Convergence and Continuous Dependence for the Brinkman–Forchheimer Equations

The Brinkman–Forchheimer equations for non-slow flow in a saturated porous medium are analyzed. It is shown that the solution depends continuously on changes in the Forchheimer coefficient, and convergence of the solution of the Brinkman–Forchheimer equations to that of the Brinkman equations is deduced, as the Forchheimer coefficient tends to zero. The next result establishes continuous dependence on changes in the Brinkman coefficient. Following this, a result is proved establishing convergence of a solution of the Brinkman–Forchheimer equations to a solution of the Darcy–Forchheimer equations, as the Brinkman coefficient (effective viscosity) tends to zero. Finally, upper and lower bounds are derived for the energy decay rate which establish that the energy decays exponentially, but not faster than this.     

Discretization in semi-infinite programming: the rate of convergence

The discretization approach for solving semi-infinite optimization problems is considered. We are interested in the convergence rate of the error between the solution of the semi-infinite problem and the solution of the discretized program depending on the discretization mesh-size. It will be shown how this rate depends on whether the minimizer is strict of order one or two and on whether the discretization includes boundary points of the index set in a specific way. This is done for ordinary and for generalized semi-infinite problems. Received: November 21, 2000 / Accepted: May 2001?Published online September 17, 2001     

关于增生算子方程解的带误差的Ishikawa迭代程序

该文在Banach空间中证明了，带误差的Ishikawa迭代序列强收敛到Lipschitz连续的增生算子方程的唯一解．而且，也给Ishikawa迭代序列提供了一般的收敛率估计．利用该结果还推得，带误差的Ishikawa迭代序列也强收敛到Lipschitz连续的强增生算子方程的唯一解．     

A generalization of column generation to accelerate convergence

This paper proposes a generalization of column generation, reformulating the master problem with fewer variables at the expense of adding more constraints; the sub-problem structure does not change. It shows both analytically and computationally that the reformulation promotes faster convergence to an optimal solution in application to a linear program and to the relaxation of an integer program at each node in the branch-and-bound tree. Further, it shows that this reformulation subsumes and generalizes prior approaches that have been shown to improve the rate of convergence in special cases.     

Accelerated Gauss-Newton algorithms for nonlinear least squares problems

Recent theoretical and practical investigations have shown that the Gauss-Newton algorithm is the method of choice for the numerical solution of nonlinear least squares parameter estimation problems. It is shown that when line searches are included, the Gauss-Newton algorithm behaves asymptotically like steepest descent, for a special choice of parameterization. Based on this a conjugate gradient acceleration is developed. It converges fast also for those large residual problems, where the original Gauss-Newton algorithm has a slow rate of convergence. Several numerical test examples are reported, verifying the applicability of the theory.     

The rate of convergence of weak greedy approximations over orthogonal dictionaries

The convergence rate of a weak orthogonal greedy algorithm is studied for the subspace ?¹ ? ?² and orthogonal dictionaries. It is shown that general results on convergence rate of weak orthogonal greedy algorithms can be essentially improved in the studied case. It is also shown that this improvement is asymptotically sharp.     

A Framework for Analyzing Local Convergence Properties with Applications to Proximal-Point Algorithms

A general algorithmic scheme for solving inclusions in a Banach space is investigated in respect to its local convergence behavior. Particular emphasis is placed on applications to certain proximal-point-type algorithms in Hilbert spaces. The assumptions do not necessarily require that a solution be isolated. In this way, results existing for the case of a locally unique solution can be extended to cases with nonisolated solutions. Besides the convergence rates for the distance of the iterates to the solution set, strong convergence to a sole solution is shown as well. As one particular application of the framework, an improved convergence rate for an important case of the inexact proximal-point algorithm is derived.     

Convergence rates toward the travelling waves for a model system of the radiating gas

The present paper is concerned with an asymptotics of a solution to the model system of radiating gas. The previous researches have shown that the solution converges to a travelling wave with a rate (1 + t)^?1/4 as time t tends to infinity provided that an initial data is given by a small perturbation from the travelling wave in the suitable Sobolev space and the perturbation is integrable. In this paper, we make more elaborate analysis under suitable assumptions on initial data in order to obtain shaper convergence rates than previous researches. The first result is that if the initial data decays at the spatial asymptotic point with a certain algebraic rate, then this rate reflects the time asymptotic convergence rate. Precisely, this convergence rate is completely same as the spatial convergence rate of the initial perturbation. The second result is that if the initial data is given by the Riemann data, an admissible weak solution, which has a discontinuity, converges to the travelling wave exponentially fast. Both of two results are proved by obtaining decay estimates in time through energy methods with suitably chosen weight functions. Copyright © 2006 John Wiley & Sons, Ltd.     

On the convergence rate of the simulated annealing algorithm

The convergence rate of the simulated annealing algorithm is examined. It is shown that, if the objective function is nonsingular, then the number of its evaluations required to obtain the desired accuracy ɛ in the solution can be a slowly (namely, logarithmically) growing function as ɛ approaches zero.     

A two-species plasma in the limit of large ion mass

Classical solutions of the relativistic Vlasov–Maxwell system are considered, describing a collisionless plasma with two species of particles. ions and electrons. It is shown that as the ion mass m tends to infinity, the corresponding solution of the relativistic Vlasov–Maxwell system tends to the solution of a system, in which the ions are given by a fixed ion background and only the electrons move. The convergence is uniform on compact time intervals, with an asymptotic convergence rate of m^?1.