Newton-like Iterative Method Research Articles

Interpolated Coefficient Mixed Finite Elements for Semilinear Time Fractional Diffusion Equations

In this paper, we consider a fully discrete interpolated coefficient mixed finite element method for semilinear time fractional reaction–diffusion equations. The classic L1 scheme based on graded meshes and new mixed finite element based on triangulation is used for the temporal and spatial discretization, respectively. The interpolation coefficient technique is used to deal with the semilinear term, and the discrete nonlinear system is solved by a Newton-like iterative method. Stability and convergence results for both the original variable and its flux are derived. Numerical experiments confirm our theoretical analysis.

A Newton-like iterative method implemented in the DelPhi for solving the nonlinear Poisson-Boltzmann equation.

DelPhi is a popular scientific program which numerically solves the Poisson-Boltzmann equation (PBE) for electrostatic potentials and energies of biomolecules immersed in water via finite difference method. It is well known for its accuracy, reliability, flexibility, and efficiency. In this work, a new edition of DelPhi that uses a novel Newton-like method to solve the nonlinear PBE, in addition to the already implemented Successive Over Relaxation (SOR) algorithm, is introduced. Our tests on various examples have shown that this new method is superior to the SOR method in terms of stability when solving the nonlinear PBE, being able to converge even for problems involving very strong nonlinearity.

The local and semilocal convergence analysis of new Newton-like iteration methods

The aim of this paper is to find new iterative Newton-like schemes inspired by the modified Newton iterative algorithm and prove that these iterations are faster than the existing ones in the literature. We further investigate their behavior and finally illustrate the results by numerical examples.

Third order derivative free SPH iterative method for solving nonlinear systems

The majority of iterative methods to find roots of a function requires the evaluation of derivatives of this function. In this paper, based on the basic principle of the SPH method’s kernel approximation, a kernel approximation was constructed to compute first and second order derivatives through Taylor series expansion. Derivatives in our proposed method were replaced in a Newton-like iterative method to obtain a derivative free SPH iterative method for solving nonlinear systems. To illustrate that the new method has the same order of convergence as the considered iterative method, some numerical examples are presented.

Construction of Newton-like iteration methods for solving nonlinear equations

In this paper, we present a simple, and yet powerful and easily applicable scheme in constructing the Newton-like iteration formulae for the computation of the solutions of nonlinear equations. The new scheme is based on the homotopy analysis method applied to equations in general form equivalent to the nonlinear equations. It provides a tool to develop new Newton-like iteration methods or to improve the existing iteration methods which contains the well-known Newton iteration formula in logic; those all improve the Newton method. The orders of convergence and corresponding error equations of the obtained iteration formulae are derived analytically or with the help of Maple. Some numerical tests are given to support the theory developed in this paper.

Pseudo-Transient Continuation for Nonsmooth Nonlinear Equations

Pseudo-transient continuation is a Newton-like iterative method for computing steady-state solutions of differential equations in cases where the initial data are far from a steady state. The iteration mimics a temporal integration scheme, with the time step being increased as steady state is approached. The iteration is an inexact Newton iteration in the terminal phase. In this paper we show how steady-state solutions to certain ordinary and differential algebraic equations with nonsmooth dynamics can be computed with the method of pseudo-transient continuation. An example of such a case is a discretized PDE with a Lipschitz continuous, but nondifferentiable, constitutive relation as part of the nonlinearity. In this case we can approximate a generalized derivative with a difference quotient. The existing theory for pseudo-transient continuation requires Lipschitz continuity of the Jacobian. Newton-like methods for nonsmooth equations have been globalized by trust-region methods, smooth approximations, and splitting methods in the past, but these approaches are not designed to find steady-state solutions of time-dependent problems. The method in this paper synthesizes the ideas from nonsmooth calculus and the method of pseudo-transient continuation.

Low-Rank Approximations with Sparse Factors II: Penalized Methods with Discrete Newton-Like Iterations

In [SIAM J. Matrix Anal. Appl., 23 (2002), pp. 706--727], we developed numerical algorithms for computing sparse low-rank approximations of matrices, and we also provided a detailed error analysis of the proposed algorithms together with some numerical experiments. The low-rank approximations are constructed in a certain factored form with the degree of sparsity of the factors controlled by some user-specified parameters. In this paper, we cast the sparse low-rank approximation problem in the framework of penalized optimization problems. We discuss various approximation schemes for the penalized optimization problem which are more amenable to numerical computations. We also include some analysis to show the relations between the original optimization problem and the reduced one. We then develop a globally convergent discrete Newton-like iterative method for solving the approximate penalized optimization problems. We also compare the reconstruction errors of the sparse low-rank approximations computed by our new methods with those obtained using the methods in the earlier paper and several other existing methods for computing sparse low-rank approximations. Numerical examples show that the penalized methods are more robust and produce approximations with factors which have fewer columns and are sparser.

On the error estimates of several Newton-like methods

In this paper, convergence theorems and error estimates for several Newton-like iteration methods under new Kantorovitch–Ostrowski type conditions using the information of higher derivatives at initial point(s) are established. We get not only the convergence speed estimates of iterative sequence { x n }, but also the estimates of f( x n+1 )/ f( x n ).

A new method of reconstruction of thermal conductivity-depth profiles from photoacoustic or photothermal measurements

A new method to reconstruct the continuously inhomogeneous thermal conductivity-depth profiles of solid samples from photoacoustic or photothermal measurements by employing a hybrid of a Newton-like iterative method and a regularization method is presented. Three different error functions are defined in order to seek the proper regularization parameters automatically. Generally, the photoacoustic or photothermal signal is proportional to the temperature variation of the sample's surface, therefore the profile's reconstruction requires only measurements of the photoacoustic or photothermal signal over a sufficiently wide frequency range. The numerical experiments demonstrate that this algorithm is effective and stable for the reconstruction of profiles from the measured data with random noise.

A numerical algorithm for inverse problems in photothermal detection

An iterative algorithm for solving nonlinear inverse problems in the photothermal detection is presented. The basic idea of the algorithm is that the nonlinear inverse problem in the frequency domain is solved by a hybrid of Newton-like iterative method, pseudoinverse technique, and finite difference method. The numerical simulation shows the algorithm is effective for profile reconstruction of the thermal conductivity in the case of errorless data.

Iterative Einschliessungen von Lösungen nichtlinearer Differentialgleichungen durch Newton-ähnliche Iterationsverfahren

In der vorliegenden Arbeit untersuchen wir monoton einschliessende Newton-ähnliche Iterationsverfahren zur näherungsweisen Lösung verschiedener Klassen vonnichtlinearen Differentialgleichungen. Die behandelten Methoden sind auch für nichtkonvexe Nichtlinearitäten anwendbar. Ferner konstruieren wir einschliessende Startnäherungen für diese Verfahren, so dass wir die Existenz der Lösungen der gegebenen Differentialgleichungen sichern können. Die Konvergenz der Verfahren wird auch für den Fall bewiesen, dass die Halbordnungskegel der betrachteten Funktionenräume nicht regulär sind. Es werden vier Beispiele angegeben.

Iterative Einschliessung von Lösungen Nichtlinearer Operatorgleichungen und Bestimmung von Startnäherungen

In the present paper some Newton-like iteration methods are developed to enclose solutions of nonlinear operator equations of the kindF(x)=0. HereF maps a certain subset of a partially ordered vector space into another partially ordered vector space. The obtained results are proved without any special properties of the orderings by taking use of a new kind of a generalized divided difference operator, so that they even hold for nonconvex operators. Furthermore a method for constructing including starting points is presented and two examples are given.

A numerical algorithm for remote sensing of density profiles of a simple ocean model by acoustic pulses

An iterative algorithm for solving nonlinear inverse problems in remote sensing of density profiles of a simple ocean model by using acoustic pulses is developed. Here the adiabatic sound velocity is assumed to be proportional to the inverse square root of the density. The basic idea of this new algorithm is that first, the original pulse problem in the time domain is reduced to a continuous wave problem in the frequency domain and then the nonlinear inverse problem in the frequency domain is solved by a hybrid of a Newton-like iterative method, Backus and Gilbert linear inversion technique, and the finite difference method. This new computational algorithm is tested by numerical simulations with given data from 10 different frequencies and is found to give excellent results. The effects of taking data from various frequency ranges and of the contaminating instrument and ambient noise on the accuracy and efficiency of numerical computation are investigated. It is found that the low frequency data are preferred over the data from the high frequency spectrum. Under favorable conditions, the maximum pointwise numerical error of the density profile ϱ( x) is less than 5% of the total variation of the density profile, ρ tv =|Max ρ( x) − Min ρ( x)|. Better result can be achieved if a large number of data are available and more efforts are made in the numerical computation.