On the applicability of two Newton methods for solving equations in Banach space

In this study we examine the applicability of Newton’s method and the modified Newton’s method for approximating a locally unique solution of a nonlinear equation in a Banach space. We assume that the Newton-Kantorovich hypothesis for Newton’s method is violated, but the corresponding condition for the modified Newton method holds. Under these conditions there is no guarantee that Newton’s method starting from the same initial guess as the modified Newton’s method converges. Hence, it seems that we must always use the modified Newton method under these conditions. However, we provide a numerical example to demonstrate that in practice this may not be a good decision.     

Use of Halley's Method in the Nonlinear Finite Element Analysis

Halley's method is a higher order iteration method for the solution of nonlinear systems of equations. Unlike Newton's method, which converges quadratically in the vicinity of the solution, Halley's method can exhibit a cubic order of convergence. The equations of Halley's method for multiple dimensions are derived using Padé approximants and inverse one-point interpolation, as proposed by Cuyt. The investigation of the performance of Halley's method concentrates on eight-node volume elements for nonlinear deformations using Staint Venant-Kirchhoff's constitutive law, as well as a geometric linear theory of von Mises plasticity. The comparison with Newton's method reveals the sensibility of Halley's method, in view of the radius of attraction but also demonstrates the advantages of Halley's method considering simulation costs and the order of convergence. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Kepler equation and accelerated Newton method

We study the efficiency of the accelerated Newton method (Garlach, SIAM Rev. 36 (1994) 272–276) for several orders of convergence versus Danby's method for the resolution of Kepler's equation; we find that the cited method of order three is competitive with Danby's method and the classical Newton's method. We also generalize the accelerated Newton method for the resolution of system of algebraic equations, obtaining a formula of order three and a proof of its convergence; its application to several examples shows that its efficiency is greater than Newton's method.     

A new smoothing quasi-Newton method for nonlinear complementarity problems

A new smoothing quasi-Newton method for nonlinear complementarity problems is presented. The method is a generalization of Thomas’ method for smooth nonlinear systems and has similar properties as Broyden's method. Local convergence is analyzed for a strictly complementary solution as well as for a degenerate solution. Presented numerical results demonstrate quite similar behavior of Thomas’ and Broyden's methods.     

Chebyshev pseudospectral method for computing numerical solution of convection–diffusion equation

A method for computing highly accurate numerical solutions of 1D convection–diffusion equations is proposed. In this method, the equation is first discretized with respect to the spatial variable, transforming the original problem into a set of ordinary differential equations, and then the resulting system is integrated in time by the fourth-order Runge–Kutta method. Spatial discretization is done by using the Chebyshev pseudospectral collocation method. Before describing the method, we review a finite difference-based method by Salkuyeh [D. Khojasteh Salkuyeh, On the finite difference approximation to the convection–diffusion equation, Appl. Math. Comput. 179 (2006) 79–86], and, contrary to the proposal of the author, we show that this method is not suitable for problems involving time dependent boundary conditions, which calls for revision. Stability analysis based on pseudoeigenvalues to determine the maximum time step for the proposed method is also carried out. Superiority of the proposed method over a revised version of Salkuyeh’s method is verified by numerical examples.     

A quadrature-based approach to improving the collocation method

Summary The collocation method is a popular method for the approximate solution of boundary integral equations, but typically does not achieve the high order of convergence reached by the Galerkin method in appropriate negative norms. In this paper a quadrature-based method for improving upon the collocation method is proposed, and developed in detail for a particular case. That case involves operators with even symbol (such as the logarithmic potential) operating on 1-periodic functions; a smooth-spline trial space of odd degree, with constant mesh spacingh=1/n; and a quadrature rule with 2n points (where ann-point quadrature rule would be equivalent to the collocation method). In this setting it is shown that a special quadrature rule (which depends on the degree of the splines and the order of the operator) can yield a maximum order of convergence two powers ofh higher than the collocation method.     

Moving planes, moving spheres, and a priori estimates

In this paper, we use the method of moving planes and the method of moving spheres to obtain a priori estimates for the solutions of semi-linear elliptic equations.Using the ‘method of moving planes’, we establish a sharper estimate on the solutions for prescribing scalar curvature equations with indefinite curvature functions and thus generalized a resent result of Lin.Applying ‘the method of moving spheres’, we give a different proof for a well-known sup+inf inequality established by Brezis, Li and Shafrir.     

Minimizing multi-homogeneous Bézout numbers by a local search method

Consider the multi-homogeneous homotopy continuation method for solving a system of polynomial equations. For any partition of variables, the multi-homogeneous Bézout number bounds the number of isolated solution curves one has to follow in the method. This paper presents a local search method for finding a partition of variables with minimal multi-homogeneous Bézout number. As with any other local search method, it may give a local minimum rather than the minimum over all possible homogenizations. Numerical examples show the efficiency of this local search method.

     

A Fourier spectral-discontinuous Galerkin method for time-dependent 3-D Schrödinger–Poisson equations with discontinuous potentials

In this paper, we propose a high order Fourier spectral-discontinuous Galerkin method for time-dependent Schrödinger–Poisson equations in 3-D spaces. The Fourier spectral Galerkin method is used for the two periodic transverse directions and a high order discontinuous Galerkin method for the longitudinal propagation direction. Such a combination results in a diagonal form for the differential operators along the transverse directions and a flexible method to handle the discontinuous potentials present in quantum heterojunction and supperlattice structures. As the derivative matrices are required for various time integration schemes such as the exponential time differencing and Crank Nicholson methods, explicit derivative matrices of the discontinuous Galerkin method of various orders are derived. Numerical results, using the proposed method with various time integration schemes, are provided to validate the method.     

An efficient gradient method with approximate optimal stepsize for large-scale unconstrained optimization

In this paper, we introduce a new concept of approximate optimal stepsize for gradient method, use it to interpret the Barzilai-Borwein (BB) method, and present an efficient gradient method with approximate optimal stepsize for large unconstrained optimization. If the objective function f is not close to a quadratic on a line segment between the current iterate x _k and the latest iterate x _k?1, we construct a conic model to generate the approximate optimal stepsize for gradient method if the conic model is suitable to be used. Otherwise, we construct a new quadratic model or two other new approximation models to generate the approximate optimal stepsize for gradient method. We analyze the convergence of the proposed method under some suitable conditions. Numerical results show the proposed method is very promising.     

Spectral Galerkin method and its application to a Cauchy problem of Helmholtz equation

Based on a mathematical model of laser beams, we present a spectral Galerkin method for solving a Cauchy problem of the Helmholtz equation in a rectangle, where the Cauchy data pairs are given at y?=?0 and boundary data are for x?=?0 and x?=?π. The solution is sought in the interval 0?<?y?<?1. The spectral Galerkin method is considered as a regularization method. We then perform an analysis on the error bound for this method. For illustration, several numerical experiments are constructed to demonstrate the feasibility and efficiency of the proposed method.     

Jacobian smoothing Brown’s method for NCP

Jacobian smoothing Brown’s method for nonlinear complementarity problems (NCP) is studied in this paper. This method is a generalization of classical Brown’s method. It belongs to the class of Jacobian smoothing methods for solving semismooth equations. Local convergence of the proposed method is proved in the case of a strictly complementary solution of NCP. Furthermore, a locally convergent hybrid method for general NCP is introduced. Some numerical experiments are also presented.     

A variant of Nitsche's method

We present a method that aims to reconcile Nitsche's method with the traditional finite element method ('weak' versus 'strong implementation' of essential boundary conditions). We retain the original idea of a variational formulation based on an extended energy, but replace the original boundary terms by domain terms involving weak derivatives. The solution of the proposed method coincides, for the Poisson problem, with the one of the traditional method, which in particular shows monotonicity under the standard angle condition for the Courant element. For more general second-order problems, it allows for the weighting of boundary terms inherent to Nitsche's method. This is of particular interest for singularly perturbed problems.     

The <Emphasis Type="Italic">Q</Emphasis> Method for Symmetric Cone Programming

The Q method of semidefinite programming, developed by Alizadeh, Haeberly and Overton, is extended to optimization problems over symmetric cones. At each iteration of the Q method, eigenvalues and Jordan frames of decision variables are updated using Newton’s method. We give an interior point and a pure Newton’s method based on the Q method. In another paper, the authors have shown that the Q method for second-order cone programming is accurate. The Q method has also been used to develop a “warm-starting” approach for second-order cone programming. The machinery of Euclidean Jordan algebra, certain subgroups of the automorphism group of symmetric cones, and the exponential map is used in the development of the Newton method. Finally we prove that in the presence of certain non-degeneracies the Jacobian of the Newton system is nonsingular at the optimum. Hence the Q method for symmetric cone programming is accurate and can be used to “warm-start” a slightly perturbed symmetric cone program.     

Statistic D-Property of Voronoi Summation Methods of Class W Q 2

We propose a general method for obtaining Tauberian theorems with remainder for one class of Voronoi summation methods for double sequences of elements of a locally convex, linear topological space. This method is a generalization of the Davydov method of C-points.     

Local limit theorems for occupancy models

We present a rather general method for proving local limit theorems, with a good rate of convergence, for sums of dependent random variables. The method is applicable when a Stein coupling can be exhibited. Our approach involves both Stein's method for distributional approximation and Stein's method for concentration. As applications, we prove local central limit theorems with rate of convergence for the number of germs with d neighbors in a germ‐grain model, and the number of degree‐d vertices in an Erd?s‐Rényi random graph. In both cases, the error rate is optimal, up to logarithmic factors.     

Inexact multisplitting methods for linear complementarity problems

We present an inexact multisplitting method for solving the linear complementarity problems, which is based on the inexact splitting method and the multisplitting method. This new method provides a specific realization for the multisplitting method and generalizes many existing matrix splitting methods for linear complementarity problems. Convergence for this new method is proved when the coefficient matrix is an _H+$H_{+}$-matrix. Then, two specific iteration forms for this inexact multisplitting method are presented, where the inner iterations are implemented either through a matrix splitting method or through a damped Newton method. Convergence properties for both these specific forms are analyzed, where the system matrix is either an _H+$H_{+}$-matrix or a symmetric matrix.     

Modified Newton’s method for systems of nonlinear equations with singular Jacobian

It is well known that Newton’s method for a nonlinear system has quadratic convergence when the Jacobian is a nonsingular matrix in a neighborhood of the solution. Here we present a modification of this method for nonlinear systems whose Jacobian matrix is singular. We prove, under certain conditions, that this modified Newton’s method has quadratic convergence. Moreover, different numerical tests confirm the theoretical results and allow us to compare this variant with the classical Newton’s method.     

A robust Lagrangian-DNN method for a class of quadratic optimization problems

The Lagrangian-doubly nonnegative (DNN) relaxation has recently been shown to provide effective lower bounds for a large class of nonconvex quadratic optimization problems (QAPs) using the bisection method combined with first-order methods by Kim et al. (Math Program 156:161–187, 2016). While the bisection method has demonstrated the computational efficiency, determining the validity of a computed lower bound for the QOP depends on a prescribed parameter \(\epsilon > 0\). To improve the performance of the bisection method for the Lagrangian-DNN relaxation, we propose a new technique that guarantees the validity of the computed lower bound at each iteration of the bisection method for any choice of \(\epsilon > 0\). It also accelerates the bisection method. Moreover, we present a method to retrieve a primal-dual pair of optimal solutions of the Lagrangian-DNN relaxation using the primal-dual interior-point method. As a result, the method provides a better lower bound and substantially increases the robustness as well as the effectiveness of the bisection method. Computational results on binary QOPs, multiple knapsack problems, maximal stable set problems, and quadratic assignment problems illustrate the robustness of the proposed method. In particular, a tight bound for QAPs with size \(n=50\) could be obtained.     

An implicitly restarted block Lanczos bidiagonalization method using Leja shifts

In this paper, we propose an implicitly restarted block Lanczos bidiagonalization (IRBLB) method for computing a few extreme or interior singular values and associated right and left singular vectors of a large matrix A. Our method combines the advantages of a block routine, implicit shifting, and the application of Leja points as shifts in the accelerating polynomial. The method neither requires factorization of A nor of a matrix that contains A. This makes the method well suited for very large matrices. For many matrices, the method presented in this paper requires less computational work and computer storage than other available approaches. Computed examples illustrate the performance of the method described.