Asymptotic Convergence Factor Research Articles

A Flexible Extended Krylov Subspace Method for Approximating Markov Functions of Matrices

A flexible extended Krylov subspace method (F-EKSM) is considered for numerical approximation of the action of a matrix function f(A) to a vector b, where the function f is of Markov type. F-EKSM has the same framework as the extended Krylov subspace method (EKSM), replacing the zero pole in EKSM with a properly chosen fixed nonzero pole. For symmetric positive definite matrices, the optimal fixed pole is derived for F-EKSM to achieve the lowest possible upper bound on the asymptotic convergence factor, which is lower than that of EKSM. The analysis is based on properties of Faber polynomials of A and (I−A/s)−1. For large and sparse matrices that can be handled efficiently by LU factorizations, numerical experiments show that F-EKSM and a variant of RKSM based on a small number of fixed poles outperform EKSM in both storage and runtime, and usually have advantages over adaptive RKSM in runtime.

Cooperative Spacing Sampled Control of Vehicle Platoon Considering Undirected Topology and Analog Fading Networks

The work on this paper studies the combined effect of vehicle dynamics, sampling period and the quality of inter-vehicle sensing and communication on the performance of platoon control. First, we induce the sampled control protocol from continuous–time linear consensus protocol by employing zero–order hold circuit and periodic sampling technology, then by virtue of the obtained sampled control protocol, the continuous–time platoon system is equivalently transformed into a discrete–time system. Second, for the inter–vehicle communication with ideal channel and under undirected information flow topology (UIFT), the stability thresholds of control gains are explicitly established by solving a discrete–time simultaneous stabilization problem, bilinear transformation and the Routh-Hurwitz stability criterion. Meanwhile, to ensure the fastest asymptotic stability of the platoon dynamics, we propose an optimal asymptotic convergence factor and a correspondingly optimal sampled control protocol. Third, For the inter–vehicle communication with identical fading networks and under UIFT, through solving a modified Riccati inequality, we provide a sampled control protocol depending not only on information flow topology (IFT) and sampling period, but also on the statistics of the inter–vehicle communication channel. Besides, employing Lyapunov inequality in probability theory, we show a necessary condition for the sampled control protocol ensuring the platoon dynamics mean square stable. Finally, simulations are performed to demonstrate the theory’s discoveries.

Spectral Coarse Spaces for the Substructured Parallel Schwarz Method

The parallel Schwarz method (PSM) is an overlapping domain decomposition (DD) method to solve partial differential equations (PDEs). Similarly to classical nonoverlapping DD methods, the PSM admits a substructured formulation, that is, it can be formulated as an iteration acting on variables defined exclusively on the interfaces of the overlapping decomposition. In this manuscript, spectral coarse spaces are considered to improve the convergence and robustness of the substructured PSM. In this framework, the coarse space functions are defined exclusively on the interfaces. This is in contrast to classical two-level volume methods, where the coarse functions are defined in volume, though with local support. The approach presented in this work has several advantages. First, it allows one to use some of the well-known efficient coarse spaces proposed in the literature, and facilitates the numerical construction of efficient coarse spaces. Second, the computational work is comparable or lower than standard volume two-level methods. Third, it opens new interesting perspectives as the analysis of the new two-level substructured method requires the development of a new convergence analysis of general two-level iterative methods. The new analysis casts light on the optimality of coarse spaces: given a fixed dimension m, the spectral coarse space made by the first m dominant eigenvectors is not necessarily the minimizer of the asymptotic convergence factor. Numerical experiments demonstrate the effectiveness of the proposed new numerical framework.

On the Asymptotic Linear Convergence Speed of Anderson Acceleration Applied to ADMM

Empirical results show that Anderson acceleration (AA) can be a powerful mechanism to improve the asymptotic linear convergence speed of the Alternating Direction Method of Multipliers (ADMM) when ADMM by itself converges linearly. However, theoretical results to quantify this improvement do not exist yet. In this paper we explain and quantify this improvement in linear asymptotic convergence speed for the special case of a stationary version of AA applied to ADMM. We do so by considering the spectral properties of the Jacobians of ADMM and the stationary version of AA evaluated at the fixed point, where the coefficients of the stationary AA method are computed such that its asymptotic linear convergence factor is optimal. The optimal linear convergence factors of this stationary AA-ADMM method are computed analytically or by optimization, based on previous work on optimal stationary AA acceleration. Using this spectral picture and those analytical results, our approach provides new insight into how and by how much the stationary AA method can improve the asymptotic linear convergence factor of ADMM. Numerical results also indicate that the optimal linear convergence factor of the stationary AA methods gives a useful estimate for the asymptotic linear convergence speed of the non-stationary AA method that is used in practice.

A modulus-based multigrid method for nonlinear complementarity problems with application to free boundary problems with nonlinear source terms

To overcome the dependence of the convergence rate on the grid size in the existing modulus-based method, we present a modulus-based multigrid method to efficiently solve the nonlinear complementarity problems. In this paper, the nonlinear complementarity problems under consideration arise from free boundary problems with nonlinear source terms. The two-grid local Fourier analysis is given to predict the asymptotic convergence factor and the optimal relaxation parameter of the presented modulus-based multigrid method, and the predictions are agreement with the experimental results. Numerical results also show that both W- and F-cycles significantly outperform the existing modulus-based method and achieve asymptotic optimality in terms of grid-independent convergence rate and linear CPU time when the grid is refined.

A modified modulus-based multigrid method for linear complementarity problems arising from free boundary problems

The linear complementarity problem arising from a free boundary problem can be equivalently reformulated as a fixed-point equation. We present a modified modulus-based multigrid method to solve this fixed-point equation. This modified method is a full approximation scheme using the modulus-based splitting iteration method as the smoother and avoids the transformation between the auxiliary and the original functions which was necessary in the existing modulus-based multigrid method. We predict its asymptotic convergence factor by applying local Fourier analysis to the corresponding two-grid case. Numerical results show that the W-cycle possesses an h-independent convergence rate and a linear elapsed CPU time, and the convergence rate of the V-cycle can be improved by increasing the smoothing steps. Compared with the existing modulus-based multigrid method, the modified method is more straightforward and is a standard full approximation scheme, which makes it more convenient and efficient in practical applications.

On the robustness of the multigrid method combining ILU and Partial Weight applied in an orthotropic diffusion problem

This paper proposes an efficient and robust algorithm for solving a physical orthotropy problem. The algorithm is based on choosing the most efficient restriction operator and on an incomplete LU decomposition suited for each orthotropy direction. Local Fourier Analysis (LFA) is carried out in order to increase the efficiency of the multigrid method. Pure diffusion with orthotropy aligned to the coordinate axis x is the model considered. Equations are discretized by Finite Difference Method with uniform grid and second-order numerical scheme. Problems are solved with geometric multigrid method, correction scheme, V-cycle and standard coarsening ratio. The asymptotic convergence factor is calculated for different multigrid components, such as restriction operators, prolongation operators and solvers. Based on the optimum components obtained by LFA, we carried out experiments to analyze the complexity and computational cost of the algorithm proposed. The main conclusion is that the methodology proposed is efficient for the resolution of problems with strong orthotropy.

Pulse-Modulated Intermittent Control in Consensus of Multiagent Systems

This paper proposes a control framework, called pulse-modulated intermittent control, which unifies impulsive control and sampled control. Specifically, the concept of pulse function is introduced to characterize the control/rest intervals and the amplitude of the control. By choosing some specified functions as the pulse function, the proposed control scheme can be reduced to sampled control or impulsive control. The proposed control framework is applied to consensus problems of multiagent systems. Using discretization approaches and stability theory, several necessary and sufficient conditions are established to ensure the consensus of the controlled system. The results show that consensus depends not only on the network topology, the sampling period and the control gains, but also the pulse function. Moreover, a lower bound of the asymptotic convergence factor is derived as well. For a given pulse function and an undirected graph, an optimal control gain is designed to achieve the fastest convergence. In addition, impulsive control and sampled control are revisited in the proposed control framework. Finally, some numerical examples are given to verify the effectiveness of theoretical results.

Modulus‐based multigrid methods for linear complementarity problems

SummaryBy employing modulus‐based matrix splitting iteration methods as smoothers, we establish modulus‐based multigrid methods for solving large sparse linear complementarity problems. The local Fourier analysis is used to quantitatively predict the asymptotic convergence factor of this class of multigrid methods. Numerical results indicate that the modulus‐based multigrid methods of the W‐cycle can achieve optimality in terms of both convergence factor and computing time, and their asymptotic convergence factors can be predicted perfectly by the local Fourier analysis of the corresponding modulus‐based two‐grid methods.

Leader-follower consensus and synchronization in numerosity-constrained networks with dynamic leadership.

In this work, we study leader-follower consensus and synchronization protocols over a stochastically switching network. The agents representing the followers can communicate with any other agent, whereas the agents serving as leaders are restricted to interact only with the other leaders. The model incorporates the phenomenon of numerosity, which limits the perceptual capacity of the agents while allowing for shuffling with whom each individual interacts at each time step. We derive closed form expressions for necessary and sufficient conditions for consensus, the rate of convergence to consensus, and conditions for stochastic synchronization in terms of the asymptotic convergence factor. We provide simulation results to validate the theoretical findings and to illustrate the dependence of this factor on system parameters. The closed form results enable us to study the factors affecting the feasibility of consensus. We show that agents' traits can be chosen for an engineered system to maximize the convergence speed and that protocol speed is enhanced as the proportion of the leaders increases in certain cases. These results may find application in the design and control of an engineered leader-follower system, where consensus or synchronization at the fastest possible rate is desired.

Ultra-fast formation control of high-order discrete-time multi-agent systems based on multi-step predictive mechanism

This paper deals with the ultra-fast formation control problem of high-order discrete-time multi-agent systems. Using the local neighbor-error knowledge, a novel ultra-fast protocol with multi-step predictive information and self-feedback term is proposed. The asymptotic convergence factor is improved by a power of q+1 compared to the routine protocol. To some extent, the ultra-fast algorithm overcomes the influence of communication topology to the convergence speed. Furthermore, some sufficient conditions are given herein. The ones decouple the design of the synchronizing gains from the detailed graph properties, and explicitly reveal how the agent dynamic and the communication graph jointly affect the ultra-fast formationability. Finally, some simulations are worked out to illustrate the effectiveness of our theoretical results.

Ultra-fast consensus of discrete-time multi-agent systems under a unified framework

This paper deals with the ultra-fast consensus problem of high-order discrete-time multi-agent systems under a unified framework. A multi-step neighbour-error predictive mechanism is established based on the future evolution of the network dynamic without predictive information. By predicting the dynamics of a network several steps ahead and adding this information into the consensus protocol, a novel ultra-fast consensus protocol with the self-feedback term is proposed. The asymptotic convergence factor is improved by a power of q + 1 compared to the routine consensus protocol. The ultra-fast consensus algorithm can overcome the influence of communication topology to the convergence speed toward consensus, and solve the difficult problem of selecting the optimal control gain which minimises the asymptotic convergence factor. Moveover, some sufficient conditions for ultra-fast consensus design are given herein. The ones decouple the design of the synchronising gains from the detailed graph properties, and explicitly reveal how the agent dynamic and the communication graph jointly affect ultra-fast consensus of discrete-time multi-agent systems. A simulation is performed to illustrate the effectiveness of the theoretical results.

Ultra-fast consensus of discrete-time multi-agent systems with multi-step predictive output feedback

This article addresses the ultra-fast consensus problem of high-order discrete-time multi-agent systems based on a unified consensus framework. A novel multi-step predictive output mechanism is proposed under a directed communication topology containing a spanning tree. By predicting the outputs of a network several steps ahead and adding this information into the consensus protocol, it is shown that the asymptotic convergence factor is improved by a power of q + 1 compared to the routine consensus. The difficult problem of selecting the optimal control gain is solved well by introducing a variable called convergence step. In addition, the ultra-fast formation achievement is studied on the basis of this new consensus protocol. Finally, the ultra-fast consensus with respect to a reference model and robust consensus is discussed. Some simulations are performed to illustrate the effectiveness of the theoretical results.

Composite convergence bounds based on Chebyshev polynomials and finite precision conjugate gradient computations

The conjugate gradient method (CG) for solving linear systems of algebraic equations represents a highly nonlinear finite process. Since the original paper of Hestenes and Stiefel published in 1952, it has been linked with the Gauss-Christoffel quadrature approximation of Riemann-Stieltjes distribution functions determined by the data, i.e., with a simplified form of the Stieltjes moment problem. This link, developed further by Vorobyev, Brezinski, Golub, Meurant and others, indicates that a general description of the CG rate of convergence using an asymptotic convergence factor has principal limitations. Moreover, CG is computationally based on short recurrences. In finite precision arithmetic its behaviour is therefore affected by a possible loss of orthogonality among the computed direction vectors. Consequently, any consideration concerning the CG rate of convergence relevant to practical computations must include analysis of effects of rounding errors. Through the example of composite convergence bounds based on Chebyshev polynomials, this paper argues that the facts mentioned above should become a part of common considerations on the CG rate of convergence. It also explains that the spectrum composed of small number of well separated tight clusters of eigenvalues does not necessarily imply a fast convergence of CG or other Krylov subspace methods.

Leader–follower consensus over numerosity-constrained random networks

In this work, we study a discrete-time consensus protocol for a group of agents which communicate over a class of stochastically switching networks inspired by fish schooling. The network model incorporates the phenomenon of numerosity, that plays a prominent role in the collective behavior of animal groups, by defining the individuals’ perception of numbers. The agents comprise leaders, which share a common state, and followers, which update their states based on information exchange among neighboring agents. We establish a closed form expression for the asymptotic convergence factor of the protocol, that measures the decay rate of disagreement among the followers’ and the leaders’ states. Handleable forms of this expression are derived for the physically relevant cases of large networks whose agents are composed of primarily leaders or followers. Numerical simulations are conducted to validate analytical results and illustrate the consensus dynamics as a function of the number of leaders in the group, the agents’ persuasibility, and the agents’ numerosity. We find that the maximum speed of convergence for a given population can be enhanced by increasing the proportion of leaders in the group or the agents’ numerosity. On the other hand, we find that increasing the numerosity has also a negative effect as it reduces the range of agents’ persuasibility for which consensus is possible. Finally, we compare the main features of this leader–follower consensus protocol with its leaderless counterpart to elucidate the benefits and drawbacks of leadership in numerosity-constrained random networks.

Consensus Over Numerosity-Constrained Random Networks

We analyze the discrete-time consensus problem for a group of agents that communicate through a stochastic directed network with fixed out-degree. This network construction models the perceptual phenomenon of numerosity observed in animal groups exhibiting collective behavior. We find necessary and sufficient conditions for mean square consentability of the averaging protocol and we derive a closed form expression for the asymptotic convergence factor. Analytical results are illustrated through simulations.

Estimates for the asymptotic convergence factor of two intervals

Let E be the union of two real intervals not containing zero. Then L n r ( E ) denotes the supremum norm of that polynomial P n of degree less than or equal to n , which is minimal with respect to the supremum norm provided that P n ( 0 ) = 1 . It is well known that the limit κ ( E ) ≔ lim n → ∞ L n r ( E ) n exists, where κ ( E ) is called the asymptotic convergence factor, since it plays a crucial role for certain iterative methods solving large-scale matrix problems. The factor κ ( E ) can be expressed with the help of Jacobi’s elliptic and theta functions, where this representation is very involved. In this paper, we give precise upper and lower bounds for κ ( E ) in terms of elementary functions of the endpoints of E .

An optimized Schwarz method with two‐sided Robin transmission conditions for the Helmholtz equation

AbstractOptimized Schwarz methods are working like classical Schwarz methods, but they are exchanging physically more valuable information between subdomains and hence have better convergence behaviour. The new transmission conditions include also derivative information, not just function values, and optimized Schwarz methods can be used without overlap. In this paper, we present a new optimized Schwarz method without overlap in the 2d case, which uses a different Robin condition for neighbouring subdomains at their common interface, and which we call two‐sided Robin condition. We optimize the parameters in the Robin conditions and show that for a fixed frequency an asymptotic convergence factor of 1 – O(h1/4) in the mesh parameter h can be achieved. If the frequency is related to the mesh parameter h, h = O(1/ωγ) for γ⩾1, then the optimized asymptotic convergence factor is 1 – O(ω(1–2γ)/8). We illustrate our analysis with 2d numerical experiments. Copyright © 2007 John Wiley & Sons, Ltd.

Alternately linearized implicit iteration methods for the minimal nonnegative solutions of the nonsymmetric algebraic Riccati equations

AbstractFor the non‐symmetric algebraic Riccati equations, we establish a class of alternately linearized implicit (ALI) iteration methods for computing its minimal non‐negative solutions by technical combination of alternate splitting and successive approximating of the algebraic Riccati operators. These methods include one iteration parameter, and suitable choices of this parameter may result in fast convergent iteration methods. Under suitable conditions, we prove the monotone convergence and estimate the asymptotic convergence factor of the ALI iteration matrix sequences. Numerical experiments show that the ALI iteration methods are feasible and effective, and can outperform the Newton iteration method and the fixed‐point iteration methods. Besides, we further generalize the known fixed‐point iterations, obtaining an extensive class of relaxed splitting iteration methods for solving the non‐symmetric algebraic Riccati equations. Copyright © 2006 John Wiley & Sons, Ltd.

Sharp Convergence Estimates for the Preconditioned Steepest Descent Method for Hermitian Eigenvalue Problems

The paper is concerned with convergence estimates for the preconditioned steepest descent method for the computation of the smallest eigenvalue of a Hermitian operator. Available estimates are reviewed and new estimates are introduced that improve on the known ones in certain respects. In addition to the estimates for the error reduction after one iteration, we consider estimates for the so-called asymptotic convergence factor defined as the upper limit of the average error reduction per iteration. The paper focuses on sharp estimates, i.e., those that cannot be improved without using additional information.