Showing papers on "Convergence (routing) published in 2001"
TL;DR: In this paper, a modified version of the minimum compliance topology optimization problem is studied, where the direct dependence of the material properties on its pointwise density is replaced by a regularization of the density field by the mean of a convolution operator.
Abstract: In this article, a modified (‘filtered’) version of the minimum compliance topology optimization problem is studied. The direct dependence of the material properties on its pointwise density is replaced by a regularization of the density field by the mean of a convolution operator. In this setting it is possible to establish the existence of solutions. Moreover, convergence of an approximation by means of finite elements can be obtained. This is illustrated through some numerical experiments. The ‘filtering’ technique is also shown to cope with two important numerical problems in topology optimization, checkerboards and mesh dependent designs. Copyright © 2001 John Wiley & Sons, Ltd.
920 citations
TL;DR: It is shown that, for most types of radial basis functions that are considered in this paper, convergence can be achieved without further assumptions on the objective function.
Abstract: We introduce a method that aims to find the global minimum of a continuous nonconvex function on a compact subset of \dRd It is assumed that function evaluations are expensive and that no additional information is available Radial basis function interpolation is used to define a utility function The maximizer of this function is the next point where the objective function is evaluated We show that, for most types of radial basis functions that are considered in this paper, convergence can be achieved without further assumptions on the objective function Besides, it turns out that our method is closely related to a statistical global optimization method, the P-algorithm A general framework for both methods is presented Finally, a few numerical examples show that on the set of Dixon-Szego test functions our method yields favourable results in comparison to other global optimization methods
793 citations
TL;DR: This paper presents a two-year study of Internet routing convergence through the experimental instrumentation of key portions of the Internet infrastructure, including both passive data collection and fault-injection machines at Internet exchange points, and describes several unexpected properties of convergence.
Abstract: This paper examines the latency in Internet path failure, failover, and repair due to the convergence properties of interdomain routing. Unlike circuit-switched paths which exhibit failover on the order of milliseconds, our experimental measurements show that interdomain routers in the packet-switched Internet may take tens of minutes to reach a consistent view of the network topology after a fault. These delays stem from temporary routing table fluctuations formed during the operation of the border gateway protocol (BGP) path selection process on the Internet backbone routers. During these periods of delayed convergence, we show that end-to-end Internet paths will experience intermittent loss of connectivity, as well as increased packet loss and latency. We present a two-year study of Internet routing convergence through the experimental instrumentation of key portions of the Internet infrastructure, including both passive data collection and fault-injection machines at major Internet exchange points. Based on data from the injection and measurement of several hundred thousand interdomain routing faults, we describe several unexpected properties of convergence and show that the measured upper bound on Internet interdomain routing convergence delay is an order of magnitude slower than previously thought. Our analysis also shows that the upper theoretic computational bound on the number of router states and control messages exchanged during the process of BGP convergence is factorial with respect to the number of autonomous systems in the Internet. Finally, we demonstrate that much of the observed convergence delay stems from specific router vendor implementation decisions and ambiguity in the BGP specification.
703 citations
TL;DR: This paper proposes query control schemes for the Zone Routing Protocol that allow ZRP to provide routes to all accessible network nodes, with less control traffic than purely proactive link state or purely reactive route discovery, and with less delay than conventional flood searching.
Abstract: In this paper, we study the performance of route query control mechanisms for the Zone Routing Protocol (ZRP) for ad hoc networks. ZRP proactively maintains routing information for a local neighborhood (routing zone), while reactively acquiring routes to destinations beyond the routing zone. This hybrid routing approach can be more efficient than traditional routing schemes. However, without proper query control techniques, the ZRP cannot provide the expected reduction in the control traffic.Our proposed query control schemes exploit the structure of the routing zone to provide enhanced detection and prevention of overlapping queries. These techniques can be applied to single- or multiple-channel ad hoc networks to improve both the delay and control traffic performance of ZRP. Our query control mechanisms allow ZRP to provide routes to all accessible network nodes, with less control traffic than purely proactive link state or purely reactive route discovery, and with less delay than conventional flood searching.
666 citations
TL;DR: This paper proposes a set of guidelines for an AS to follow in setting its routing policies, without requiring coordination with other ASs, and proves that following these guidelines guarantees route convergence.
Abstract: The Border Gateway Protocol (BGP) allows an autonomous system (AS) to apply diverse local policies for selecting routes and propagating reachability information to other domains. However, the BGP permits ASs to have conflicting policies that can lead to routing instability. This paper proposes a set of guidelines for an AS to follow in setting its routing policies, without requiring coordination with other ASs. Our approach exploits the Internet's hierarchical structure and the commercial relationships between ASs to impose a partial order on the set of routes to each destination. The guidelines conform to conventional traffic-engineering practices of ISPs, and provide each AS with significant flexibility in selecting its local policies. Furthermore, the guidelines ensure route convergence even under changes in the topology and routing policies. Drawing on a formal model of BGP, we prove that following our proposed policy guidelines guarantees route convergence. We also describe how our methodology can be applied to new types of relationships between ASs, how to verify the hierarchical AS relationships, and how to realize our policy guidelines. Our approach has significant practical value since it preserves the ability of each AS to apply complex local policies without divulging its BGP configurations to others.
521 citations
13 Jul 2001
TL;DR: The level set method and fast sweeping and tagging methods are used to reconstruct surfaces from a scattered data set and the reconstructed surface is smoother than piecewise linear and has a natural scaling in the regularization that allows varying flexibility according to the local sampling density.
Abstract: We describe new formulations and develop fast algorithms for implicit surface reconstruction based on variational and partial differential equation (PDE) methods. In particular we use the level set method and fast sweeping and tagging methods to reconstruct surfaces from a scattered data set. The data set might consist of points, curves and/or surface patches. A weighted minimal surface-like model is constructed and its variational level set formulation is implemented with optimal efficiency. The reconstructed surface is smoother than piecewise linear and has a natural scaling in the regularization that allows varying flexibility according to the local sampling density. As is usual with the level set method we can handle complicated topology and deformations, as well as noisy or highly nonuniform data sets easily. The method is based on a simple rectangular grid, although adaptive and triangular grids are also possible. Some consequences, such as hole filling capability, are demonstrated, as well as the viability and convergence of our new fast tagging algorithm.
456 citations
01 Jan 2001
TL;DR: In this article, a deterministic differential equation can be associated with a recursive algorithm, and problems like convergence with probality one, possible convergence points and asymptotic behavior of the algorithm can all be studied in terms of this differential equation.
Abstract: Recursive algorithms where random observations enter are studied in a fairly general framework. An important feature is that the observations may depend on previous ?outputs? of the algorithm. The considered class of algorithms contains, e.g., stochastic approximation algorithms, recursive identification algorithms, and algorithms for adaptive control of linear systems. It is shown how a deterministic differential equation can be associated with the algorithm. Problems like convergence with probality one, possible convergence points and asymptotic behavior of the algorithm can all be studied in terms of this differential equation. Theorems stating the precise relationships between the differential equation and the algorithm are given as well as examples of applications of the results to problems in identification and adaptive control.
423 citations
TL;DR: In this article, a modification of the BFGS method for unconstrained optimization is proposed, which possesses a global convergence property even without convexity assumption on the objective function.
381 citations
01 Jan 2001
TL;DR: This work constructs a simple and efficient adaptive FEM for elliptic partial differential equations and proves that this algorithm converges with linear rate without any preliminary mesh adaptation nor explicit knowledge of constants.
Abstract: Adaptive finite element methods (FEM) have been widely used in applications for over 20 years now. In practice, they converge starting from coarse grids, although no mathematical theory has been able to prove this assertion. Ensuring an error reduction rate based on a posteriori error estimators, together with a reduction rate of data oscillation (information missed by the underlying averaging process), we construct a simple and efficient adaptive FEM for elliptic partial differential equations. We prove that this algorithm converges with linear rate without any preliminary mesh adaptation nor explicit knowledge of constants. Any prescribed error tolerance is thus achieved in a finite number of steps. A number of numerical experiments in two and three dimensions yield quasi-optimal meshes along with a competitive performance. Extensions to higher order elements and applications to saddle point problems are discussed as well.
Keywords: A posteriori error estimators, data oscillation, adaptive mesh refinement, convergence, Stokes, Uzawa
AMS Subject Classifications: 65N12, 65N15, 65N30, 65N50, 65Y20
Published: SIAM Review, 44 (2002) 631--658.
337 citations
TL;DR: The convergence of a penalty method for solving the discrete regularized American option valuation problem is studied and it is observed that an implicit treatment of the American constraint does not converge quadratically if constant timesteps are used.
Abstract: The convergence of a penalty method for solving the discrete regularized American option valuation problem is studied. Sufficient conditions are derived which both guarantee convergence of the nonlinear penalty iteration and ensure that the iterates converge monotonically to the solution. These conditions also ensure that the solution of the penalty problem is an approximate solution to the discrete linear complementarity problem. The efficiency and quality of solutions obtained using the implicit penalty method are compared with those produced with the commonly used technique of handling the American constraint explicitly. Convergence rates are studied as the timestep and mesh size tend to zero. It is observed that an implicit treatment of the American constraint does not converge quadratically (as the timestep is reduced) if constant timesteps are used. A timestep selector is suggested which restores quadratic convergence.
319 citations
22 Apr 2001
TL;DR: The time for end-to-end Internet convergence depends on the length of the longest possible backup autonomous system path between a source and destination node, and significant variation in the convergence behavior of Internet service providers is demonstrated, with the larger providers exhibiting the fastest convergence latencies.
Abstract: This paper examines the role inter-domain topology and routing policy play in the process of delayed Internet routing convergence. In previous work, we showed that the Internet lacks effective inter-domain path fail-over. Unlike circuit-switched networks which exhibit fail-over on the order of milliseconds, we found Internet backbone routers may take tens of minutes to reach a consistent view of the network topology after a fault. In this paper, we expand an our earlier work by exploring the impact of specific Internet provider policies and topologies on the speed of routing convergence. Based on data from the experimental injection and measurement of several hundred thousand inter-domain routing faults, we show that the time for end-to-end Internet convergence depends on the length of the longest possible backup autonomous system path between a source and destination node. We also demonstrate significant variation in the convergence behavior of Internet service providers, with the larger providers exhibiting the fastest convergence latencies. Finally, we discuss possible modifications to BGP and provider routing policies which if deployed, would improve inter-domain routing convergence.
TL;DR: In this paper, the convergence of an implicit iteration process to a common fixed point of a finite family of non-pansive mappings in a Hilbert space was proved, and it was shown that the convergence to a fixed point can be achieved by a finite iterative process.
Abstract: We prove the convergence of an implicit iteration process to a common fixed point of a finite family of nonexpansive mappings in a Hilbert space.
01 Jan 2001
TL;DR: A quasi-Fejer sequence is a sequence which satisfies the standard Fejer monotonicityproperty to within an additional error term and is shown to provide a powerful framework to analyze the convergence of a wide range of optimization algorithms in a systematic fashion.
Abstract: A quasi-Fejer sequence is a sequence which satisfies the standard Fejer monotonicityproperty to within an additional error term. This notion is studied in detail in a Hilbert space setting and shown to provide a powerful framework to analyze the convergence of a wide range of optimization algorithms in a systematic fashion. A number of convergence theorems covering and extending existing results are thus established. Special emphasis is placed on the design and the analysis of parallel algorithms.
TL;DR: The convergence and divergence regions for some algorithms are given, and the new algorithms are applied to solve the Stokes equations as well.
Abstract: Several SOR-like methods are proposed for solving augmented systems. These have many different applications in scientific computing, for example, constrained optimization and the finite element method for solving the Stokes equation. The convergence and the choice of optimal parameter for these algorithms are studied. The convergence and divergence regions for some algorithms are given, and the new algorithms are applied to solve the Stokes equations as well.
TL;DR: The asymptotic convergence of the algorithm used by the software SVM(light) and other later implementation is proved and the size of the working set can be any even number.
Abstract: The decomposition method is currently one of the major methods for solving support vector machines (SVM). Its convergence properties have not been fully understood. The general asymptotic convergence was first proposed by Chang et al. However, their working set selection does not coincide with existing implementation. A later breakthrough by Keerthi and Gilbert (2000, 2002) proved the convergence finite termination for practical cases while the size of the working set is restricted to two. In this paper, we prove the asymptotic convergence of the algorithm used by the software SVM/sup light/ and other later implementation. The size of the working set can be any even number. Extensions to other SVM formulations are also discussed.
22 Apr 2001
TL;DR: A new distributed routing algorithm that performs dynamic load-balancing for wireless access networks and constructs a load-balanced backbone tree, which simplifies routing and avoids per-destination state for routing and per-flow state for QoS reservations.
Abstract: The widespread use of wireless devices presents new challenges for network operators, who need to provide service to ever larger numbers of mobile end users, while ensuring quality-of-service guarantees. We describe a new distributed routing algorithm that performs dynamic load-balancing for wireless access networks. The algorithm constructs a load-balanced backbone tree, which simplifies routing and avoids per-destination state for routing and per-flow state for QoS reservations. We evaluate the performance of the algorithm using several metrics including adaptation to mobility, degree of load-balance, bandwidth blocking rate, and convergence speed. We find that the algorithm achieves better network utilization by lowering bandwidth blocking rates than other methods.
15 Jul 2001
TL;DR: The modified Levenberg-Marquardt algorithm for feedforward neural networks gives a better convergence rate compared to the standard LM method and is less computationally intensive and requires less memory.
Abstract: In this work, two modifications on Levenberg-Marquardt (LM) algorithm for feedforward neural networks are studied. One modification is made on performance index, while the other one is on calculating gradient information. The modified algorithm gives a better convergence rate compared to the standard LM method and is less computationally intensive and requires less memory. The performance of the algorithm has been checked on several example problems.
22 Apr 2001
TL;DR: This work presents the first distance-vector routing algorithm, MDVA, that uses a set of loop-free invariants to prevent the count-to-infinity problem, and shows how such loops can be used in traffic load-balancing and minimizing delays, which otherwise are impossible to perform in current single-path routing algorithms.
Abstract: Routing protocols using the distributed Bellman-Ford (DBF) algorithm converge very slowly to the correct routes when link costs increase, and in the case when a set of link failures results in a network partition, DBF simply fails to converge, a problem which is commonly referred to as the count-to-infinity problem. We present the first distance-vector routing algorithm, MDVA, that uses a set of loop-free invariants to prevent the count-to-infinity problem. MDVA, in addition, computes multipaths that are loop-free at every instant. In our earlier work we shows how such loop-free multipaths can be used in traffic load-balancing and minimizing delays, which otherwise are impossible to perform in current single-path routing algorithms.
25 Jun 2001
TL;DR: An approach to estimation for continuous-time and discrete-time linear systems is proposed that is based on the idea of using switching observers, and a method is described to design a switching observer that aims to minimize the upper bound to the estimation cost function.
Abstract: An approach to estimation for continuous-time and discrete-time linear systems is proposed that is based on the idea of using switching observers. Convergence conditions have been found to ensure the stability of the error dynamics; in addition, they guarantee the existence of an upper bound to a quadratic cost function of the estimation error. The observer gains may be selected by solving a set of linear matrix inequalities (LMIs). A method is described to design a switching observer that aims to minimize the upper bound to the estimation cost function. Moreover, such a design may be efficiently accomplished by using an LMI algorithm.
TL;DR: A more general model of CNNs with unbounded delay is proposed, which may have potential applications in processing such motion related phenomena as moving images, and global convergence properties of this model are studied.
Abstract: Cellular neural networks (CNNs) have been successfully applied in many areas such as classification of patterns, image processing, associative memories, etc. Since they are inherently local in nature, they can be easily implemented in very large scale integration. In the processing of static images, CNNs without delay are often applied whereas in the processing of moving images, CNNs with delay have been found more suitable. This paper proposes a more general model of CNNs with unbounded delay, which may have potential applications in processing such motion related phenomena as moving images, and studies global convergence properties of this model. The dynamic behaviors of CNNs, especially their convergence properties, play important roles in applications. This paper: (1) introduces a class of CNNs with unbounded delay; (2) gives some interesting properties of a network's output function; (3) establishes relationships between a network's state stability and its output stability; and (4) obtains simple and easily checkable conditions for global convergence by functional differential equation methods.
TL;DR: It is shown that close to a regular solution of a monotone system of semismooth equations, two Newton iterations are sufficient to solve the proximal subproblem within the required error tolerance.
Abstract: We present a unified framework for the design and convergence analysis of a class of algorithms based on approximate solution of proximal point subproblems. Our development further enhances the constructive approximation approach of the recently proposed hybrid projection–proximal and extragradient–proximal methods. Specifically, we introduce an even more flexible error tolerance criterion, as well as provide a unified view of these two algorithms. Our general method possesses global convergence and local (super)linear rate of convergence under standard assumptions, while using a constructive approximation criterion suitable for a number of specific implementations. For example, we show that close to a regular solution of a monotone system of semismooth equations, two Newton iterations are sufficient to solve the proximal subproblem within the required error tolerance. Such systems of equations arise naturally when reformulating the nonlinear complementarity problem. *Research of the first author is suppo...
04 Dec 2001
TL;DR: In this article, the optimal control of switched piecewise linear autonomous systems is studied, where the objective is to minimize a quadratic performance index over an infinite time horizon, assuming that the switching sequence and the corresponding jump matrix sequence are known, while the unknown switching times are the optimization parameters.
Abstract: The paper deals with the optimal control of switched piecewise linear autonomous systems, where the objective is that of minimizing a quadratic performance index over an infinite time horizon. We assume that the switching sequence and the corresponding jump matrix sequence is known, while the unknown switching times are the optimization parameters. The optimal control for this class of systems, assuming a switching sequence of finite length, takes the form of a homogeneous state feedback, i.e., it is possible to identify a homogeneous region of the state space such that an optimal switch should occur if and only if the present state belongs to this region. We show how such a region can be computed with a numerical procedure. As the number of allowed switches goes to infinity, we study the stability of the system and discuss some preliminary results related to the convergence of the state feedback law.
TL;DR: In this paper, the convergence of a dynamic iteration method with a (small) finite number of iteration steps in each window is studied for quasilinear problems from electrical circuit simulation and for index-3 systems arising in multibody dynamics.
Abstract: The network approach to the modelling of complex technical systems results frequently in a set of differential-algebraic systems that are connected by coupling conditions. A common approach to the numerical solution of such coupled problems is based on the coupling of standard time integration methods for the subsystems. As a unified framework for the convergence analysis of such multi-rate, multi-method or dynamic iteration approaches we study in the present paper the convergence of a dynamic iteration method with a (small) finite number of iteration steps in each window. Preconditioning is used to guarantee stability of the coupled numerical methods. The theoretical results are applied to quasilinear problems from electrical circuit simulation and to index-3 systems arising in multibody dynamics.
04 Nov 2001
TL;DR: A novel gridlessDetailed routing approach based on multilevel optimization with recursive coarsening and refinement in a "V-shaped" flow that allows efficient scaling of the gridless detailed router to very large designs is presented.
Abstract: This paper presents a novel gridless detailed routing approach based on multilevel optimization. The multilevel framework with recursive coarsening and refinement in a "V-shaped" flow allows efficient scaling of our gridless detailed router to very large designs. The downward pass of recursive coarsening builds the representations of routing regions at different levels, while the upward pass of iterative refinement allows a gradual convergence to a globally optimized solution. The use of a multicommodity flow-based routing algorithm for the initial routing at the coarsest level and a modified maze algorithm for the refinement at each level considerably improves the quality of gridless routing results. Compared with the recently published gridless detailed routing algorithm using wire planning [1], our multilevel gridless routing algorithm is 3× to 75× faster. We also compared our multilevel framework with a recently developed three-level routing approach [1] and a traditional hierarchical routing approach. Our multilevel algorithm generates better detailed routing results with higher completion rates. To our knowledge, this is the first time that multilevel optimization has been applied to IC routing.
08 Jan 2001
TL;DR: Initial experiments confirm the promise of the harmonic balance method to achieve significant reductions in computational cost, and the computational efficiency is compared with dual time stepping algorithms.
Abstract: This paper presents a technique used to accelerate the convergence of unsteady calculations of time-periodic flows to a periodic steady state. The basis of the procedure is the use of the discrete Fourier transform in time, and is similar to the harmonic balance procedure that has been pursued by Hall et. al. The technique is amenable to parallel processing, and convergence acceleration techniques such as multi-grid and implicit residual averaging. The computational efficiency of this method is compared with dual time stepping algorithms. Sample calculations are provided, and a comparison between solutions with varying temporal resolution is presented. The results show that the computational efficiency of the harmonic balance technique is largely a function of the temporal resolution. Initial experiments confirm the promise of the harmonic balance method to achieve significant reductions in computational cost.
01 Nov 2001
TL;DR: This paper presents black-box methods for estimating and trending delays for key internal tasks in OSPF: processing Link State Advertisements, performing Shortest Path First calculations, updating the Forwarding Information Base, and flooding LSAs.
Abstract: OSPF (Open Shortest Path First) is a widely used intra-domain routing protocol in IP networks. Internal processing delays in OSPF implementations impact the speed at which updates propagate in the network, the load on individual routers, and the time needed for both intra-domain and inter-domain routing to reconverge following an internal topology or a configuration change. An OSPF user, such as an Internet Service Provider, typically has no access to the software implementation, and no way to estimate these delays directly. In this paper, we present black-box methods (i.e., measurements that rely only on external observations) for estimating and trending delays for key internal tasks in OSPF: processing Link State Advertisements (LSAs), performing Shortest Path First calculations, updating the Forwarding Information Base, and flooding LSAs. Corresponding measurements are reported for production routers from Cisco Systems. To help validate the methodology, black-box and white-box (i.e., measurements that rely on internal instrumentation) are reported for a open source OSPF implementation, GateD.
TL;DR: An original method to synthesize the dimensions of a planar mechanism (linkage) whose function is to generate a trajectory shape using a neural network and shows on the four-bar linkage example the good quality of the synthesized solutions.
25 Jul 2001
TL;DR: Several mapping rules and heuristics for inferring the ASs of border routers are presented and results showing the effectiveness and validity of these rules andHeuristics are reported on.
Abstract: A number of recent studies characterize AS-level topology of the Internet by exploiting connectivity information contained in BGP routing tables. In this paper, we present an alternative method for discovering AS connectivity by inferring individual AS connections from the Internet's router-level topology. This methodology has several advantages over using BGP routing tables. First, it allows us to obtain AS-level connectivity information at a finer granularity (e.g., multiple connections between a pair of ASs); second, we can discover ASs aggregated in BGP routing tables; and third, we can identify AS border routers, which may allow us to further characterize inter-AS connections. Since border routers have, by definition, multiple interfaces, each with an address in a potentially different AS, a major challenge of our approach is to properly map border routers to their corresponding ASs. To this end, we present in this paper several mapping rules and heuristics for inferring the ASs of border routers and report on results showing the effectiveness and validity of these rules and heuristics.© (2001) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.