ON SEMILOCAL CONVERGENCE OF INEXACT NEWTON METHODS

Inexact Newton methods are constructed by combining Newton＇s method with another iterative method that is used to solve the Newton equations inexactly. In this paper, we establish two semilocal convergence theorems for the inexact Newton methods. When these two theorems are specified to Newton＇s method, we obtain a different Newton-Kantorovich theorem about Newton＇s method. When the iterative method for solving the Newton equations is specified to be the splitting method, we get two estimates about the iteration steps for the special inexact Newton methods.     

BLOCK BASED NEWTON-LIKE BLENDING INTERPOLATION

Newton＇s polynomial interpolation may be the favourite linear interpolation in the sense that it is built up by means of the divided differences which can be calculated recursively and produce useful intermediate results. However Newton interpolation is in fact point based interpolation since a new interpolating polynomial with one more degree is obtained by adding a new support point into the current set of support points once at a time. In this paper we extend the point based interpolation to the block based interpolation. Inspired by the idea of the modern architectural design, we first divide the original set of support points into some subsets （blocks）, then construct each block by using whatever interpolation means, linear or rational and finally assemble these blocks by Newton＇s method to shape the whole interpolation scheme. Clearly our method offers many flexible interpolation schemes for choices which include the classical Newton＇s polynomial interpolation as its special case. A bivariate analogy is also discussed and numerical examples are given to show the effectiveness of our method.     

CONSTRUCTION OF GEOMETRIC PARTIAL DIFFERENTIAL EQUATIONS FOR LEVEL SETS

Geometric partial differential equations of level-set form are usually constructed by a variational method using either Dirac delta function or co-area formula in the energy functional to be minimized. However, the equations derived by these two approaches are not consistent. In this paper, we present a third approach for constructing the level-set form equations. By representing various differential geometry quantities and differential geometry operators in terms of the implicit surface, we are able to reformulate three classes of parametric geometric partial differential equations （second-order, fourth-order and sixth- order） into the level-set forms. The reformulation of the equations is generic and simple, and the resulting equations are consistent with their parametric form counterparts. We further prove that the equations derived using co-area formula are also consistent with the parametric forms. However, these equations are of much complicated forms than these given by the equations we derived.     

一类整函数的牛顿法的拟直接盘和直接盘

In this paper, we consider Newton＇s method for a class of entire functions with infinite order. By using theory of dynamics of functions meromorphic outside a small set, we find there are some series of virtual immediate basins in which the dynamics converges to infinity and a series of immediate basins with finite area in the Fatou sets of Newton＇s method.     

Large deviation principle for the fourth-order stochastic heat equations with fractional noises

In this paper, we shall study a fourth-order stochastic heat equation driven by a fractional noise, which is fractional in time and white in space. We will discuss the existence and uniqueness of the solution to the equation. Furthermore, the regularity of the solution will be obtained. On the other hand, the large deviation principle for the equation with a small perturbation will be established through developing a classical method.     

Balanced and Unbalanced Components of Moist Atmospheric Flows with Phase Changes

Atmospheric variables(temperature, velocity, etc.) are often decomposed into balanced and unbalanced components that represent low-frequency and high-frequency waves, respectively. Such decompositions can be defined, for instance, in terms of eigenmodes of a linear operator. Traditionally these decompositions ignore phase changes of water since phase changes create a piecewise-linear operator that differs in different phases(cloudy versus non-cloudy). Here we investigate the following question: How can a balanced–unbalanced decomposition be performed in the presence of phase changes? A method is described here motivated by the case of small Froude and Rossby numbers,in which case the asymptotic limit yields precipitating quasi-geostrophic equations with phase changes. Facilitated by its zero-frequency eigenvalue, the balanced component can be found by potential vorticity(PV) inversion, by solving an elliptic partial differential equation(PDE), which includes Heaviside discontinuities due to phase changes. The method is also compared with two simpler methods: one which neglects phase changes, and one which simply treats the raw pressure data as a streamfunction. Tests are shown for both synthetic, idealized data and data from Weather Research and Forecasting(WRF) model simulations. In comparisons, the phase-change method and no-phase-change method produce substantial differences within cloudy regions, of approximately 5K in potential temperature, due to the presence of clouds and phase changes in the data. A theoretical justification is also derived in the form of a elliptic PDE for the differences in the two streamfunctions.     

A FINITE DIFFERENCE SCHEME FOR SOLVING THE NONLINEAR POISSON-BOLTZMANN EQUATION MODELING CHARGED SPHERES

In this work, we propose an efficient numerical method for computing the electrostaticinteraction between two like-charged spherical particles which is governed by the nonlinearPoisson-Boltzmann equation. The nonlinear problem is solved by a monotone iterativemethod which leads to a sequence of linearized equations. A modified central finite differ-ence scheme is developed to solve the linearized equations on an exterior irregular domainusing a uniform Cartesian grid. With uniform grids, the method is simple, and as aconsequence, multigrid solvers can be employed to speed up the convergence. Numericalexperiments on cases with two isolated spheres and two spheres confined in a chargedcylindrical pore are carried out using the proposed method. Our numerical schemes arefound efficient and the numerical results are found in good agreement with the previouspublished results.     

Reflected SPDEs Driven by Fractional Noises

In this paper,we study the stochastic partial differential equation with two reflecting smooth walls h^1 and h^2,driven by a fractional noise,which is fractional in time and white in space.The large deviation principle for the law of the solution to this equation,will be established through developing a classical method.Furthermore,we obtain the H?lder continuity of the solution.     

Cauchy problem of semi-linear parabolic equations with weak data in homogeneous space and application to the Navier-Stokes equations

In this paper we study the Cauchy problem for a class of semi-linear parabolic type equations withweak data n the homogeneous spaces.We give a method which can be used to construct local mild solutionsof the abstract Cauchy problem in C(σ,s,p)and L~q([O,T);H~(s,p)by introducing the concept of both admissiblequintuptet and compatible space and establishing estblishing time-space estimates for solutions to the linear parabolic typeequations For the small data,we prove that these results can be extended globally in time. We also study the     

具有非连续性系数的两相不可压缩可混溶问题的有限元法

Two phase ,miscible,incompressible flow in porous media is governed by a system of nonlinear partial differential equations. Many numerical methods have been given by didierent authors to this system,hut these methods need very high regularity conditions. Actualty,in most practical applications these regularity conditions couldn‘t be satisfied. In this paper,the problem of discontinuous coefficients with lower regularity conditions is considered and the error estimates are demonstrated.     

A stable approach to Newton's method for general mathematical programming problems inR n

The usual approach to Newton's method for mathematical programming problems with equality constraints leads to the solution of linear systems ofn +m equations inn +m unknowns, wheren is the dimension of the space andm is the number of constraints. Moreover, these linear systems are never positive definite. It is our feeling that this approach is somewhat artificial, since in the unconstrained case the linear systems are very often positive definite. With this in mind, we present an alternate Newton-like approach for the constrained problem in which all the linear systems are of order less than or equal ton. Furthermore, when the Hessian of the Lagrangian at the solution is positive definite (a situation frequently occurring), all our systems will be positive definite. Hence, in all cases, our Newton-like method offers greater numerical stability. We demonstrate that the convergence properties of this Newton-like method are superior to those of the standard approach to Newton's method. The operation count for the new method using Gaussian elimination is of the same order as the operation count for the standard method. However, if the Hessian of the Lagrangian at the solution is positive definite and we use Cholesky decomposition, then the order of the operation count for the new method is half that for the standard approach to Newton's method. This theory is generalized to problems with both equality and inequality constraints.     

An efficient method for fractional nonlinear differential equations by quasi‐Newton's method and simplified reproducing kernel method

An efficient method for nonlinear fractional differential equations is proposed in this paper. This method consists of 2 steps. First, we linearize the nonlinear operator equation by quasi‐Newton's method, which is based on Fréchet derivative. Then we solve the linear fractional differential equations by the simplified reproducing kernel method. The convergence of the quasi‐Newton's method is discussed for the general nonlinear case as well. Finally, some numerical examples are presented to illustrate accuracy, efficiency, and simplicity of the method.     

Majorizing Sequences for Nonlinear Fredholm–Hammerstein Integral Equations

We use the method of majorizing sequences to study the applicability of Newton's method to solve nonlinear Fredholm–Hammerstein integral equations. For this, we use center convergence conditions on points different from the starting point of Newton's method, which is the point usually used by other authors until now when center conditions are required. In addition, the theoretical significance of the method is used to draw conclusions about the existence and uniqueness of solutions and about the region in which they are located. As a result, we modify the domain of starting points for Newton's method.     

On Newton's Method for Solving Nonlinear Equations and Function Splitting

We provided in [14] and [15] a semilocal convergence analysis for Newton's method on a Banach space setting, by splitting the given operator. In this study, we improve the error bounds, order of convergence, and simplify the sufficient convergence conditions. Our results compare favorably with the Newton-Kantorovich theorem for solving equations.     

A TWO-GRID METHOD FOR THE C0 INTERIOR PENALTY DISCRETIZATION OF THE MONGE-AMP(E)RE EQUATION

The purpose of this paper is to analyze an efficient method for the solution of the nonlinear system resulting from the discretization of the elliptic Monge-Ampère equation by a $C^0$ interior penalty method with Lagrange finite elements. We consider the two-grid method for nonlinear equations which consists in solving the discrete nonlinear system on a coarse mesh and using that solution as initial guess for one iteration of Newton's method on a finer mesh. Thus both steps are inexpensive. We give quasi-optimal $W^{1,\infty}$ error estimates for the discretization and estimate the difference between the interior penalty solution and the two-grid numerical solution. Numerical experiments confirm the computational efficiency of the approach compared to Newton's method on the fine mesh.     

Numerical simulation of (nearly) incompressible material,using adaptive mixed finite element method

In this paper we present a way to numerically simulate large deformations of incompressible material and the corresponding hydrostatic pressure by using a mixed, adaptive finite element method (FEM). Starting from the system of differential equations we will derive the weak nonlinear formulation, which can be solved with a Newton's method. In every iteration step this will lead to a saddle point problem. By using Taylor Hood finite elements we will obtain a discrete, indefinite problem, which can be handled with the Bramble Pasciak conjugate gradient method. Finally we give a numerical example. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On a Newton-Kurchatov-type Iterative Process

Newton's method and Kurchatov's method are iterative processes known for their fast speed of convergence. We construct from both methods an iterative method to approximate solutions of nonlinear equations given by a nondifferentiable operator, and we study its semilocal convergence in Banach spaces. Finally, we consider several applications of this new iterative process.     

Error estimate and regularity for the compressible Navier‐Stokes equations by Newton's method

The finite element discretization error estimate and H¹ regularity are shown for the solution generated by Newton's method to the stationary compressible Navier‐Stokes equations by interpreting Newton's method as an equivalent iterative method. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 511–524, 2003     

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)     

A new family of Schröder's method and its variants based on power means for multiple roots of nonlinear equations

In this article, we derive one-parameter family of Schröder's method based on Gupta et al.'s (K.C. Gupta, V. Kanwar, and S. Kumar, A family of ellipse methods for solving non-linear equations, Int. J. Math. Educ. Sci. Technol. 40 (2009), pp. 571–575) family of ellipse methods for the solution of nonlinear equations. Further, we introduce new families of Schröder-type methods for multiple roots with cubic convergence. Proposed families are derived from modified Newton's method for multiple roots and one-parameter family of Schröder's method. Numerical examples are also provided to show that these new methods are competitive to other known methods for multiple roots.