New Synchronization Algorithm and Analysis of Its Convergence Rate for Clock Oscillators in Dynamical Network with Time-Delays

New synchronization algorithm and analysis of its convergence rate for clock oscillators in dynamical network with time-delays are presented.A network of nodes equipped with hardware clock oscillators with bounded drift is considered.Firstly,a dynamic synchronization algorithm based on consensus control strategy,namely fast averaging synchronization algorithm (FASA),is presented to find the solutions to the synchronization problem.By FASA,each node computes the logical clock value based on its value of hardware clock and message exchange.The goal is to synchronize all the nodes' logical clocks as closely as possible.Secondly,the convergence rate of FASA is analyzed that proves it is related to the bound by a nondecreasing function of the uncertainty in message delay and network parameters.Then,FASA's convergence rate is proven by means of the robust optimal design.Meanwhile,several practical applications for FASA,especially the application to inverse global positioning system (IGPS) base station network are discussed.Finally,numerical simulation results demonstrate the correctness and efficiency of the proposed FASA.Compared FASA with traditional clock synchronization algorithms (CSAs),the convergence rate of the proposed algorithm converges faster than that of the CSAs evidently.

CENTRAL LIMIT THEOREM AND CONVERGENCE RATES FOR A SUPERCRITICAL BRANCHING PROCESS WITH IMMIGRATION IN A RANDOM ENVIRONMENT

We are interested in the convergence rates of the submartingale Wn=Z_(n)/Π_(n)to its limit W,where(Π_(n))is the usually used norming sequence and(Z_(n))is a supercritical branching process with immigration(Y_(n))in a stationary and ergodic environmentξ.Under suitable conditions,we establish the following central limit theorems and results about the rates of convergence in probability or in law:(i)W-W_(n) with suitable normalization converges to the normal law N(0,1),and similar results also hold for W_(n+k)-W_(n) for each fixed k∈N^(*);(ii)for a branching process with immigration in a finite state random environment,if W_(1) has a finite exponential moment,then so does W,and the decay rate of P(|W-W_(n)|>ε)is supergeometric;(iii)there are normalizing constants an(ξ)(that we calculate explicitly)such that a_(n)(ξ)(W-W_(n))converges in law to a mixture of the Gaussian law.

The convergence rate and necessary-and-sufficient condition for the consistency of isogeometric collocation method

Although the isogeometric collocation(IGA-C)method has been successfully utilized in practical applications due to its simplicity and efficiency,only a little theoretical results have been established on the numerical analysis of the IGA-C method.In this paper,we deduce the convergence rate of the consistency of the IGA-C method.Moreover,based on the formula of the convergence rate,the necessary and sufficient condition for the consistency of the IGA-C method is developed.These results advance the numerical analysis of the IGA-C method.

Convergence Rates for Elliptic Homogenization Problems in Two-dimensional Domain

In this paper, we study the convergence rates of solutions for second order elliptic equations with rapidly oscillating periodic coefficients in two-dimensional domain. We use an extension of the "mixed formulation" approach to obtain the representation formula satisfied by the oscillatory solution and homogenized solution by means of the particularity of solutions for equations in two-dimensional case. Then we utilize this formula in combination with the asymptotic estimates of Green or Neumann functions for operators and uniform regularity estimates of solutions to obtain convergence rates in L^p for solutions as well as gradient error estimates for Dirichlet or Neumann problems respectively.

CONVERGENCE RATES FOR LACUNARY TRIGONOMETRIC INTERPOLATION IN _(2π)~p SPACES

The saturation rate and class of (0,m1,m2, …,mq) trigonometric inter polation operators in . spaces have been determined by Cavaretta and Selvaraj. In this paper, we consider the convergence and saturation problems of these operators in (1≤p≤∞) and obtain complete results.

Error Estimations, Error Computations, and Convergence Rates in FEM for BVPs

This paper presents derivation of a priori error estimates and convergence rates of finite element processes for boundary value problems (BVPs) described by self adjoint, non-self adjoint, and nonlinear differential operators. A posteriori error estimates are discussed in context with local approximations in higher order scalar product spaces. A posteriori error computational framework (without the knowledge of theoretical solution) is presented for all BVPs regardless of the method of approximation employed in constructing the integral form. This enables computations of local errors as well as the global errors in the computed finite element solutions. The two most significant and essential aspects of the research presented in this paper that enable all of the features described above are: 1) ensuring variational consistency of the integral form(s) resulting from the methods of approximation for self adjoint, non-self adjoint, and nonlinear differential operators and 2) choosing local approximations for the elements of a discretization in a subspace of a higher order scalar product space that is minimally conforming, hence ensuring desired global differentiability of the approximations over the discretizations. It is shown that when the theoretical solution of a BVP is analytic, the a priori error estimate (in the asymptotic range, discussed in a later section of the paper) is independent of the method of approximation or the nature of the differential operator provided the resulting integral form is variationally consistent. Thus, the finite element processes utilizing integral forms based on different methods of approximation but resulting in VC integral forms result in the same a priori error estimate and convergence rate. It is shown that a variationally consistent (VC) integral form has best approximation property in some norm, conversely an integral form with best approximation property in some norm is variationally consistent. That is best approximation property of the integral form and the VC of the integral form is equivalent, one cannot exist without the other, hence can be used interchangeably. Dimensional model problems consisting of diffusion equation, convection-diffusion equation, and Burgers equation described by self adjoint, non-self adjoint, and nonlinear differential operators are considered to present extensive numerical studies using Galerkin method with weak form (GM/WF) and least squares process (LSP) to determine computed convergence rates of various error norms and present comparisons with the theoretical convergence rates.

An Adaptive Load Stepping Algorithm for Path-Dependent Problems Based on Estimated Convergence Rates

A new adaptive(automatic)time stepping algorithm,called RCA(Rate of Convergence Algorithm)is presented.The new algorithm was applied in nonlinear finite element analysis of path-dependent problems.The step size is adjusted by monitoring the estimated convergence rate of the nonlinear iterative process.The RCA algorithm is relatively simple to implement,robust and its performance is comparable to,and in some cases better than,the automatic load incrementaion algorithm existent in commercial codes.Discussions about the convergence rate of nonlinear iterative processes,an estimation of the rate and a study of the parameters of the RCA algorithm are presented.To show the capacity of the algorithm to adjust the increment size,detailed discussions based on results for different limit load analyses are presented.The results obtained by RCA algorithm are compared with those by ABAQUS?,one of the most powerful nonlinear FEA(Finite Element Analysis)commercial software,in order to verify the capability of RCA algorithm to adjust the increment size along nonlinear analyses.

The Convergence Rate of Fréchet Distribution under the Second-Order Regular Variation Condition

In this article we consider the asymptotic behavior of extreme distribution with the extreme value index γ>0 . The rates of uniform convergence for Fréchet distribution are constructed under the second-order regular variation condition.

A.S.Convergence Rate and L^(p)-Convergence of Bisexual Branching Processes in a Random Environment and Varying Environment

Let(Z_(n))be a supercritical bisexual branching process in a random environmentξ.We study the almost sure(a.s.)convergence rate of the submartingale W_(n)=Z_(n)/In to its limit W,where(In)is an usually used norming sequence.We prove that under a moment condition of order p∈(1,2),W-W_(n)=o(e^(-na))a.s.for some a>0 that we find explicitly;assuming the logarithmic moment condition holds,we haveW-W_(n)=o(n^(-α))a.s..In order to obtain these results,we provide the L^(p)-convergence of(W_(n));similar conclusions hold for a bisexual branching process in a varying environment.

Convergence Rate of Solutions to a Hyperbolic Equation with p(x)-Laplacian Operator and Non-Autonomous Damping

This paper concerns the convergence rate of solutions to a hyperbolic equation with p(x)-Laplacian operator and non-autonomous damping.We apply the Faedo-Galerkin method to establish the existence of global solutions,and then use some ideas from the study of second order dynamical system to get the strong convergence relationship between the global solutions and the steady solution.Some differential inequality arguments and a new Lyapunov functional are proved to show the explicit convergence rate of the trajectories.

Uniform convergence rates for spot volatility estimation

This study presents the uniform convergence rate for spot volatility estimators based on delta sequences.Kernel and Fourier-based estimators are examples of this type of estimator.We also present the uniform convergence rates for kernel and Fourier-based estimators of spot volatility as applications of the main result.

On the Sublinear Convergence Rate of Multi-block ADMM

The alternating direction method of multipliers(ADMM)is widely used in solving structured convex optimization problems.Despite its success in practice,the convergence of the standard ADMM for minimizing the sum of N(N≥3)convex functions,whose variables are linked by linear constraints,has remained unclear for a very long time.Recently,Chen et al.(Math Program,doi:10.1007/s10107-014-0826-5,2014)provided a counter-example showing that the ADMM for N≥3 may fail to converge without further conditions.Since the ADMM for N≥3 has been very successful when applied to many problems arising from real practice,it is worth further investigating under what kind of sufficient conditions it can be guaranteed to converge.In this paper,we present such sufficient conditions that can guarantee the sublinear convergence rate for the ADMM for N≥3.Specifically,we show that if one of the functions is convex(not necessarily strongly convex)and the other N-1 functions are strongly convex,and the penalty parameter lies in a certain region,the ADMM converges with rate O(1/t)in a certain ergodic sense and o(1/t)in a certain non-ergodic sense,where t denotes the number of iterations.As a by-product,we also provide a simple proof for the O(1/t)convergence rate of two-blockADMMin terms of both objective error and constraint violation,without assuming any condition on the penalty parameter and strong convexity on the functions.

Convergence Rate Analysis for Deep Ritz Method

Using deep neural networks to solve PDEs has attracted a lot of attentions recently.However,why the deep learning method works is falling far behind its empirical success.In this paper,we provide a rigorous numerical analysis on deep Ritz method(DRM)[47]for second order elliptic equations with Neumann boundary conditions.We establish the first nonasymptotic convergence rate in H^(1)norm for DRM using deep networks with ReLU^(2)activation functions.In addition to providing a theoretical justification of DRM,our study also shed light on how to set the hyperparameter of depth and width to achieve the desired convergence rate in terms of number of training samples.Technically,we derive bound on the approximation error of deep ReLU^(2)network in C^(1)norm and bound on the Rademacher complexity of the non-Lipschitz composition of gradient norm and ReLU^(2)network,both of which are of independent interest.

Convergence Rate for Galerkin Approximation of the Stochastic Allen–Cahn Equations on 2D Torus

In this paper we discuss the convergence rate for Galerkin approximation of the stochastic Allen–Cahn equations driven by space-time white noise on T^(2). First we prove that the convergence rate for stochastic 2D heat equation is of order α-δ in Besov space C^(-α) for α∈(0, 1) and δ > 0 arbitrarily small. Then we obtain the convergence rate for Galerkin approximation of the stochastic Allen–Cahn equations of order α-δ in C^(-α) for α∈(0, 2/9) and δ > 0 arbitrarily small.

On the Convergence Rate of a Class of Proximal-Based Decomposition Methods for Monotone Variational Inequalities

A unified efficient algorithm framework of proximal-based decomposition methods has been proposed for monotone variational inequalities in 2012,while only global convergence is proved at the same time.In this paper,we give a unified proof on theO(1/t)iteration complexity,together with the linear convergence rate for this kind of proximal-based decomposition methods.Besides theε-optimal iteration complexity result defined by variational inequality,the non-ergodic relative error of adjacent iteration points is also proved to decrease in the same order.Further,the linear convergence rate of this algorithm framework can be constructed based on some special variational inequality properties,without necessary strong monotone conditions.

On the Convergence Rate of an Inexact Proximal Point Algorithm for Quasiconvex Minimization on Hadamard Manifolds

In this paper,we present an analysis about the rate of convergence of an inexact proximal point algorithm to solve minimization problems for quasiconvex objective functions on Hadamard manifolds.We prove that under natural assumptions the sequence generated by the algorithm converges linearly or superlinearly to a critical point of the problem.

The Convergence Rate from Discrete to Continuous Optimal Investment Stopping Problem

The author studies the optimal investment stopping problem in both continuous and discrete cases, where the investor needs to choose the optimal trading strategy and optimal stopping time concurrently to maximize the expected utility of terminal wealth.Based on the work of Hu et al.(2018) with an additional stochastic payoff function,the author characterizes the value function for the continuous problem via the theory of quadratic reflected backward stochastic differential equations(BSDEs for short) with unbounded terminal condition. In regard to the discrete problem, she gets the discretization form composed of piecewise quadratic BSDEs recursively under Markovian framework and the assumption of bounded obstacle, and provides some useful a priori estimates about the solutions with the help of an auxiliary forward-backward SDE system and Malliavin calculus. Finally, she obtains the uniform convergence and relevant rate from discretely to continuously quadratic reflected BSDE, which arise from corresponding optimal investment stopping problem through above characterization.

On the Convergence Rate of a Proximal Point Algorithm for Vector Function on Hadamard Manifolds

The proximal point algorithm has many interesting applications,such as signal recovery,signal processing and others.In recent years,the proximal point method has been extended to Riemannian manifolds.The main advantages of these extensions are that nonconvex problems in classic sense may become geodesic convex by introducing an appropriate Riemannian metric,constrained optimization problems may be seen as unconstrained ones.In this paper,we propose an inexact proximal point algorithm for geodesic convex vector function on Hadamard manifolds.Under the assumption that the objective function is coercive,the sequence generated by this algorithm converges to a Pareto critical point.When the objective function is coercive and strictly geodesic convex,the sequence generated by this algorithm converges to a Pareto optimal point.Furthermore,under the weaker growth condition,we prove that the inexact proximal point algorithm has linear/superlinear convergence rate.

Adaptive Retransmission Design for Wireless Federated Edge Learning

As a popular distributed machine learning framework,wireless federated edge learning(FEEL)can keep original data local,while uploading model training updates to protect privacy and prevent data silos.However,since wireless channels are usually unreliable,there is no guarantee that the model updates uploaded by local devices are correct,thus greatly degrading the performance of the wireless FEEL.Conventional retransmission schemes designed for wireless systems generally aim to maximize the system throughput or minimize the packet error rate,which is not suitable for the FEEL system.A novel retransmission scheme is proposed for the FEEL system to make a tradeoff between model training accuracy and retransmission latency.In the proposed scheme,a retransmission device selection criterion is first designed based on the channel condition,the number of local data,and the importance of model updates.In addition,we design the air interface signaling under this retransmission scheme to facilitate the implementation of the proposed scheme in practical scenarios.Finally,the effectiveness of the proposed retransmission scheme is validated through simulation experiments.