A novel method for robust minimisation of univariate functions with quadratic convergence

We describe a novel method for minimisation of univariate functions which exhibits an essentially quadratic convergence and whose convergence interval is only limited by the existence of near maxima. Minimisation is achieved through a fixed-point iterative algorithm, involving only the first and second-order derivatives, that eliminates the effects of near inflexion points on convergence, as usually observed in other minimisation methods based on the quadratic approximation. Comparative numerical studies against the standard quadratic and Brent's methods demonstrate clearly the high robustness, high precision and convergence rate of the new method, even when a finite difference approximation is used in the evaluation of the second-order derivative.     

A priori and a posteriori error analyses in the study of viscoelastic problems

In this work, the numerical approximation of a viscoelastic problem is studied. A fully discrete scheme is introduced by using the finite element method to approximate the spatial variable and an Euler scheme to discretize time derivatives. Then, two numerical analyses are presented. First, a priori estimates are proved from which the linear convergence of the algorithm is derived under suitable regularity conditions. Secondly, an a posteriori error analysis is provided extending some preliminary results obtained in the study of the heat equation. Upper and lower error bounds are obtained.     

Point estimation and some applications to iterative methods

We consider one of the crucial problems in solving polynomial equations concerning the construction of such initial conditions which provide a safe convergence of simultaneous zero-finding methods. In the first part we deal with the localization of polynomial zeros using disks in the complex plane. These disks are used for the construction of initial inclusion disks which, under suitable conditions, provide the convergence of the Gargantini-Henrici interval method. They also play a key role in the convergence analysis of the fourth order Ehrlich-Aberth method with Newton's correction for the simultaneous approximation of all zeros of a polynomial. For this method we state the initial condition which enables the safe convergence. The initial condition is computationally verifiable since it depends only on initial approximations, which is of practical importance.     

Approximation of a laminated microstructure for a rotationally invariant,double well energy density

Summary. We give error estimates for the approximation of a laminated microstructure which minimizes the energy for a rotationally invariant, double well energy density . We present error estimates for the convergence of the deformation in the convergence of directional derivatives of the deformation in the “twin planes,” the weak convergence of the deformation gradient, the convergence of the microstructure (or Young measure) of the deformation gradients, and the convergence of nonlinear integrals of the deformation gradient. Received July 25, 1995 / Revised version received November 20, 1995     

Error analysis for optimal control problem governed by convection diffusion equations: DG method

In the current paper, we study the convergence properties of the DGFE approximation of optimal control problem governed by convection-diffusion equations. We derive a posteriori error estimates and a priori error estimates for both the states, ad-joint and the control variable approximation. For the optimal control problem, these estimates are apparently not available in the literature.     

An efficient higher order family of root finders

A one parameter family of iterative methods for the simultaneous approximation of simple complex zeros of a polynomial, based on a cubically convergent Hansen–Patrick's family, is studied. We show that the convergence of the basic family of the fourth order can be increased to five and six using Newton's and Halley's corrections, respectively. Since these corrections use the already calculated values, the computational efficiency of the accelerated methods is significantly increased. Further acceleration is achieved by applying the Gauss–Seidel approach (single-step mode). One of the most important problems in solving nonlinear equations, the construction of initial conditions which provide both the guaranteed and fast convergence, is considered for the proposed accelerated family. These conditions are computationally verifiable; they depend only on the polynomial coefficients, its degree and initial approximations, which is of practical importance. Some modifications of the considered family, providing the computation of multiple zeros of polynomials and simple zeros of a wide class of analytic functions, are also studied. Numerical examples demonstrate the convergence properties of the presented family of root-finding methods.     

On the numerical solution of a logarithmic integral equation of the first kind for the Helmholtz equation

Summary We describe a quadrature method for the numerical solution of the logarithmic integral equation of the first kind arising from the single-layer approach to the Dirichlet problem for the two-dimensional Helmholtz equation in smooth domains. We develop an error analysis in a Sobolev space setting and prove fast convergence rates for smooth boundary data.     

Approximation by interpolating variational splines

In this paper we present an approximation problem of parametric curves and surfaces from a Lagrange or Hermite data set. In particular, we study an interpolation problem by minimizing some functional on a Sobolev space that produces the new notion of interpolating variational spline. We carefully establish a convergence result. Some specific cases illustrate the generality of this work.     

A posteriori error estimations of residual type for Signorini's problem

This paper presents an a posteriori error analysis for the linear finite element approximation of the Signorini problem in two space dimensions. A posteriori estimations of residual type are defined and upper and lower bounds of the discretization error are obtained. We perform several numerical experiments in order to compare the convergence of the terms in the error estimator with the discretization error.     

Defect-based local error estimators for splitting methods,with application to Schrödinger equations,Part I: The linear case

We introduce a defect correction principle for exponential operator splitting methods applied to time-dependent linear Schrödinger equations and construct a posteriori local error estimators for the Lie–Trotter and Strang splitting methods. Under natural commutator bounds on the involved operators we prove asymptotical correctness of the local error estimators, and along the way recover the known a priori convergence bounds. Numerical examples illustrate the theoretical local and global error estimates.     

A posteriori error bound methods for the inclusion of polynomial zeros

Using Carstensen's results from 1991 we state a theorem concerning the localization of polynomial zeros and derive two a posteriori error bound methods with the convergence order 3 and 4. These methods possess useful property of inclusion methods to produce disks containing all simple zeros of a polynomial. We establish computationally verifiable initial conditions that guarantee the convergence of these methods. Some computational aspects and the possibility of implementation on parallel computers are considered, including two numerical examples. A comparison of a posteriori error bound methods with the corresponding circular interval methods, regarding the computational costs and sizes of produced inclusion disks, were given.     

Strong predictor–corrector Euler–Maruyama methods for stochastic differential equations with Markovian switching

In this paper numerical methods for solving stochastic differential equations with Markovian switching (SDEwMSs) are developed by pathwise approximation. The proposed family of strong predictor–corrector Euler–Maruyama methods is designed to overcome the propagation of errors during the simulation of an approximate path. This paper not only shows the strong convergence of the numerical solution to the exact solution but also reveals the order of the error under some conditions on the coefficient functions. A natural analogue of the p$p$-stability criterion is studied. Numerical examples are given to illustrate the computational efficiency of the new predictor–corrector Euler–Maruyama approximation.     

Error bounds and estimates for eigenvalues of integral equations

Summary Approximate solutions of the linear integral equation eigenvalue problem can be obtained by the replacement of the integral by a numerical quadrature formula and then collocation to obtain a linear algebraic eigenvalue problem. This method is often called the Nyström method and its convergence was discussed in [7]. In this paper computable error bounds and dominant error terms are derived for the approximation of simple eigenvalues of nonsymmetric kernels.     

Uniform convergence of derivatives of extended Lagrange interpolation

Summary The authors construct some extended interpolation formulae to approximate the derivatives of a function in uniform norm. They prove theorems on uniform convergence and give estimates of pointwise type and of simultaneous approximation.This material is based upon work supported by the Italian Research Council (first and second authors), and by the Ministero dell'Università e della Ricerca Scientifica e Tecnologica (second and third author).     

Newton's method for a class of nonsmooth functions

This paper presents and justifies a Newton iterative process for finding zeros of functions admitting a certain type of approximation. This class includes smooth functions as well as nonsmooth reformulations of variational inequalities. We prove for this method an analogue of the fundamental local convergence theorem of Kantorovich including optimal error bounds.The research reported here was sponsored by the National Science Foundation under Grants CCR-8801489 and CCR-9109345, by the Air Force Systems Command, USAF, under Grants AFOSR-88-0090 and F49620-93-1-0068, by the U. S. Army Research Office under Grant No. DAAL03-92-G-0408, and by the U. S. Army Space and Strategic Defense Command under Contract No. DASG60-91-C-0144. The U. S. Government has certain rights in this material, and is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon.     

Exponential stability of numerical solutions for a class of stochastic age-dependent capital system with Poisson jumps

Recently, numerical solutions of stochastic differential equations have received a great deal of attention. Numerical approximation schemes are invaluable tools for exploring their properties. In this paper, we introduce a class of stochastic age-dependent (vintage) capital system with Poisson jumps. We also give the discrete approximate solution with an implicit Euler scheme in time discretization. Using Gronwall’s lemma and Barkholder-Davis-Gundy’s inequality, some criteria are obtained for the exponential stability of numerical solutions to the stochastic age-dependent capital system with Poisson jumps. It is proved that the numerical approximation solutions converge to the analytic solutions of the equations under the given conditions, where information on the order of approximation is provided. These error bounds imply strong convergence as the timestep tends to zero. A numerical example is used to illustrate the theoretical results.     

Lagrange interpolation for continuous piecewise smooth functions

This note is devoted to Lagrange interpolation for continuous piecewise smooth functions. A new family of interpolatory functions with explicit approximation error bounds is obtained. We apply the theory to the classical Lagrange interpolation.     

BiCGStab,VPAStab and an adaptation to mildly nonlinear systems

The key equations of BiCGStab are summarised to show its connections with Padé and vector-Padé approximation. These considerations lead naturally to stabilised vector-Padé approximation of a vector-valued function (VPAStab), and an algorithm for the acceleration of convergence of a linearly generated sequence of vectors. A generalisation of this algorithm for the acceleration of convergence of a nonlinearly generated system is proposed here, and comparative numerical results are given.     

A collocation method for high-frequency scattering by convex polygons

We consider the problem of scattering of a time-harmonic acoustic incident plane wave by a sound soft convex polygon. For standard boundary or finite element methods, with a piecewise polynomial approximation space, the computational cost required to achieve a prescribed level of accuracy grows linearly with respect to the frequency of the incident wave. Recently Chandler–Wilde and Langdon proposed a novel Galerkin boundary element method for this problem for which, by incorporating the products of plane wave basis functions with piecewise polynomials supported on a graded mesh into the approximation space, they were able to demonstrate that the number of degrees of freedom required to achieve a prescribed level of accuracy grows only logarithmically with respect to the frequency. Here we propose a related collocation method, using the same approximation space, for which we demonstrate via numerical experiments a convergence rate identical to that achieved with the Galerkin scheme, but with a substantially reduced computational cost.     

Multivariate approximation by PDE splines

This work deals with an approximation method for multivariate functions from data constituted by a given data point set and a partial differential equation (PDE). The solution of our problem is called a PDE spline. We establish a variational characterization of the PDE spline and a convergence result of it to the function which the data are obtained. We estimate the order of the approximation error and finally, we present an example to illustrate the fitting method.