‐based matrix splitting iteration methods for a class of implicit complementarity problem are presented, and their convergence analysis is given. Numerical experiments confirm the theoretical analysis and show that the proposed methods are efficient. Copyright © 2016 John Wiley & Sons, Ltd.     

Variational approach to shape derivatives for a class of Bernoulli problems

The shape derivative of a functional related to a Bernoulli problem is derived without using the shape derivative of the state. The gradient information is combined with level set ideas in a steepest descent algorithm. Numerical examples show the feasibility of the approach.     

Preconditioned sor methods for generalized least-squares problems

1.IntroductionThegeneralizedleastsquaresproblem,definedasmin(Ax--b)"W--'(Ax--b),(1.1)xacwhereAERm",m>n,bERm,andWERm'misasymmetricandpositivedefinitematrix,isfrequentlyfoundwhensolvingproblemsinstatistics,engineeringandeconomics.Forexample,wegetgeneralizedleastsquaresproblemswhensolvingnonlinearregressionanalysisbyquasi-likelihoodestimation,imagereconstructionproblemsandeconomicmodelsobtainedbythemaximumlikelihoodmethod(of.[1,21).Paige[3,4]investigatestheproblemexplicitly.Hechangestheorig…     

Approximation of inverse operators by a new family of high‐order iterative methods

The main goal of this paper is to approximate inverse operators by high‐order Newton‐type methods with the important feature of not using inverse operators. We analyse the semilocal convergence, the speed of convergence, and the efficiency of these methods. We determine that Chebyshev's method is the most efficient method and test it on two problems: one associated to the heat equation and the other one to a boundary value problem. We consider examples with matrices that are close to be singular and/or are badly conditioned. We check the robustness and the stability of the methods by considering situations with many steps and noised data. Copyright © 2013 John Wiley & Sons, Ltd.     

A CLASS OF SINGULAR PERTURBATIONS FOR SECOND ORDER QUASI－LINEAR BOUNDARY VALUE PROBLEMS ON INFINITE INTERVAL

ＡＣＬＡＳＳＯＦＳＩＮＧＵＬＡＲＰＥＲＴＵＲＢＡＴＩＯＮＳＦＯＲＳＥＣＯＮＤＯＲＤＥＲＱＵＡＳＩ－ＬＩＮＥＡＲＢＯＵＮＤＡＲＹＶＡＬＵＥＰＲＯＢＬＥＭＳＯＮＩＮＦＩＮＩＴＥＩＮＴＥＲＶＡＬＺＨＡＯＷＥＩＬＩ（赵为礼）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈ...     

Some new analysis results for a class of interface problems

Interface problems modeled by differential equations have many applications in mathematical biology, fluid mechanics, material sciences, and many other areas. Typically, interface problems are characterized by discontinuities in the coefficients and/or the Dirac delta function singularities in the source term. Because of these irregularities, solutions to the differential equations are not smooth or discontinuous. In this paper, some new results on the jump conditions of the solution across the interface are derived using the distribution theory and the theory of weak solutions. Some theoretical results on the boundary singularity in which the singular delta function is at the boundary are obtained. Finally, the proof of the convergency of the immersed boundary (IB) method is presented. The IB method is shown to be first‐order convergent in L ^∞ norm. Copyright © 2013 John Wiley & Sons, Ltd.     

Improved generalized differentiability conditions for Newton-like methods

We provide a semilocal convergence analysis for Newton-like methods using the ω$\omega$-versions of the famous Newton–Kantorovich theorem (Argyros (2004) [1], Argyros (2007) [3], Kantorovich and Akilov (1982) [13]). In the special case of Newton’s method, our results have the following advantages over the corresponding ones (Ezquerro and Hernaández (2002) [10], Proinov (2010) [17]) under the same information and computational cost: finer error estimates on the distances involved; at least as precise information on the location of the solution, and weaker sufficient convergence conditions.     

A NEW GENERALIZED GRADIENT PROJECTION TYPEALGORITHM FOR LINEARLY CONSTRAINED PROBLEMS

In this paper, we provide a new generalized gradient projection algorithm for nonlinear programming problems with linear constraints. This algorithm has simple structure and is very practical and stable. Under the weaker assumptions, we have proved the global convergence of our algorithm.     

Mortar element methods for parabolic problems

In this article a standard mortar finite element method and a mortar element method with Lagrange multiplier are used for spatial discretization of a class of parabolic initial‐boundary value problems. Optimal error estimates in L^∞(L²) and L^∞(H¹)‐norms for semidiscrete methods for both the cases are established. The key feature that we have adopted here is to introduce a modified elliptic projection. In the standard mortar element method, a completely discrete scheme using backward Euler scheme is discussed and optimal error estimates are derived. The results of numerical experiments support the theoretical results obtained in this article. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

A note on the existence of solutions for difference equations via variational methods

We improve the results of Li by using less restrictive assumptions.     

Convergence criteria for generalized gradient methods of solving locally Lipschitz feasibility problems

We prove the convergence of a class of iterative algorithms for solving locally Lipschitz feasibility problems, that is, finite systems of inequalities f _i (x)0, (i I), where each f _i is a locally Lipschitz functional on ⁿ . We also obtain a new convergence criterion for the so-called block-iterative projection methods of finding common points of finite families of convex closed subsets of ⁿ as defined by Aharoni and Censor ([3]).The work of Dan Butnariu was done while visiting the Department of Mathematics of the University of Texas at Arlington.     

一类四阶奇异边值问题的正解存在的充分必要条件

利用上下解方法和极大值原理给出了一般边界条件下四阶微分方程的奇异迫值问题有C^2[0，1]和C^3[0，1]正解存在的充分必要条件．推广了韦忠礼(1999)的结果。     

Stabilization of the trial method for the Bernoulli problem in case of prescribed Dirichlet data

We apply the trial method for the solution of Bernoulli's free boundary problem when the Dirichlet boundary condition is imposed for the solution of the underlying Laplace equation, and the free boundary is updated according to the Neumann boundary condition. The Dirichlet boundary value problem for the Laplacian is solved by an exponentially convergent boundary element method. The update rule for the free boundary is derived from the linearization of the Neumann data around the actual free boundary. With the help of shape sensitivity analysis and Banach's fixed‐point theorem, we shed light on the convergence of the respective trial method. Especially, we derive a stabilized version of this trial method. Numerical examples validate the theoretical findings.Copyright © 2014 John Wiley & Sons, Ltd.     

A mixed hp FEM for the approximation of fourth‐order singularly perturbed problems on smooth domains

We consider fourth‐order singularly perturbed problems posed on smooth domains and the approximation of their solution by a mixed Finite Element Method on the so‐called Spectral Boundary Layer Mesh. We show that the method converges uniformly, with respect to the singular perturbation parameter, at an exponential rate when the error is measured in the energy norm. Numerical examples illustrate our theoretical findings.