Rational approximation to trigonometric operators

We consider the approximation of trigonometric operator functions that arise in the numerical solution of wave equations by trigonometric integrators. It is well known that Krylov subspace methods for matrix functions without exponential decay show superlinear convergence behavior if the number of steps is larger than the norm of the operator. Thus, Krylov approximations may fail to converge for unbounded operators. In this paper, we propose and analyze a rational Krylov subspace method which converges not only for finite element or finite difference approximations to differential operators but even for abstract, unbounded operators. In contrast to standard Krylov methods, the convergence will be independent of the norm of the operator and thus of its spatial discretization. We will discuss efficient implementations for finite element discretizations and illustrate our analysis with numerical experiments. AMS subject classification (2000)  65F10, 65L60, 65M60, 65N22     

An effective first order reliability method based on Barzilai–Borwein step

In nonlinear problems, the Hasofer–Lind–Rackwitz–Fiessler algorithm of the first order reliability method sometimes is puzzled by its non-convergence. A new Hasofer–Lind–Rackwitz–Fiessler algorithm incorporating Barzilai–Borwein step is investigated in this paper to speed up the rate of convergence and performs in a stable manner. The algorithm is essentially established on the basis of the global Barzilai–Borwein gradient method, which is dealt with two stages. The first stage, implemented by the traditional steepest descent method with specific decayed step sizes, prepares a good initial point for the global Barzilai–Borwein gradient algorithm in the second stage, which takes the merit function as the objective to locate the most probable failure point. The efficiency and convergence of the proposed method and some other reliability analysis methods are presented and discussed in details by several numerical examples. It is found that the proposed method is stable and very efficient in the nonlinear problems except those super nonlinear ones, even more accurate than the descent direction method with step sizes following the fixed exponential decay strategy.     

Summation of Some Trigonometric and Schlömilch Series

In this paper we consider trigonometric series in terms of the Riemann zeta function and related functions of reciprocal powers. The obtained closed form formulas we apply to the evaluation of the Riemann zeta function and related functions of reciprocal powers. One can establish recursive relations for them and relations between any two of those functions. These closed formulas enable us also to find sums of some Schlömilch series. We give an example which shows how the convergence of a trigonometric series can be accelerated by applying Krylov's method and our formula (7).     

An affine-scaling interior-point CBB method for box-constrained optimization

We develop an affine-scaling algorithm for box-constrained optimization which has the property that each iterate is a scaled cyclic Barzilai–Borwein (CBB) gradient iterate that lies in the interior of the feasible set. Global convergence is established for a nonmonotone line search, while there is local R-linear convergence at a nondegenerate local minimizer where the second-order sufficient optimality conditions are satisfied. Numerical experiments show that the convergence speed is insensitive to problem conditioning. The algorithm is particularly well suited for image restoration problems which arise in positron emission tomography where the cost function can be infinite on the boundary of the feasible set. This material is based upon work supported by the National Science Foundation under Grants 0203270, 0619080, and 0620286.     

An adaptive nonmonotone global Barzilai–Borwein gradient method for unconstrained optimization

The Barzilai–Borwein (BB) gradient method has received many studies due to its simplicity and numerical efficiency. By incorporating a nonmonotone line search, Raydan (SIAM J Optim. 1997;7:26–33) has successfully extended the BB gradient method for solving general unconstrained optimization problems so that it is competitive with conjugate gradient methods. However, the numerical results reported by Raydan are poor for very ill-conditioned problems because the effect of the degree of nonmonotonicity may be noticeable. In this paper, we focus more on the nonmonotone line search technique used in the global Barzilai–Borwein (GBB) gradient method. We improve the performance of the GBB gradient method by proposing an adaptive nonmonotone line search based on the morphology of the objective function. We also prove the global convergence and the R-linear convergence rate of the proposed method under reasonable assumptions. Finally, we give some numerical experiments made on a set of unconstrained optimization test problems of the CUTEr collection. The results show the efficiency of the proposed method in the sense of the performance profile introduced (Math Program. 2002;91:201–213) by Dolan and Moré.     

A Barzilai–Borwein type method for stochastic linear complementarity problems

We consider the expected residual minimization (ERM) formulation of stochastic linear complementarity problem (SLCP). By employing the Barzilai–Borwein (BB) stepsize and active set strategy, we present a BB type method for solving the ERM problem. The global convergence of the proposed method is proved under mild conditions. Preliminary numerical results show that the method is promising.     

Epigraphical nesting: A unifying theory for the convergence of algorithms

In an earlier paper, the authors introduced epigraphical nesting of objective functions as a means to characterize the convergence of global optimization algorithms. Epigraphical nesting of objective functions may be looked upon as a relaxation of epigraphical convergence of objective functions, whereby one ensures that epigraphs of the approximations contain asymptotically the epigraph of the objective function of the original optimization problem. In this paper, we show that, for algorithms which seek only a stationary point, convergence can be assured by objective function approximations whose directional derivatives attain an epigraphical nesting property. We demonstrate that epigraphical nesting provides a unifying thread that ties together a number of different algorithms, including those for the solution of variational inequalities and smooth as well as nonsmooth optimization. We show that the Newton method and its variants for unconstrained optimization, successive quadratic programming methods for constrained optimization, and proximal point algorithms (deterministic and stochastic) all construct approximations in which the directional derivatives attain an epigraphical nesting property, even though the approximations themselves fail to attain such a property.This work was supported in part by Grants NSF-DDM-89-10046 and NSF-DDM-91-14352 from the National Science Foundation.     

一类常微分方程的非多项式样条函数逼近解

本文提出一种新的数值方法来获得三阶带障碍边值问题的常微分方程的数值解,这类方法的基函数包括三角函数、指数函数、多项式函数。本文将给出一数值例子说明这种方法优于其他差分法、配置法、多项式样条函数法。     

Hybrid spectral gradient method for the unconstrained minimization problem

We present a hybrid algorithm that combines a genetic algorithm with the Barzilai–Borwein gradient method. Under specific assumptions the new method guarantees the convergence to a stationary point of a continuously differentiable function, from any arbitrary initial point. Our preliminary numerical results indicate that the new methodology finds efficiently and frequently the global minimum, in comparison with the globalized Barzilai–Borwein method and the genetic algorithm of the Toolbox of Genetic Algorithms of MatLab.      

An efficient gradient method with approximate optimal stepsize for the strictly convex quadratic minimization problem

In this paper, a new type of stepsize, approximate optimal stepsize, for gradient method is introduced to interpret the Barzilai–Borwein (BB) method, and an efficient gradient method with an approximate optimal stepsize for the strictly convex quadratic minimization problem is presented. Based on a multi-step quasi-Newton condition, we construct a new quadratic approximation model to generate an approximate optimal stepsize. We then use the two well-known BB stepsizes to truncate it for improving numerical effects and treat the resulted approximate optimal stepsize as the new stepsize for gradient method. We establish the global convergence and R-linear convergence of the proposed method. Numerical results show that the proposed method outperforms some well-known gradient methods.     

Best approximation and moduli of smoothness: Computation and equivalence theorems

In this paper we investigate the trigonometric series with the β-general monotone coefficients. First, we study the uniform convergence criterion. The estimates of best approximations and moduli of smoothness of the series in uniform metrics are obtained in terms of coefficients. These results imply several important relations between moduli of smoothness of different orders (in particular, Marchaud-type inequality) and best approximations.     

Postprocessing and higher order convergence for the mixed finite element approximations of the eigenvalue problem

In this paper, we propose a method to improve the convergence rate of the lowest order Raviart-Thomas mixed finite element approximations for the second order elliptic eigenvalue problem. Here, we prove a supercloseness result for the eigenfunction approximations and use a type of finite element postprocessing operator to construct an auxiliary source problem. Then solving the auxiliary additional source problem on an augmented mixed finite element space constructed by refining the mesh or by using the same mesh but increasing the order of corresponding mixed finite element space, we can increase the convergence order of the eigenpair approximation. This postprocessing method costs less computation than solving the eigenvalue problem on the finer mesh directly. Some numerical results are used to confirm the theoretical analysis.     

Parametric Disjunctive Programming: One-Sided Differentiability of the Value Function

We consider a countable family of one-parameter convex programs and give sufficient conditions for the one-sided differentiability of its optimal value function. The analysis is based on the Borwein dual problem for a family of convex programs (a convex disjunctive program). We give conditions that assure stability of the situation of perfect duality in the Borwein theory.For the reader's convenience, we start with a review of duality results for families of convex programs. A parametric family of dual problems is introduced that contains the dual problems of Balas and Borwein as special cases. In addition, a vector optimization problem is defined as a dual problem. This generalizes a result by Helbig about families of linear programs.     

On the summation of Schlömilch's series

ABSTRACT

Schlömilch's series is named after the German mathematician Oscar Xavier Schlömilch, who derived it in 1857 as a Fourier series type expansion in terms of the Bessel function of the first kind. However, except for Bessel functions, here we consider an expansion in terms of Struve functions or Bessel and Struve integrals as well. The method for obtaining a sum of Schlömilch's series in terms of the Bessel or Struve functions is based on the summation of trigonometric series, which can be represented in terms of the Riemann zeta and related functions of reciprocal powers and in certain cases can be brought in the closed form, meaning that the infinite series are represented by finite sums. By using Krylov's method we obtain the convergence acceleration of the trigonometric series.     

On the convergence of infinite compositions of entire functions

Some trigonometric functions can be expressed by the infinite composition of polynomials. First we need to consider the convergence problem to study infinite compositions of functions. This article discusses the convergence of infinite compositions of entire functions whose constant terms are not necessarily zero.     

Convergence of weighted averages of relaxed projections

The convergence of the algorithm for solving convex feasibility problem is studied by the method of sequential averaged and relaxed projections. Some results of H. H. Bauschke and J. M. Borwein are generalized by introducing new methods. Examples illustrating these generalizations are given.     

A new adaptive Barzilai and Borwein method for unconstrained optimization

In this paper we view the Barzilai and Borwein (BB) method from a new angle, and present a new adaptive Barzilai and Borwein (NABB) method with a nonmonotone line search for general unconstrained optimization. In the proposed method, the scalar approximation to the Hessian matrix is updated by the Broyden class formula to generate an adaptive stepsize. It is remarkable that the new stepsize is chosen adaptively in the interval which contains the two well-known BB stepsizes. Moreover, for the negative curvature direction, a strategy for the choice of the stepsize is designed to accelerate the convergence rate of the NABB method. Furthermore, we apply the NABB method without any line search to strictly convex quadratic minimization. The numerical experiments show the NABB method is very promising.     

On the Barzilai and Borwein choice of steplength for the gradient method

In a recent paper, Barzilai and Borwein presented a new choiceof steplength for the gradient method. Their choice does notguarantee descent in the objective function and greatly speedsup the convergence of the method. They presented a convergenceanalysis of their method only in the two-dimensional quadraticcase. We establish the convergence of the Barzilai and Borweingradient method when applied to the minimization of a strictlyconvex quadratic function of any number of variables.     

Classical solution of a one-dimensional parabolic boundary value problem with nonlinear boundary conditions and moving boundary

We consider a nonlinear parabolic boundary value problem of the Stefan type with one space variable, which generalizes the model of hydride formation under constant conditions. We suggest a grid method for constructing approximations to the unknown boundary and to the concentration distribution. We prove the uniform convergence of the interpolation approximations to a classical solution of the boundary value problem. (The boundary is smooth, and the concentration distribution has the necessary derivatives.) Thus, we prove the theorem on the existence of a solution, and the proof is given in constructive form: the suggested convergent grid method can be used for numerical experiments.     

Hermite spectral and pseudospectral methods for numerical differentiation

A numerical differentiation problem for a given function with noisy data is discussed in this paper. A mollification method based on spanned by Hermite functions is proposed and the mollification parameter is chosen by a discrepancy principle. The convergence estimates of the derivatives are obtained. To get a practical approach, we also derive corresponding results for pseudospectral (Hermite-Gauss interpolation) approximations. Numerical examples are given to show the efficiency of the method.