Uniform convergent monotone iterates for semilinear singularly perturbed parabolic problems

This paper deals with discrete monotone iterative methods for solving semilinear singularly perturbed parabolic problems. Monotone sequences, based on the accelerated monotone iterative method, are constructed for a nonlinear difference scheme which approximates the semilinear parabolic problem. This monotone convergence leads to the existence-uniqueness theorem. An analysis of uniform convergence of the monotone iterative method to the solutions of the nonlinear difference scheme and continuous problem is given. Numerical experiments are presented.     

Monotone iterates with quadratic convergence rate for solving weighted average approximations to semilinear parabolic problems

This paper deals with discrete monotone iterative methods for solving semilinear singularly perturbed parabolic problems. Monotone sequences, based on the accelerated monotone iterative method, are constructed for a nonlinear difference scheme which approximates the semilinear parabolic problem. This monotone convergence leads to the existence-uniqueness theorem. An analysis of convergence of the monotone iterative method to the solutions of the nonlinear difference scheme is given. Numerical experiments are presented.     

Uniform quadratic convergence of monotone iterates for nonlinear singularly perturbed parabolic problems

This paper deals with a monotone iterative method for solving nonlinear singularly perturbed parabolic problems. Monotone sequences, based on the method of upper and lower solutions, are constructed for a nonlinear difference scheme which approximates the nonlinear parabolic problem. This monotone convergence leads to the existence-uniqueness theorem. The monotone sequences possess quadratic convergence rate. An analysis of uniform convergence of the monotone iterative method to the solutions of the nonlinear difference scheme and to the continuous problem is given. Numerical experiments are presented.     

Thermally coupled quasi-Newtonian flows: Analysis and computation

In this paper, we consider an incompressible quasi-Newtonian flow with a temperature dependent viscosity obeying a power law, and the thermal balance includes viscous heating. Some mathematical results such as the existence and uniqueness are established, finite element approximation based on an iterative solution scheme is proposed, and convergence analysis is presented.     

Iterative methods for a forward-backward heat equation in two-dimension

A finite difference method is introduced to solve the forward-backward heat equation in two space dimensions. In this procedure, the backward and forward difference scheme in two subdomains and a coarse-mesh second-order central difference scheme at the middle interface are used. Maximum norm error estimate for the procedure is derived. Then an iterative method based on domain decomposition is presented for the numerical scheme and the convergence of the given method is established. Then numerical experiments are presented to support the theoretical analysis.     

A fast iterative method to find the matrix geometric mean of two HPD matrices

The purpose of this research is to present a novel scheme based on a quick iterative scheme for calculating the matrix geometric mean of two Hermitian positive definite (HPD) matrices. To do this, an iterative scheme with global convergence is constructed for the sign function using a novel three‐step root‐solver. It is proved that the new scheme is convergent and shown to have global convergence behavior for this target, when square matrices having no pure imaginary eigenvalues. Next, the constructed scheme is used and extended through a well‐known identity for the calculation of the matrix geometric mean of two HPD matrices. Ultimately, several experiments are collected to show its usefulness.     

Quadrature Based Optimal Iterative Methods with Applications in High-Precision Computing

We present a simple yet effective and applicable scheme, based on quadrature, for constructing optimal iterative methods. According to the, still unproved, Kung-Traub conjecture an optimal iterative method based on $n+1$ evaluations could achieve a maximum convergence order of $2^n$. Through quadrature, we develop optimal iterative methods of orders four and eight. The scheme can further be applied to develop iterative methods of even higher orders. Computational results demonstrate that the developed methods are efficient as compared with many well known methods.     

Convergence of Three-Step Taylor Galerkin Finite Element Scheme Based Monotone Schwarz Iterative Method for Singularly Perturbed Differential-Difference Equation

Monotone Schwarz iterative methods for parabolic partial differential equations are well known for their advantage of eliminating the search for an initial solution. In this article, we propose a monotone Schwarz iterative method for singularly perturbed parabolic retarded differential-difference equations based on a three-step Taylor Galerkin finite element scheme. The stability and ε-uniform convergence of the three-step Taylor Galerkin finite element method have been discussed. Further, by using maximum principle and induction hypothesis, the convergence of the proposed monotone Schwarz iterative method has been established.     

A Regularized Iterative Scheme for Solving Nonlinear Ill-Posed Problems

In this article, we consider a regularized iterative scheme for solving nonlinear ill-posed problems. The convergence analysis and error estimates are derived by choosing the regularization parameter according to both a priori and a posteriori methods. The iterative scheme is stopped using an a posteriori stopping rule, and we prove that the scheme converges to the solution of the well-known Lavrentiev scheme. The salient features of the proposed scheme are: (i) convergence and error estimate analysis require only weaker assumptions compared to standard assumptions followed in literature, and (ii) consideration of an adaptive a posteriori stopping rule and a parameter choice strategy that gives the same convergence rate as that of an a priori method without using the smallness assumption, the source condition. The above features are very useful from theory and application points of view. We also supply the numerical results to illustrate that the method is adaptable. Further, we compare the numerical result of the proposed method with the standard approach to demonstrate that our scheme is stable and achieves good computational output.     

Analysis of a symplectic difference scheme for a coupled nonlinear Schrödinger system

In general, proofs of convergence and stability are difficult for symplectic schemes of nonlinear equations. In this paper, a symplectic difference scheme is proposed for an initial-boundary value problem of a coupled nonlinear Schrödinger system. An important lemma and an induction argument are used to prove the unique solvability, convergence and stability of numerical solutions. An iterative algorithm is also proposed for the symplectic scheme and its convergence is proved. Numerical examples show the efficiency of the symplectic scheme and the correction of our numerical analysis.     

Convergence and spectral analysis of the Frontini-Sormani family of multipoint third order methods from quadrature rule

Point of attraction theory is an important tool to analyze the local convergence of iterative methods for solving systems of nonlinear equations. In this work, we prove a generalized form of Ortega-Rheinbolt result based on point of attraction theory. The new result guarantees that the solution of the nonlinear system is a point of attraction of iterative scheme, especially multipoint iterations. We then apply it to study the attraction theorem of the Frontini-Sormani family of multipoint third order methods from Quadrature Rule. Error estimates are given and compared with existing ones. We also obtain the radius of convergence of the special members of the family. Two numerical examples are provided to illustrate the theory. Further, a spectral analysis of the Discrete Fourier Transform of the numerical errors is conducted in order to find the best method of the family. The convergence and the spectral analysis of a multistep version of one of the special member of the family are studied.     

Algorithm for forming derivative-free optimal methods

We develop a simple yet effective and applicable scheme for constructing derivative free optimal iterative methods, consisting of one parameter, for solving nonlinear equations. According to the, still unproved, Kung-Traub conjecture an optimal iterative method based on k+1 evaluations could achieve a maximum convergence order of $2^{k}$ . Through the scheme, we construct derivative free optimal iterative methods of orders two, four and eight which request evaluations of two, three and four functions, respectively. The scheme can be further applied to develop iterative methods of even higher orders. An optimal value of the free-parameter is obtained through optimization and this optimal value is applied adaptively to enhance the convergence order without increasing the functional evaluations. Computational results demonstrate that the developed methods are efficient and robust as compared with many well known methods.     

ITERATIVE METHODS FOR THE FORWARD-BACKWARD HEAT EQUATION

In this paper we propose the finite difference method for the forward-backward heatequation.We use a coarse-mesh second-order central difference scheme at the middleline mesh points and derive the error estimate.Then we discuss the iterative methodbased on the domain decomposition for our scheme and derive the bounds for the rates ofconvergence.Finally we present some numerical experiments to support our analysis.     

Tangential frequency filtering decompositions for symmetric matrices

Summary. The tangential frequency filtering decomposition (TFFD) is introduced. The convergence theory of an iterative scheme based on the TFFD for symmetric matrices is the focus of this paper. The existence of the TFFD and the convergence of the induced iterative algorithm is shown for symmetric and positive definite matrices. Convergence rates independent of the number of unknowns are proven for a smaller class of matrices. Using this framework, the convergence independent of the number of unknowns is shown for Wittum's frequency filtering decomposition. Some characteristic properties of the TFFD are illustrated and results of several numerical experiments are presented. Received April 1, 1996 / Revised version July 4, 1996     

Landweber iterative methods for angle-limited image reconstruction

We introduce a general iterative scheme for angle-limited image reconstruction based on Landwebet＇s method. We derive a representation formula for this scheme and consequently establish its convergence conditions. Our results suggest certain relaxation strategies for an accelerated convergence for angle-limited image reconstruction in L^2-norm comparing with alternative projection methods. The convolution-backprojection algorithm is given for this iterative process.     

Convergence of a nonlinear finite difference scheme for the Kuramoto–Tsuzuki equation

A nonlinear finite difference scheme is studied for solving the Kuramoto–Tsuzuki equation. Because the maximum estimate of the numerical solution can not be obtained directly, it is difficult to prove the stability and convergence of the scheme. In this paper, we introduce the Brouwer-type fixed point theorem and induction argument to prove the unique existence and convergence of the nonlinear scheme. An iterative algorithm is proposed for solving the nonlinear scheme, and its convergence is proved. Based on the iterative algorithm, some linearized schemes are presented. Numerical examples are carried out to verify the correction of the theory analysis. The extrapolation technique is applied to improve the accuracy of the schemes, and some interesting results are obtained.     

A numerical method for pricing European options with proportional transaction costs

In the paper, we propose a numerical technique based on a finite difference scheme in space and an implicit time-stepping scheme for solving the Hamilton–Jacobi–Bellman (HJB) equation arising from the penalty formulation of the valuation of European options with proportional transaction costs. We show that the approximate solution from the numerical scheme converges to the viscosity solution of the HJB equation as the mesh sizes in space and time approach zero. We also propose an iterative scheme for solving the nonlinear algebraic system arising from the discretization and establish a convergence theory for the iterative scheme. Numerical experiments are presented to demonstrate the robustness and accuracy of the method.     

A spectral algorithm for large-scale systems of nonlinear monotone equations

A derivative-free iterative scheme that uses the residual vector as search direction for solving large-scale systems of nonlinear monotone equations is presented. It is closely related to two recently proposed spectral residual methods for nonlinear systems which use a nonmonotone line-search globalization strategy and a step-size based on the Barzilai-Borwein choice. The global convergence analysis is presented. In order to study the numerical behavior of the algorithm, it is included an extensive series of numerical experiments. Our computational experiments show that the new algorithm is computationally efficient.     

Strong convergence theorems of iterative scheme based on the extragradient method for mixed equilibrium problems and fixed point problems

In this paper, a new iterative scheme based on the extragradient method for finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of a family of finitely nonexpansive mappings and the set of solutions of the variational inequality for a monotone, Lipschitz continuous mapping is proposed. A strong convergence theorem for this iterative scheme in Hilbert spaces is established. Applications to optimization problems are given.     

Convergence of hybrid viscosity and steepest-descent methods for pseudocontractive mappings and nonlinear Hammerstein equations

In this article, we first introduce an iterative method based on the hybrid viscosity approximation method and the hybrid steepest-descent method for finding a fixed point of a Lipschitz pseudocontractive mapping (assuming existence) and prove that our proposed scheme has strong convergence under some mild conditions imposed on algorithm parameters in real Hilbert spaces. Next, we introduce a new iterative method for a solution of a nonlinear integral equation of Hammerstein type and obtain strong convergence in real Hilbert spaces. Our results presented in this article generalize and extend the corresponding results on Lipschitz pseudocontractive mapping and nonlinear integral equation of Hammerstein type reported by some authors recently. We compare our iterative scheme numerically with other iterative scheme for solving non-linear integral equation of Hammerstein type to verify the efficiency and implementation of our new method.