Stabilization theory of iterative operator-splitting methods

In this paper we present a stabilization theory of iterative operator-splitting methods for linear and nonlinear differential equations. Continuous formulation is described and also the stability for linear and nonlinear cases. We apply linearization techniques to adduce proof of the linear theory. Iterative methods are applied to linearize and couple operator equations. A general theory is derived for linear and nonlinear iterative-splitting methods. Test examples verify the underlying theoretical results.     

Explicit group versus implicit line iterative methods

In this paper, a new explicit 4 point and 9 point block overrelaxation scheme for the numerical solution of the sparse linear systems derived from the discretisation of self-adjoint elliptic partial differential equations by finite difference/element techniques. A comparison with the implicit 1-line and 2‐line block S.O.R. schemes for the model problem clearly demonstrates the new techniques to be computationally superior.     

The numerov alternating group explicit (numage) method:

In this paper, more accurate solutions are obtained by applying the Numerov formula to the Alternating Group Explicit (AGE) method in the solution of linear and non-linear two point boundary value problems.     

A new iterative method for large sparse saddle point problems

In this short note, the convergence of a new iterative method for the Saddle Point Problem is presented.     

The alternating group explicit (AGE) iterative method for solving a Ladyzhenskaya model for stationary incompressible viscous flow

In this paper, the alternating group explicit (AGE) iterative method is applied to a nonlinear fourth-order PDE describing the flow of an incompressible fluid. This equation is a Ladyzhenskaya equation. The AGE method is shown to be extremely powerful and flexible and affords its users many advantages. Computational results are obtained to demonstrate the applicability of the method on some problems with known solutions. This paper demonstrates that the AGE method can be implemented to approximate solutions efficiently to the Navier–Stokes equations and the Ladyzhenskaya equations. Problems with a known solution are considered to test the method and to compare the computed results with the exact values. Streamfunction contours and some plots are displayed showing the main features of the solution.     

New iterative method for solving non-linear equations with fourth-order convergence

In this work, we introduce an extension of the classical Newton's method for solving non-linear equations. This method is free from second derivative. Similar to Newton's method, the proposed method will only require function and first derivative evaluations. The order of convergence of the introduced method for a simple root is four. Numerical results show that the new method can be of practical interest.     

The iterative alternating decomposition explicit (iade) method to solve second order parabolic equations with periodic boundary conditions

In [1] the Iterative Alternating Decomposition Explicit (IADE) method was introduced for the x solution of second order parabolic equations in one-space dimension with Dirichlet boundary conditions. Its versatility as a fast, convergent, stable and highly accurate method is now extended to the parabolic equation with periodic boundary conditions. The new method is shown to retain its high order of accuracy and the special structure of the constituent decomposed matrices reduces substantially its storage requirement.     

Higher-order simultaneous methods for the determination of polynomial multiple zeros

Starting from Laguerre's method and using Newton's and Halley's corrections for a multiple zero, new simultaneous methods of Laguerre's type for finding multiple (real or complex) zeros of polynomials are constructed. The convergence order of the proposed methods is five and six, respectively. By applying the Gauss–Seidel approach, these methods are further accelerated. The lower bounds of the R-order of convergence of the improved (single-step) methods are derived. Faster convergence of all proposed methods is attained with negligible number of additional operations, which provides a high computational efficiency of these methods. A detailed convergence analysis and numerical results are given.     

New point and group iterative methods for solving three-dimensional elliptic equations

In this paper we construct a rotated finite differnce formula in 3-dimensions analogous to the rotated finite difference formula in 2-dimensions (Dahlquist and Bjorck, 1974; Vichnevetsky, 1981). Further, an explicit 2×2×2 de-coupled group (EDG) iterative method for the solution of problems arising in 3-dimensional elliptic partial differential equations is developed.Performance results for the two algorithms are presented and a comparison with the 7 point S.O.R. scheme in 3-dimensions confirm the new methods to be computationally superior.     

On the convergence of iterative methods for stabilized saddle point problems

The convergence of the inexact Uzawa method for stabilized saddle point problems was analysed in a recent paper by Cao, Evans and Qin. We show that this method converges under conditions weaker than those stated in their paper.     

The iterative transformation method: numerical solution of one-dimensional parabolic moving boundary problems

The main contribution of this paper is the application of the iterative transformation method to the numerical solution of the sequence of free boundary problems obtained from one-dimensional parabolic moving boundary problems via the implicit Euler's method. The combination of the two methods represents a numerical approach to the solution of those problems. Three parabolic moving boundary problems, two with explicit and one with implicit moving boundary conditions, are solved in order to test the validity of the proposed approach. As far as the moving boundary position is concerned the obtained numerical results are found to be in agreement with those available in literature.     

Numerical solution of functional differential equations: a Green's function-based iterative approach

In this article, a recently introduced iterative method based on Green's functions and fixed-point iteration schemes is presented for the approximate solutions of delay and functional differential equations. The approach is especially suited for handling boundary value problems (BVPs). The algorithm is illustrated through a number of examples that confirm the high accuracy and efficiency of the strategy. The results of the test examples show excellent agreement with exact solutions and outperforms other existing numerical iterative schemes.     

On the local convergence of secant-type methods

In this article, we carry out a local convergence study for Secant-type methods. Our goal is to enlarge the radius of convergence, without increasing the necessary hypothesis. Finally, some numerical tests and comparisons with early results are analyzed.     

On the optimum pyramidal‐horn design methods

In this article the optimum pyramidal‐horn design methods are considered. The validity of comparing the different methods with respect to design accuracy is examined in light of the fundamental accuracy limits of Schelkonoff's horn‐gain formulas. © 2006 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2006.     

An investigation of Newton-Sketch and subsampled Newton methods

ABSTRACT

Sketching, a dimensionality reduction technique, has received much attention in the statistics community. In this paper, we study sketching in the context of Newton's method for solving finite-sum optimization problems in which the number of variables and data points are both large. We study two forms of sketching that perform dimensionality reduction in data space: Hessian subsampling and randomized Hadamard transformations. Each has its own advantages, and their relative tradeoffs have not been investigated in the optimization literature. Our study focuses on practical versions of the two methods in which the resulting linear systems of equations are solved approximately, at every iteration, using an iterative solver. The advantages of using the conjugate gradient method vs. a stochastic gradient iteration are revealed through a set of numerical experiments, and a complexity analysis of the Hessian subsampling method is presented.     

On a third order family of methods for solving nonlinear equations

In this article we present a third-order family of methods for solving nonlinear equations. Some well-known methods belong to our family, for example Halley's method, method (24) from [M. Basto, V. Semiao, and F.L. Calheiros, A new iterative method to compute nonlinear equations, Appl. Math. Comput. 173 (2006), pp. 468–483] and the super-Halley method from [J.M. Gutierrez and M.A. Hernandez, An acceleration of Newton's method: Super-Halley method, Appl. Math. Comput. 117 (2001), pp. 223–239]. The convergence analysis shows the third order of our family. We also give sufficient conditions for the stopping inequality |x _n+1?α|≤|x _n+1?x _n | for this family. Comparison of the family members shows that there are no significant differences between them. Several examples are presented and compared.     

A family of Newton-type methods for solving nonlinear equations

A family of Newton-type methods free from second and higher order derivatives for solving nonlinear equations is presented. The order of the convergence of this family depends on a function. Under a condition on this function this family converge cubically and by imposing one condition more on this function one can obtain methods of order four. It has been shown that this family covers many of the available iterative methods. From this family two new iterative methods are obtained. Numerical experiments are also included.     

An iterative method for suboptimal control of a class of nonlinear time-delayed systems

This work presents an approximate solution method for the infinite-horizon nonlinear time-delay optimal control problem. A variational iteration method (VIM) is applied to design feedforward and feedback optimal controllers. By using the VIM, the original optimal control is transformed into a sequence of nonhomogeneous linear two-point boundary value problems (TPBVPs). The existence and uniqueness of the optimal control law are proved. The optimal control law obtained consists of an accurate linear feedback term and a nonlinear compensation term which is the limit of an adjoint vector sequence. The feedback term is determined by solving Riccati matrix differential equation. By using the finite-step iteration of a nonlinear compensation sequence, we can obtain a suboptimal control law. Simulation results demonstrate the validity and applicability of the VIM.     

Computationally efficient simultaneous policy update algorithm for nonlinear H∞ state feedback control with Galerkin's method

The main bottleneck for the application of H_∞ control theory on practical nonlinear systems is the need to solve the Hamilton–Jacobi–Isaacs (HJI) equation. The HJI equation is a nonlinear partial differential equation (PDE) that has proven to be impossible to solve analytically, even the approximate solution is still difficult to obtain. In this paper, we propose a simultaneous policy update algorithm (SPUA), in which the nonlinear HJI equation is solved by iteratively solving a sequence of Lyapunov function equations that are linear PDEs. By constructing a fixed point equation, the convergence of the SPUA is established rigorously by proving that it is essentially a Newton's iteration method for finding the fixed point. Subsequently, a computationally efficient SPUA (CESPUA) based on Galerkin's method, is developed to solve Lyapunov function equations in each iterative step of SPUA. The CESPUA is simple for implementation because only one iterative loop is included. Through the simulation studies on three examples, the results demonstrate that the proposed CESPUA is valid and efficient. Copyright © 2012 John Wiley & Sons, Ltd.     

The solution of elliptic partial differential equations by a new block over-relaxation technique

In this paper, a new explicit 4-pint block over-relaxation scheme is presented for the numerical solution of the sparse linear systems derived from the discretization of self-adjoint elliptic partial differential equations. A comparison with the implicit line and 2-line block SOR schemes for the model problem shows the new technique to be competitive