Showing papers on "Convergence (routing) published in 1999"
TL;DR: The bounds for convergence on learning-rate and momentum parameters are derived, and it is demonstrated that the momentum term can increase the range of learning rate over which the system converges.
2,033 citations
30 Aug 1999
TL;DR: This work presents an abstract model of BGP and uses it to define several global sanity conditions on routing policies that are related to BGP convergence/divergence, and shows that the complexity of statically checking it is either NP-complete or NP-hard.
Abstract: The Border Gateway Protocol (BGP) is the de facto inter-domain routing protocol used to exchange reachability information between Autonomous Systems in the global Internet. BGP is a path-vector protocol that allows each Autonomous System to override distance-based metrics with policy-based metrics when choosing best routes. Varadhan et al. [18] have shown that it is possible for a group of Autonomous Systems to independently define BGP policies that together lead to BGP protocol oscillations that never converge on a stable routing. One approach to addressing this problem is based on static analysis of routing policies to determine if they are safe. We explore the worst-case complexity for convergence-oriented static analysis of BGP routing policies. We present an abstract model of BGP and use it to define several global sanity conditions on routing policies that are related to BGP convergence/divergence. For each condition we show that the complexity of statically checking it is either NP-complete or NP-hard.
421 citations
01 Jan 1999
TL;DR: In this paper, a collection of control algorithms which are able to give rise to 2-sliding modes have been presented, and for each of them the sufficient convergence conditions are given.
Abstract: In this paper a collection of control algorithms which are able to give rise to 2-sliding modes have been presented, and for each of them the sufficient convergence conditions are given. Furthermore, the real sliding behaviour is briefly considered, and, in some cases, the upper bound of the convergence time is given.
302 citations
21 Mar 1999
TL;DR: The network routing messages exchanged between core Internet backbone routers are examined to show that as a result of specific router vendor software changes suggested by earlier analysis, the volume of Internet routing updates has decreased by an order of magnitude.
Abstract: This paper examines the network routing messages exchanged between core Internet backbone routers. Internet routing instability, or the rapid fluctuation of network reachability information, is an important problem currently facing the Internet engineering community. High levels of network instability can lead to packet loss, increased network latency and time to convergence. At the extreme, high levels of routing instability have led to the loss of internal connectivity in wide-area, national networks. In an earlier study of inter-domain routing, we described widespread, significant pathological behaviour in the routing information exchanged between backbone service providers at the major US public Internet exchange points. These pathologies included several orders of magnitude more routing updates in the Internet core than anticipated, large numbers of duplicate routing messages, and unexpected frequency components between routing instability events. The work described in this paper extends our earlier analysis by identifying the origins of several of these observed pathological Internet routing behaviour. We show that as a result of specific router vendor software changes suggested by our earlier analysis, the volume of Internet routing updates has decreased by an order of magnitude. We also describe additional router software changes that can decrease the volume of routing updates exchanged in the Internet core by an additional 30 percent or more. We conclude with a discussion of trends in the evolution of Internet architecture and policy that may lead to a rise in Internet routing instability.
207 citations
TL;DR: In this paper, the convergence behavior of computed electronic energies, in particular electronic enthalpies of reaction, as a function of the one-electron space is considered and extrapolations to the limit of a complete oneelectron basis are possible.
Abstract: The highly accurate calculation of molecular electronic structure requires the expansion of the molecular electronic wavefunction to be as nearly complete as possible both in one- and n- electron space. In this review, we consider the convergence behaviour of computed electronic energies, in particular electronic enthalpies of reaction, as a function of the one-electron space. Based on the convergence behaviour, extrapolations to the limit of a complete one-electron basis are possible and such extrapolations are compared with the direct computation of electronic energies near the basis-set limit by means of explicitly correlated methods. The most elaborate and accurate computations are put into perspective with respect to standard and—from a computational point of view—inexpensive density functional, complete basis set (CBS) and Gaussian-2 calculations. Using the explicitly correlated coupled-cluster method including singles, doubles and non-iterative triples replacements, it is possible to compute (the electronic part of) enthalpies of reaction accurate to within 1 kJ mol 1 . To achieve this level of accuracy with standard coupled-cluster methods, large basis sets or extrapolations to the basis-set limit are necessary to exploit fully the intrinsic accuracy of the coupled-cluster methods.
204 citations
01 Jan 1999
TL;DR: In this article, a multi-grid, flux-difference-split, finite-volume code, VULCAN, is presented for solving the elliptic and parabolized form of the equations governing three-dimensional, turbulent, calorically perfect and non-equilibrium chemically reacting flows.
Abstract: A multi-grid, flux-difference-split, finite-volume code, VULCAN, is presented for solving the elliptic and parabolized form of the equations governing three-dimensional, turbulent, calorically perfect and non-equilibrium chemically reacting flows. The space marching algorithms developed to improve convergence rate and or reduce computational cost are emphasized. The algorithms presented are extensions to the class of implicit pseudo-time iterative, upwind space-marching schemes. A full approximate storage, full multi-grid scheme is also described which is used to accelerate the convergence of a Gauss-Seidel relaxation method. The multi-grid algorithm is shown to significantly improve convergence on high aspect ratio grids.
165 citations
TL;DR: In this article, a new numerical method for solving geometric moving interface problems is presented, which combines a level set approach and a semi-Lagrangian time stepping scheme which is explicit yet unconditionally stable.
143 citations
TL;DR: In this paper, theoretical properties of the slice sampler have been analyzed, and it has been shown that the algorithm is stochastically monotone, and analytic bounds on the total variation distance from stationarity of the method by using Foster-Lyapunov drift condition methodology.
Abstract: We analyse theoretical properties of the slice sampler. We find that the algorithm has extremely robust geometric ergodicity properties. For the case of just one auxiliary variable, we demonstrate that the algorithm is stochastically monotone, and we deduce analytic bounds on the total variation distance from stationarity of the method by using Foster–Lyapunov drift condition methodology.
141 citations
01 May 1999
TL;DR: This paper proposes a new method for parameter adaptation in stochastic optimization, applicable to a wide range of objective functions, as well as to a large set of local optimization techniques.
Abstract: Optimization is an important operation in many domains of science and technology. Local optimization techniques typically employ some form of iterative procedure, based on derivatives of the function to be optimized (objective function). These techniques normally involve parameters that must be set by the user, often by trial and error. Those parameters can have a strong influence on the convergence speed of the optimization. In several cases, a significant speed advantage could be gained if one could vary these parameters during the optimization, to reflect the local characteristics of the function being optimized. Some parameter adaptation methods have been proposed for this purpose, for deterministic optimization situations. For stochastic (also called on-line) optimization situations, there appears to be no simple and effective parameter adaptation method. This paper proposes a new method for parameter adaptation in stochastic optimization. The method is applicable to a wide range of objective functions, as well as to a large set of local optimization techniques. We present the derivation of the method, details of its application to gradient descent and to some of its variants, and examples of its use in the gradient optimization of several functions, as well as in the training of a multilayer perceptron by on-line backpropagation. Introduction Optimization is an operation that is often used in several different domains of science and technology. It normally consists of maximizing or minimizing a given function (called objective function ), that is chosen to represent the quality of a given system. The system may be physical, (mechanical, chemical, etc.), a mathematical model, a computer program, etc., or even a mixture of several of these.
117 citations
112 citations
TL;DR: It is proved that for a given linear programming problem and sufficiently large penalty parameters, any trajectory of the neural network converges in finite time to its solution set.
Abstract: A class of neural networks that solve linear programming problems is analyzed. The neural networks considered are modeled by dynamic gradient systems that are constructed using a parametric family of exact (nondifferentiable) penalty functions. It is proved that for a given linear programming problem and sufficiently large penalty parameters, any trajectory of the neural network converges in finite time to its solution set. For the analysis, Lyapunov-type theorems are developed for finite time convergence of nonsmooth sliding mode dynamic systems to invariant sets. The results are illustrated via numerical simulation examples.
01 Jan 1999
TL;DR: In this paper, a class of numerical schemes for the Isaacs equation of pursuit-evasion games is presented, where the solution is interpreted in the viscosity sense, as well as discontinuous value functions, and the convergence of the approximation scheme to the value function of the game is proved.
Abstract: We present a class of numerical schemes for the Isaacs equation of pursuit-evasion games. We consider continuous value functions, where the solution is interpreted in the viscosity sense, as well as discontinuous value functions, where the notion of viscosity envelope-solution is needed. The convergence of the approximation scheme to the value function of the game is proved in both cases. A priori estimates of the convergence in L∞ are established when the value function is Holder continuous. We also treat problems with state constraints and discuss several issues concerning the implementation of the approximation scheme, the synthesis of approximate feedback controls, and the approximation of optimal trajectories. The efficiency of the algorithm is illustrated by a number of numerical tests, either in the case of one player (i.e., minimum time problem) or for some 2-players games.
TL;DR: Estimate of exponential convergence rate and exponential stability are studied for a class of neural networks which includes the Hopfield neural networks and the cellular neural networks, both local and global exponential convergence.
Abstract: Estimates of exponential convergence rate and exponential stability are studied for a class of neural networks which includes Hopfield neural networks and cellular neural networks. Both local and global exponential convergence are discussed. Theorems for estimation of the exponential convergence rate are established and the bounds on the rate of convergence are given. The domains of attraction in the case of local exponential convergence are obtained. Simple conditions are presented for checking exponential stability of the neural networks.
TL;DR: These schemes provide the first examples of nonoscillatory second-order Godunov-type schemes based on global projection operators based onglobal projection operators, and it is shown that they are nonoscilledatory in the sense of satisfying the maximum principle.
Abstract: In this paper, we construct second-order central schemes for multidimensional Hamilton--Jacobi equations and we show that they are nonoscillatory in the sense of satisfying the maximum principle. Thus, these schemes provide the first examples of nonoscillatory second-order Godunov-type schemes based on global projection operators. Numerical experiments are performed; $L^1$/$L^{\infty}$-errors and convergence rates are calculated. For convex Hamiltonians, numerical evidence confirms that our central schemes converge with second-order rates, when measured in the L1-norm advocated in our recent paper [Numer. Math, to appear]. The standard $L^{\infty}$-norm, however, fails to detect this second-order rate.
TL;DR: In this paper, a Newton type method based on an active set strategy defined by means of the dual variables is developed and analyzed, and numerical examples for blocky signals and images perturbed by very high noise are included.
Abstract: Lagrangian and augmented Lagrangian methods for nondifferentiable
optimization problems that arise from the total bounded variation formulation of image restoration problems are analyzed. Conditional convergence of the
Uzawa algorithm and unconditional convergence of the first order augmented
Lagrangian schemes are discussed. A Newton type method based on an active
set strategy defined by means of the dual variables is developed and
analyzed. Numerical examples for blocky signals and images perturbed
by very high noise are included.
TL;DR: A rigorous error analysis is given for the immersed interface method (IIM) applied to elliptic problems with discontinuities and singularities, and second-order convergence of IIM is indicated by the analysis.
Abstract: A rigorous error analysis is given for the immersed interface method (IIM) applied to elliptic problems with discontinuities and singularities. The finite difference scheme using IIM is shown to satisfy the conditions of a maximum principle for a certain class of problems. Comparison functions are constructed to obtain error bounds for some of the approximate solutions. The asymptotic error expansion provides further useful insights and details of the behaviour and convergence properties of IIM, which leads to a sharper estimate of the error bound. Second-order convergence of IIM is indicated by the analysis. Numerical examples are also given to support the analytical results.
TL;DR: The convergence rates of genetic algorithms are discussed by using the minorization condition in the Markov chain theory to obtain the bound on its convergence rate on the general state space and on the finite state space.
TL;DR: In this article, a multizone decomposition technique with multiquadric scheme was proposed for numerical solutions of a time-dependent problem. But the computational efficiency was not improved substantially with faster convergence without significant degradation in accuracy.
Abstract: This paper discusses the application of the multizone decomposition technique with multiquadric scheme for the numerical solutions of a time-dependent problem. The construction of the multizone algorithm is based on a domain decomposition technique to subdivide the global region into a number of finite subdomains. The reduction of ill-conditioning and the improvement of the computational efficiency can be achieved with a smaller resulting matrix on each subdomain. The proposed scheme is applied to a hypothetical linear two-dimensional hydrodynamic model as well as a real-life nonlinear two-dimensional hydrodynamic model in the Tolo Harbour of Hong Kong to simulate the water flow circulation patterns. To illustrate the computational efficiency and accuracy of the technique, the numerical results are compared with those solutions obtained from the same problem using a single domain multiquadric scheme. The computational efficiency of the multizone technique is improved substantially with faster convergence without significant degradation in accuracy.
20 Mar 1999
TL;DR: The idea of node's individual view of the network with the concept of visibility sets is formalized and a routing approach based on the notion of a feasible path is introduced, i.e., such path in the node's visibility set that satisfies certain specified restrictions.
Abstract: This paper is focused on the problem of hop-by-hop routing in a network where different nodes have different views of the network topology. In particular, each node may be aware of just a subset of the network links, perceiving the rest as if their cost was infinite. We formalize the idea of node's individual view of the network with the concept of visibility sets and introduce a routing approach based on the notion of a feasible path, i.e., such path in the node's visibility set that satisfies certain specified restrictions. It is shown that, in a network with general visibility sets, forwarding the packet along an optimal feasible path is necessary and sufficient to guarantee its eventual delivery to destination without being dropped or routed to the same node twice. Based on the proposed approach, we derive the precise routing policy and formulate an efficient algorithm to search for a family of one-to-all optimal feasible paths in a network with embedded visibility sets. We then proceed to prove the correctness of the algorithm. The new routing method provides for execution of multiple dynamic routing protocols, possibly overlaying each other in the same address space, within a network with common kinds of metrics of arbitrary complexity. It solves the problem of interoperability when new metrics or novel link properties are being introduced and eliminates the necessity to run different protocols and protocol versions within disjoint routing domains.
TL;DR: Local convergence properties of inexact Newton and Newton-like methods for systems of nonlinear equations are investigated, and new sufficient conditions for linear convergence in an arbitrary vector norm are provided.
Abstract: In this paper we investigate local convergence properties of inexact Newton and Newton-like methods for systems of nonlinear equations. Processes with modified relative residual control are considered, and new sufficient conditions for linear convergence in an arbitrary vector norm are provided. For a special case the results are affine invariant.
TL;DR: An analytic procedure is developed which allows a specific nonlinear system and a specific reference signal to determine a priori whether or not the parameter estimates will converge to their true values, simply by checking the linear independence of the rows of a constant real matrix.
Abstract: Following the development of a parameter convergence analysis procedure for output-feedback nonlinear systems (1995, 1998), the authors shift their attention to strict feedback nonlinear systems in this paper. They develop an analytic procedure which allows us, given a specific nonlinear system and a specific reference signal, to determine a priori whether or not the parameter estimates will converge to their true values, simply by checking the linear independence of the rows of a constant real matrix. Moreover, the authors show that this convergence is exponential. Finally, they prove that even if the rows of this constant matrix are not linearly independent, partial parameter convergence is still achieved, in the sense that the parameter error vector converges asymptotically to the left nullspace of this matrix.
TL;DR: This paper analyzes the convergence rates of a class of recursive algorithms for global optimization via Monte Carlo methods by using perturbed Liapunov function methods and examining the asymptotic properties of suitably scaled estimation error sequences.
Abstract: Inspired and motivated by the recent advances in simulated annealing algorithms, this paper analyzes the convergence rates of a class of recursive algorithms for global optimization via Monte Carlo methods. By using perturbed Liapunov function methods, stability results of the algorithms are established. Then the rates of convergence are ascertained by examining the asymptotic properties of suitably scaled estimation error sequences.
TL;DR: Third-order linear and non-linear boundary-value problems are solved using fourth-degree B-splines and the method is tested on two problems from the literature.
Abstract: Third-order linear and non-linear boundary-value problems are solved using fourth-degree B-splines. The convergence of the method is discussed. The method is tested on two problems from the literature
TL;DR: The author proposes to use the post stabilization approach, well known when dealing with differential systems with invariant submanifolds, which enables to increase the sampling period while keeping very good precision.
Abstract: Optimization based non-linear observers are those where the integrated output prediction error is used to define the dynamic of the observer's state. Their main advantage is genericity and the natural assumptions that are needed for the related convergence results to hold. Here, we present some new theoretical results concerning these observers. Despite the title we use, our observer is not based on the search for a global minimum of a cost function. It is based on a descent-like approach without the use of the Hessian matrix. A nice feature is that the resulting observer takes a classical form of in the sense that it is defined by a set of ordinary differential equations. Besides the theoretical results, some implementation issues are discussed. In particular, we propose to use the post stabilization approach, well known when dealing with differential systems with invariant submanifolds. This enables to increase the sampling period while keeping very good precision. Two illustrative examples are presente...
TL;DR: It is shown how to sum the successive approximations analytically to find a single antidiffusive velocity that represents the effects of an arbitrary number of passes, leading to successively more accurate solutions to the advection equation.
Abstract: Multidimensional positive definite advection transport algorithm (MPDATA) is an iterative process for approximating the advection equation, which uses a donor cell approximation to compensate for the truncation error of the originally specified donor cell scheme. This step may be repeated an arbitrary number of times, leading to successively more accurate solutions to the advection equation. In this paper, we show how to sum the successive approximations analytically to find a single antidiffusive velocity that represents the effects of an arbitrary number of passes. The analysis is first done in one dimension to illustrate the method and then is repeated in two dimensions. The existence of cross terms in the truncation analysis of the two-dimensional equations introduces an extra complication into the calculation. We discuss the implementation of our new antidiffusive velocities and provide some examples of applications, including a third-order accurate scheme.
TL;DR: In this paper, the authors studied the limiting behavior of the solutions to a class of conservation laws with vanishing nonlinear diffusion and dispersion terms and proved the convergence to the entropy solution of the first order problem under a condition on the relative size of the diffusion and the dispersion.
Abstract: We study the limiting behavior of the solutions to a class of conservation laws with vanishing nonlinear diffusion and dispersion terms. We prove the convergence to the entropy solution of the first order problem under a condition on the relative size of the diffusion and the dispersion terms. This work is motivated by the pseudo-viscosity approximation introduced by Von Neumann in the 50's.
TL;DR: In this article, the convergence of iterative methods for bearing only tracking (BOT) was analyzed and sufficient conditions for convergence were derived for maximizing the likelihood functional of the likelihood function.
Abstract: This paper deals with the analysis of the convergence of iterative methods for bearings only tracking (BOT). A geometric and unified framework is developed. Explicit sufficient conditions ensuring convergence of iterative methods for maximization of the likelihood functional are then derived.
TL;DR: D discrete-time Lyapunov functions are constructed for the cases of exact and approximate doubly stochastic constraint satisfaction, which show convergence to a fixed point.
Abstract: The softassign quadratic assignment algorithm is a discrete-time, continuous-state, synchronous updating optimizing neural network. While its effectiveness has been shown in the traveling salesman problem, graph matching, and graph partitioning in thousands of simulations, its convergence properties have not been studied. Here, we construct discrete-time Lyapunov functions for the cases of exact and approximate doubly stochastic constraint satisfaction, which show convergence to a fixed point. The combination of good convergence properties and experimental success makes the softassign algorithm an excellent choice for neural quadratic assignment optimization.
01 Jan 1999
TL;DR: A sample of an arbitraryorder sliding mode attracting trajectories in finite time and featuring the utmost accuracy is presented for the first time in the present paper.
Abstract: Sliding mode is used in order to retain a dynamic system accurately at a given constraint and is the main operation mode in variable structure systems. Such mode is a motion on a discontinuity set of a dynamic system and features theoretically-infinite-frequency switching. The standard sliding modes are known to feature finite time convergence, precise keeping of the constraint and robustness with respect to internal and external disturbances. In realization their sliding precision is proportional to the time interval between measurements. Having generalized the notion of sliding mode, higher order sliding modes preserve or generalize its main properties and remove the chattering effect. With discrete measurements rth-order sliding mode realization may provide up to the rth order of sliding precision with respect to the measurement interval. Such finite-time convergence modes were known only for r = 1, 2, 3. A sample of an arbitraryorder sliding mode attracting trajectories in finite time and featuring the above-mentioned utmost accuracy is presented for the first time in the present paper. The proposed controllers provide for full real-time control of the output variable if the relative degree r of the uncertain dynamic
01 Jun 1999
TL;DR: This paper analyzes and compares eigenvalues with the eigenvector condition number, the field of values, and pseudospectra as the basis for convergence bounds for minimum residual Krylov subspace methods applied to non-normal coefficient matrices.
Abstract: Eigenvalues with the eigenvector condition number, the field of values, and pseudospectra have all been suggested as the basis for convergence bounds for minimum residual Krylov subspace methods applied to non-normal coefficient matrices. This paper analyzes and compares these bounds, illustrating with six examples the success and failure of each one. Refined bounds based on eigenvalues and the field of values are suggested to handle low-dimensional non-normality. It is observed that pseudospectral bounds can capture multiple convergence stages. Unfortunately, computation of pseudospectra can be rather expensive. This motivates an adaptive technique for estimating GMRES convergence based on approximate pseudospectra taken from the Arnoldi process that is the basis for GMRES.