带参数共轭梯度法簇的全局收敛性

共轭梯度法是最优化中最常用的方法之一,广泛地应用于求解大规模优化问题,其中参数β_k的不同选取可以构成不同的共轭梯度法.给出了一类含有三个参数的共轭梯度算法,这种算法能够在给定的条件下证明选定的β_k在每一步都能产生一个下降方向,同时在强Wolfe线搜索下,这种算法具有全局收敛性.     

A Quadratic Hybridization of Polak–Ribière–Polyak and Fletcher–Reeves Conjugate Gradient Methods

In order to take advantage of the attractive features of Polak–Ribière–Polyak and Fletcher–Reeves conjugate gradient methods, two hybridizations of these methods are suggested, using a quadratic relaxation of a hybrid conjugate gradient parameter proposed by Gilbert and Nocedal. In the suggested methods, the hybridization parameter is computed based on a conjugacy condition. Under proper conditions, it is shown that the proposed methods are globally convergent for general objective functions. Numerical results are reported; they demonstrate the efficiency of one of the proposed methods in the sense of the performance profile introduced by Dolan and Moré.     

Self-adaptive operator splitting methods for monotone variational inequalities

Summary. Solving a variational inequality problem VI(Ω,F) is equivalent to finding a solution of a system of nonsmooth equations (a hard problem). The Peaceman-Rachford and /or Douglas-Rachford operator splitting methods are advantageous when they are applied to solve variational inequality problems, because they solve the original problem via solving a series of systems of nonlinear smooth equations (a series of easy problems). Although the solution of VI(Ω,F) is invariant under multiplying F by some positive scalar β, yet the numerical experiment has shown that the number of iterations depends significantly on the positive parameter β which is a constant in the original operator splitting methods. In general, it is difficult to choose a proper parameter β for individual problems. In this paper, we present a modified operator splitting method which adjusts the scalar parameter automatically per iteration based on the message of the iterates. Exact and inexact forms of the modified method with self-adaptive variable parameter are suggested and proved to be convergent under mild assumptions. Finally, preliminary numerical tests show that the self-adaptive adjustment rule is proper and necessary in practice.     

Old and new parameter choice rules for discrete ill-posed problems

Linear discrete ill-posed problems are difficult to solve numerically because their solution is very sensitive to perturbations, which may stem from errors in the data and from round-off errors introduced during the solution process. The computation of a meaningful approximate solution requires that the given problem be replaced by a nearby problem that is less sensitive to disturbances. This replacement is known as regularization. A regularization parameter determines how much the regularized problem differs from the original one. The proper choice of this parameter is important for the quality of the computed solution. This paper studies the performance of known and new approaches to choosing a suitable value of the regularization parameter for the truncated singular value decomposition method and for the LSQR iterative Krylov subspace method in the situation when no accurate estimate of the norm of the error in the data is available. The regularization parameter choice rules considered include several L-curve methods, Regińska’s method and a modification thereof, extrapolation methods, the quasi-optimality criterion, rules designed for use with LSQR, as well as hybrid methods.     

Interface procedures for finite difference approximations of the advection-diffusion equation

We investigate several existing interface procedures for finite difference methods applied to advection-diffusion problems. The accuracy, stiffness and reflecting properties of various interface procedures are investigated.The analysis and numerical experiments show that there are only minor differences between various methods once a proper parameter choice has been made.     

Interface procedures for finite difference approximations of the advection–diffusion equation

We investigate several existing interface procedures for finite difference methods applied to advection–diffusion problems. The accuracy, stiffness and reflecting properties of various interface procedures are investigated.The analysis and numerical experiments show that there are only minor differences between various methods once a proper parameter choice has been made.     

A hybrid conjugate gradient method based on a quadratic relaxation of the Dai–Yuan hybrid conjugate gradient parameter

To take advantage of the attractive features of the Hestenes–Stiefel and Dai–Yuan conjugate gradient (CG) methods, we suggest a hybridization of these methods using a quadratic relaxation of a hybrid CG parameter proposed by Dai and Yuan. In the proposed method, the hybridization parameter is computed based on a conjugacy condition. Under proper conditions, we show that our method is globally convergent for uniformly convex functions. We give a numerical comparison of the implementations of our method and two efficient hybrid CG methods proposed by Dai and Yuan using a set of unconstrained optimization test problems from the CUTEr collection. Numerical results show the efficiency of the proposed hybrid CG method in the sense of the performance profile introduced by Dolan and Moré.     

A note on a quadratic formulation for linear complementarity problems

In this note, we discuss some properties of a quadratic formulation for linear complementarity problems. Projected SOR methods proposed by Mangasarian apply to symmetric matrices only. The quadratic formulation discussed here makes it possible to use these SOR methods for solving nonsymmetric LCPs. SOR schemes based on this formulation preserve sparsity. For proper choice of a free parameter, this quadratic formulation also preserves convexity. The value of the quadratic function for the solution of original LCP is also known.     

A class of adaptive Dai–Liao conjugate gradient methods based on the scaled memoryless BFGS update

Minimizing the distance between search direction matrix of the Dai–Liao method and the scaled memoryless BFGS update in the Frobenius norm, and using Powell’s nonnegative restriction of the conjugate gradient parameters, a one-parameter class of nonlinear conjugate gradient methods is proposed. Then, a brief global convergence analysis is made with and without convexity assumption on the objective function. Preliminary numerical results are reported; they demonstrate a proper choice for the parameter of the proposed class of conjugate gradient methods may lead to promising numerical performance.     

Multigrid methods for a parameter dependent problem in primal variables

Summary. In this paper we consider multigrid methods for the parameter dependent problem of nearly incompressible materials. We construct and analyze multilevel-projection algorithms, which can be applied to the mixed as well as to the equivalent, non-conforming finite element scheme in primal variables. For proper norms, we prove that the smoothing property and the approximation property hold with constants that are independent of the small parameter. Thus we obtain robust and optimal convergence rates for the W-cycle and the variable V-cycle multigrid methods. The numerical results pretty well conform the robustness and optimality of the multigrid methods proposed. Received June 17, 1998 / Revised version received October 26, 1998 / Published online September 7, 1999     

Optimizing the smoothed bootstrap

In this paper we develop the technique of a generalized rescaling in the smoothed bootstrap, extending Silverman and Young's idea of shrinking. Unlike most existing methods of smoothing, with a proper choice of the rescaling parameter the rescaled smoothed bootstrap method produces estimators that have the asymptotic minimum mean (integrated) squared error, asymptotically improving existing bootstrap methods, both smoothed and unsmoothed. In fact, the new method includes existing smoothed bootstrap methods as special cases. This unified approach is investigated in the problems of estimation of global and local functionals and kernel density estimation. The emphasis of this investigation is on theoretical improvements which in some cases offer practical potential.     

A singularly perturbed Cauchy problem

Singular perturbations are considered for the initial value problem for second order hyperbolic equations with a small positive parameter multiplying the second order time derivative term. Thus, in contrast to recent work of de Jager and Geel, the reduced equation is of the same order as the original equation but of a different type. Asymptotic expansions are constructed and shown to be uniformly asymptotically valid on sets bounded in the time direction. The proof uses energy estimates which require some delicacy due to the dependence of the characteristics on the small parameter. The problem includes that of a vibrating string in a highly viscous medium when appropriate scaling is made. The proper initial conditions differ from those treated by Zlamal and the methods employed here are different as well.     

Improving the compression rate versus L 1 error ratio in cell-average error control algorithms

Multiresolution representations of data are powerful tools in data compression. A common framework for applications is the cell-average setting. For a proper adaptation to singularities, it is interesting to develop nonlinear methods. Thus, one needs to control the stability of these representations. We introduce a generalization, depending on a parameter λ, of the classical cell-average error-control algorithms and we study the choice of the parameter to get the best relation quality vs. ratio of compression. It turns out that λ can be chosen nonlinearly and that for the L ¹ norm we can get a significant improvement over the classical error-control algorithms.     

Application of M-PML reflectionless boundary conditions to the numerical simulation of wave propagation in anisotropic media. Part I: Reflectivity

This paper presents a detailed study of the construction of reflectionless boundary conditions for anisotropic elastic problems. A Multiaxial Perfectly Matched Layer (M-PML) approach is considered. With a proper stabilization parameter, the M-PML ensures solution stability for arbitrary anisotropic media. It is proved that this M-PML modification is not perfectly matched, and the reflectivity of the M-PML exceeds that of the standard PML. Moreover, the reflection coefficient linearly depends on the stabilization parameter. A problem of constructing an optimal stabilization parameter is formulated as follows: find a minimum possible parameter that ensures stability. This problem is considered in the second part of this work.     

Vector extrapolation enhanced TSVD for linear discrete ill-posed problems

The truncated singular value decomposition (TSVD) is a popular solution method for small to moderately sized linear ill-posed problems. The truncation index can be thought of as a regularization parameter; its value affects the quality of the computed approximate solution. The choice of a suitable value of the truncation index generally is important, but can be difficult without auxiliary information about the problem being solved. This paper describes how vector extrapolation methods can be combined with TSVD, and illustrates that the determination of the proper value of the truncation index is less critical for the combined extrapolation-TSVD method than for TSVD alone. The numerical performance of the combined method suggests a new way to determine the truncation index. In memory of Gene H. Golub.     

Non-linear model parameter estimation–estimating a feasible parameter set with respect to model use

This article deals with non-linear model parameter estimation from experimental data. As for non-linear models a rigorous identifiability analysis is difficult to perform, parameter estimation is performed in such a way that uncertainty in the estimated parameter values is represented by the range of model use results when the model is used for a certain purpose. Using this approach, the article presents a simulation study where the objective is to discover whether the estimation of model parameters can be improved, so that a small enough range of model use results is obtained. The results of the study indicate that from plant measurements available for the estimation of model parameters, it is possible to extract data that are important for the estimation of model parameters relative to a certain model use. If these data are improved by a proper measurement campaign (e.g. proper choice of measured variables, better accuracy, higher measurement frequency) it is to be expected that a valid model for a certain model use will be obtained. The simulation study is performed for an activated sludge model from wastewater treatment, while the estimation of model parameters is done by Monte Carlo simulation.     

A regularization smoothing method for second-order cone complementarity problem

In this paper, the second-order cone complementarity problem is studied. Based on the Fischer–Burmeister function with a perturbed parameter, which is also called smoothing parameter, a regularization smoothing Newton method is presented for solving the sequence of regularized problems of the second-order cone complementarity problem. Under proper conditions, the global convergence and local superlinear convergence of the proposed algorithm are obtained. Moreover, the local superlinear convergence is established without strict complementarity conditions. Preliminary numerical results suggest the effectiveness of the algorithm.     

Markov Teams — An analytical approach to process migration in distributed computing systems

A process migration mechanism offers a means to exploit the performance reserves present in networks of workstations used as personal computers by allowing the processors to migrate processes from overload ones to underused ones. Several distributed operating systems provide such a facility, the benefits of its use depending on the specification of a proper process migration policy. This work proposes an analytical model, the Markov Team Model, to assist the design of such a policy. Besides deriving this model from results of classical Team Theory and Markov Decision Processes, we study the special case of homogeneous distributed computing systems and present methods for parameter estimation. Numerical examples are used to demonstrate the benefits of using this model.     

The interaction between a penny-shaped crack and a spherical inclusion under torsion

The axisymmetric interaction problem of an elastic spherical inclusion with a penny-shaped crack in an elastic space under torsion is considered. The superposition and reflection methods [3]-[4] are used to solve the mixed boundary value problem in question. With the help of the dual integral equations technique and appropriate re-expansion of the eigenfunction, the problem is reduced to an infinite system of linear algebraic equations of the second kind. The matrix elements of that system decrease exponentially along the rows and the columns. Its unique solution is proved to exist in a proper class of sequences and is shown to be represented by a convergent, in the vicinity of the origin, power series in a geometric parameter, equal to the ratio of the radius of the inclusion to its distance from the crack. This procedure provides an efficient formula for the stress intensity factor.     

A type III phase-field system with a logarithmic potential

Our aim in this note is to study the existence and uniqueness of solutions to a phase-field system, based on type III heat conduction, with a logarithmic potential. The main difficulty is to prove that the order parameter remains in the physically relevant range and this is achieved by deriving proper a priori estimates.