Topic
Rate of convergence
About: Rate of convergence is a research topic. Over the lifetime, 31257 publications have been published within this topic receiving 795334 citations. The topic is also known as: convergence rate.
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TL;DR: In this article, an a-posteriors strategy for choosing the regularization parameter in Tikhonov regularization for solving nonlinear ill-posed problems is proposed and the convergence rate obtained with this strategy is optimal.
Abstract: The authors propose an a-posteriors strategy for choosing the regularization parameter in Tikhonov regularization for solving nonlinear ill-posed problems and show that under certain conditions, the convergence rate obtained with this strategy is optimal. As a by-product, a new stability estimate for the regularized solutions is given which applies to a class of parameter identification problems. The authors compare the parameter choice strategy with Morozov’s Discrepancy Principle. Finally, numerical results are presented.
145 citations
01 Oct 2007
TL;DR: A decreasing coefficient to the updating principle is introduced, so that PSO can be viewed as a regular stochastic approximation algorithm, to improve exploration ability and to emphasize the role of this additional velocity.
Abstract: Particle swarm optimization (PSO) is a population-based stochastic recursion procedure, which simulates the social behavior of a swarm of ants or a school of fish. Based upon the general representation of individual particles, this paper introduces a decreasing coefficient to the updating principle, so that PSO can be viewed as a regular stochastic approximation algorithm. To improve exploration ability, a random velocity is added to the velocity updating in order to balance exploration behavior and convergence rate with respect to different optimization problems. To emphasize the role of this additional velocity, the modified PSO paradigm is named PSO with controllable random exploration velocity (PSO-CREV). Its convergence is proved using Lyapunov theory on stochastic process. From the proof, some properties brought by the stochastic components are obtained such as ldquodivergence before convergencerdquo and ldquocontrollable exploration.rdquo Finally, a series of benchmarks is proposed to verify the feasibility of PSO-CREV.
145 citations
TL;DR: In this paper, the authors present three numerical schemes for solving the Falkner-Skan equation with positive or negative wall shear, using Newton's method with the aid of variational equations and yields quadratic convergence.
144 citations
TL;DR: In this article, a new relationship between regeneration time T and τC was developed, and this gave a bound on the tail of T, based on e,λ and b, which is a strict improvement on existing results.
144 citations
TL;DR: In the class of quasi-contractive operators satisfying Zamfirescu's conditions, the most used fixed point iterative methods, that is, the Picard, Mann, and Ishikawa iterations, are all known to be convergent to the unique fixed point.
Abstract: In the class of quasi-contractive operators satisfying Zamfirescu's conditions, the most used fixed point iterative methods, that is, the Picard, Mann, and Ishikawa iterations, are all known to be convergent to the unique fixed point. In this paper, the comparison of the first two methods with respect to their convergence rate is obtained.
144 citations