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, a differentially private Laplacian consensus algorithm was proposed for the multi-agent average consensus problem under the requirement of differential privacy of the agents initial states against an adversary that has access to all the messages.
183 citations
TL;DR: Another tree reconstruction method, the witness-antiwitness method (WAM), is presented, which is faster than DCM, especially on random trees, and converges to the true tree topology at the same rate as DCM.
183 citations
TL;DR: This paper proposes to modify the Newton method for variational inequality problems by using a certain differentiable merit function to determine a suitable step length and shows that the method is globally convergent and the rate of convergence is quadratic.
Abstract: Variational inequality problems have been used to formulate and study equilibrium problems, which arise in many fields including economics, operations research and regional sciences. For solving variational inequality problems, various iterative methods such as projection methods and the nonlinear Jacobi method have been developed. These methods are convergent to a solution under certain conditions, but their rates of convergence are typically linear. In this paper we propose to modify the Newton method for variational inequality problems by using a certain differentiable merit function to determine a suitable step length. The purpose of introducing this merit function is to provide some measure of the discrepancy between the solution and the current iterate. It is then shown that, under the strong monotonicity assumption, the method is globally convergent and, under some additional assumptions, the rate of convergence is quadratic. Limited computational experience indicates the high efficiency of the proposed method.
183 citations
03 Apr 2003
TL;DR: A reachability method for systems with input is developed, based on the relation between such systems and the corresponding autonomous systems in terms of reachable sets, which allows to compute conservative approximations with as great degree of accuracy as desired.
Abstract: In this paper we present an approach to approximate reachability computation for nonlinear continuous systems. Rather than studying a complex nonlinear system x = g(x), we study an approximating system x = f(x) which is easier to handle. The class of approximating systems we consider in this paper is piecewise linear, obtained by interpolating g over a mesh. In order to be conservative, we add a bounded input in the approximating system to account for the interpolation error. We thus develop a reachability method for systems with input, based on the relation between such systems and the corresponding autonomous systems in terms of reachable sets. This method is then extended to the approximate piecewise linear systems arising in our construction. The final result is a reachability algorithm for nonlinear continuous systems which allows to compute conservative approximations with as great degree of accuracy as desired, and more importantly, it has good convergence rate. If g is a C2 function, our method is of order 2. Furthermore, the method can be straightforwardly extended to hybrid systems.
183 citations
TL;DR: The asymptotic properties of estimates obtained using Laplace's approximation for nonlinear mixed-effects models are investigated in this article, where the Laplace approximation is applied only to the random effects of the integrated likelihood.
Abstract: The asymptotic properties of estimates obtained using Laplace's approximation for nonlinear mixed-effects models are investigated. Unlike the restricted maximum likelihood approach, e.g. Wolfinger (1993), here the Laplace approximation is applied only to the random effects of the integrated likelihood. This results in approximate maximum likelihood estimation. The resulting estimates are shown to be consistent with the rate of convergence depending on both the number of individuals and the number of observations per individual. Conditions under which the leading term Laplace approximation should be avoided are discussed.
182 citations