convergence of consensus models with stochastic disturbances we consider consensus algorithms in their most general setting and provide conditions under which such algorithms are guaranteed to converge almost surely to a consensus let a t b t rn n be possibly stochastic nonstationary matrices and x t m t 6 rn 1 be state and perturbation vectors respectively for any consensus algorithm of the form x t 1 a t x t b t m t we provide conditions under which consensus is achieved almost surely i e pr limt x t c1 1 for some c r moreover we show that this general result subsumes recently reported results for specific consensus algorithms classes including sum preserving nonsum preserving quantized and noisy gossip algorithms also provided are the e converging time for any such converging iterative algorithm i e the earliest time at which the vector x t is e close to consensus and sufficient conditions for convergence in expectation to the average of the initial node measurements finally mean square error bounds of any consensus algorithm of the form discussed above are presented

convergence and nonlinear stability of the lagrange galerkin method for the navier stokes equations the lagrange galerkin method is a numerical technique for solving convection dominated diffusion problems based on combining a special discretisation of the lagrangian material derivative along particle trajectories with a galerkin finite element method we present optimal error estimates for the lagrange galerkin mixed finite element approximation of the navier stokes equations in a velocity pressure formulation the method is shown to be nonlinearly stable

distributed average consensus with dithered quantization in this paper we develop algorithms for distributed computation of averages of the node data over networks with bandwidth power constraints or large volumes of data distributed averaging algorithms fail to achieve consensus when deterministic uniform quantization is adopted we propose a distributed algorithm in which the nodes utilize probabilistically quantized information i e dithered quantization to communicate with each other the algorithm we develop is a dynamical system that generates sequences achieving a consensus at one of the quantization values almost surely in addition we show that the expected value of the consensus is equal to the average of the original sensor data we derive an upper bound on the mean square error performance of the probabilistically quantized distributed averaging pqda moreover we show that the convergence of the pqda is monotonic by studying the evolution of the minimum length interval containing the node values we reveal that the length of this interval is a monotonically nonincreasing function with limit zero we also demonstrate that all the node values in the worst case converge to the final two quantization bins at the same rate as standard unquantized consensus finally we report the results of simulations conducted to evaluate the behavior and the effectiveness of the proposed algorithm in various scenarios

analysis of an algorithm for the galerkin characteristic method it is shown that the interpretation of the galerkin characteristic method for the scalar advection equation in the framework of particle methods yields a computationally efficient algorithm such an algorithm consists of updating the dependent variable at the grid points by cubic spline interpolation at the feet of the characteristic curves the algorithm is unconditionally stable the error analysis in the maximum norm shows that for sufficiently smooth functions the feet of the characteristic curves are points of high order convergence

average consensus with packet drop communication average consensus consists in the problem of determining the average of some quantities by means of a distributed algorithm it is a simple instance of problems arising when designing estimation algorithms operating on data produced by sensor networks simple solutions based on linear estimation algorithms have already been proposed in the literature and their performance has been analyzed in detail if the communication links which allow the data exchange between the sensors have some loss then the estimation performance will degrade in this contribution the performance degradation due to this data loss is evaluated

finite time median related group consensus over directed networks in this paper we investigate a novel finite time median related group consensus problem where the finial consensus value can be identified as a desired function of the median of initial states in

nonuniqueness for the heat flow of harmonic maps on the disk we prove that for suitable initial data the heat flow of harmonic maps from the disk to the sphere admits infinitely many solutions characterised by backward bubbling at some arbitrarily large time all having uniformly bounded energy

on the transport diffusion algorithm and its applications to the navier stokes equations this paper deals with an algorithm for the solution of diffusion and or convection equations where we mixed the method of characteristics and the finite element method globally it looks like one does one step of transport plus one step of diffusion or projection but the mathematics show that it is also an implicit time discretization of thepde in lagrangian form we give an error bound h t h h t in the interesting case that holds also for the navier stokes equations even when the reynolds number is infinite euler equation

blowup dynamics for smooth data equivariant solutions to the critical schrodinger map problem we consider the energy critical schrodinger map problem with the 2 sphere target for equivariant initial data of homotopy index k 1 we show the existence of a codimension one set of smooth well localized initial data arbitrarily close to the ground state harmonic map in the energy critical norm which generates finite time blowup solutions we give a sharp description of the corresponding singularity formation which occurs by concentration of a universal bubble of energy

distributed averaging with random network graphs and noises we consider a discrete time distributed averaging algorithm over multi agent networks with measurement noises and time varying random graphs each agent updates its state by a weighted sum of pairwise state differences between its neighbors and itself with both additive and multiplicative measurement noises the network structure is modeled by a sequence of time varying random digraphs which may be spatially and temporally dependent by stochastic lyapunov method and the combination of algebraic graph theory and martingale convergence theory we obtain sufficient conditions for stochastic approximation type algorithms to achieve mean square and almost sure average consensus we prove that all states of the agents converge to a common random variable whose mathematical expectation is the average of initial values in mean square and almost surely if the sequence of digraphs is conditionally balanced and uniformly conditionally jointly connected an upper bound of the variance of the limit random variable that is the mean square steady state error for stochastic average consensus is given quantitatively related to the weights the algorithm gain and the energy level of the noises

network based consensus averaging with general noisy channels this paper focuses on the consensus averaging problem on graphs under general imperfect communications we study a particular class of distributed consensus algorithms based on damped updates and using the ordinary differential equation method we prove that the updates converge almost surely to the consensus average for various models of perturbation of data exchanged between nodes the convergence is not asymptotic in the size of the network our analysis applies to various types of stochastic disturbances to the updates including errors in update calculations dithered quantization and imperfect data exchange among nodes under a suitable stability condition we prove that the error between estimated and true averages is asymptotically gaussian and we show how the asymptotic covariance is specified by the graph laplacian for additive perturbations we show how the scaling of the asymptotic mse is controlled by the spectral gap of the laplacian

convergence of a semi lagrangian scheme for the one dimensional vlasov poisson system a semi lagrangian scheme is proposed for solving the periodic one dimensional vlasov poisson system in phase space on unstructured meshes the distribution function f t x v and the electric field e t x are shown to converge to the exact solution values in the l infty norm the rate of convergence is in o h4 3

numerical methods for convection dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures finite element and finite difference methods are combined with the method of characteristics to treat a parabolic problem of the form cu t bu x au x x f optimal order error estimates in l 2 and w 1 2 are derived for the finite element procedure various error estimates are presented for a variety of finite difference methods the estimates show that for convection dominated problems b gg a these schemes have much smaller time truncation errors than those of standard methods extensions to n space variables and time dependent or nonlinear coefficients are indicated along with applications of the concepts to certain problems described by systems of differential equations

distributed localization of coverage holes using topological persistence we develop distributed algorithms to detect and localize coverage holes in sensor networks we neither assume coordinate information of the nodes neither any distances between the nodes we use algebraic topological methods to define a coverage hole and develop provably correct algorithm to detect a hole we then partition the network into smaller subnetworks while ensuring that the holes are preserved and checking for holes in each we show that repeating this process leads to localizing the coverage holes we demonstrate the improved complexity of our algorithm using simulations

arbitrary order time accurate semi lagrangian spectral approximations of the vlasov poisson system abstract the vlasov poisson system modeling the evolution of non collisional plasmas in the electrostatic limit is approximated by a semi lagrangian technique spectral methods of periodic type are implemented through a collocation approach groups of particles are represented by the fourier lagrangian basis and evolve for a single timestep along a high order accurate representation of the local characteristic lines the time advancing technique is based on truncated taylor series that can be in principle of any order of accuracy a variant is obtained by coupling the phase space discretization with high order accurate backward differentiation formulas bdf at each timestep particle displacements are reinterpolated and expressed in the original basis to guarantee the order of accuracy in all the variables at relatively low costs thus these techniques combine the excellent features of spectral approximations with high order time integration the resulting method has excellent conservation properties indeed it can be proven that the total number of particles proportional to the total mass and charge is conserved up to the machine precision series of numerical experiments are performed in order to assess the real performance in particular comparisons with standard benchmarks are examined

finite element methods for linear hyperbolic problems abstract we give a survey of some recent work by the authors on finite element methods for convectiondiffusion problems and first order linear hyperbolic problems

a finite element semi lagrangian explicit runge kutta chebyshev method for convection dominated reaction diffusion problems explicit runge kutta chebyshev methods have proved to be efficient for reaction diffusion problems of moderate stiffness in this paper we extend such an efficiency to convection dominated reaction diffusion problems by giving a formulation of these methods in a semi lagrangian framework using c0 finite elements of degree m 2 as the space discretization method we also study the convergence in the l2 norm of the methods proposed in this paper

distributed consensus algorithms in sensor networks with imperfect communication link failures and channel noise the paper studies average consensus with random topologies intermittent links and noisy channels consensus with noise in the network links leads to the bias variance dilemma running consensus for long reduces the bias of the final average estimate but increases its variance we present two different compromises to this tradeoff the a nd algorithm modifies conventional consensus by forcing the weights to satisfy a persistence condition slowly decaying to zero and the a nc algorithm where the weights are constant but consensus is run for a fixed number of iterations iota then it is restarted and rerun for a total of p runs and at the end averages the final states of the p runs monte carlo averaging we use controlled markov processes and stochastic approximation arguments to prove almost sure convergence of a nd to a finite consensus limit and compute explicitly the mean square error mse variance of the consensus limit we show that a nd represents the best of both worlds zero bias and low variance at the cost of a slow convergence rate rescaling the weights balances the variance versus the rate of bias reduction convergence rate in contrast a nc because of its constant weights converges fast but presents a different bias variance tradeoff for the same number of iterations iota p shorter runs smaller iota lead to high bias but smaller variance larger number p of runs to average over for a static nonrandom network with gaussian noise we compute the optimal gain for a nc to reach in the shortest number of iterations iota p with high probability 1 delta epsiv delta consensus epsiv residual bias our results hold under fairly general assumptions on the random link failures and communication noise

bubbling of the heat flows for harmonic maps from surfaces in this article we prove that any palais smale sequence of the energy functional on surfaces with uniformly l2 bounded tension fields converges pointwise by taking a subsequence if necessary to a map from connected possibly singular surfaces which consist of the original surfaces and finitely many bubble trees we therefore get the corresponding results about how the solutions of heat flow for harmonic maps from surfaces form singularities at infinite time 1997 john wiley sons inc

convergence of a high order semi lagrangian scheme with propagation of gradients for the one dimensional vlasov poisson system in this paper we give a proof of convergence of a new numerical method introduced in n besse and e sonnendrucker j comput phys 191 2003 pp 341 376 for the vlasov equation the numerical method is based on the semi lagrangian principle and the transport of the gradient of the statistical distribution function in order to get a high order and stable reconstruction these kinds of new schemes have been successfully implemented on unstructured meshes of four dimensional phase space cf n besse etude mathematique et numerique de l equation de vlasov sur des maillages non structures de l espace des phases these de l universite louis pasteur strasbourg france 2003 n besse j segre and e sonnendrucker transport theory statist phys 34 2005 pp 311 332 in order to make a rigorous proof of convergence of this method and simplify the convergence analysis we have considered the periodic one dimensional vlasov poisson system in phase space on a grid the distribution f t x v and the electric field are shown to converge to the exact solution values in h 1 norm the rate of convergence is of mathcal o delta t 2 frac delta x 4 alpha delta t frac delta v 4 alpha delta t alpha in n 2 alpha leq 1

asymptotic agreement in distributed estimation each of several agents updates his estimate of the same random variable whenever he makes a new observation or receives the estimate made by another agent in turn each agent transmits his estimate to a randomly chosen subset of the other agents a subset of agents forms a communicating ring if for every pair underline m underline p of ring members there is a sequence of ring members underline m underline m 1 underline m 2 underline m n 1 underline p such that underline m i sends his estimate to underline m i 1 infinitely often if each ring member knows that he is a ring member then the estimates of all the ring members asymptotically agree however this common limit can depend upon the order in which estimates are transmitted

broadcast based consensus with non zero mean stochastic perturbations bolstered by the growing interest in building wireless sensor and ad hoc networks with applications ranging across different engineering disciplines distributed consensus algorithms have recently seen a new revival since their inception in the early 1980s of particular interest is the recently developed broadcast based consensus algorithm which is one special type of randomized consensus algorithms and is amenable to practical implementation in wireless networks this paper focuses on the performance analysis of this broadcast based consensus algorithm in the presence of non zero mean stochastic perturbations it is demonstrated that as the algorithm proceeds the deviation of the node states from their average will converge in expectation to a fixed value which is determined by the laplacian matrix of the network the mixing parameter and the mean of the stochastic perturbations asymptotic upper and lower bounds on the total mean square deviation are derived which describe the range of distances over which the node states deviate from consensus these bounds can facilitate evaluation of the applicability of this algorithm in practice results are also provided on the algorithm s e converging time i e the earliest time at which the deviation is e close to its steady value and on the mean and mean square behaviors of the displacement of node states from their initial states at large iteration number as a special case study performance of the broadcast based consensus algorithm under zero mean stochastic disturbances is analyzed and results regarding its convergence mean square deviation and mean square displacement are given the theoretical results presented in this study hold true regardless of the statistics of the stochastic disturbances and are valid for arbitrary network topology as long as the topology is connected

a galerkin characteristic algorithm for transport diffusion equations we introduce in this paper an algorithm for integration of the transport diffusion equations by combining the galerkin finite element method with the discretization of the total derivative along th

stable blowup dynamics for the 1 corotational energy critical harmonic heat flow we exhibit a stable finite time blowup regime for the 1 corotational energy critical harmonic heat flow from 2 into a smooth compact revolution surface of 3 that reduces to the semilinear parabolic problem for a suitable class of functions f the corresponding initial data can be chosen smooth well localized and arbitrarily close to the ground state harmonic map in the energy critical topology we give sharp asymptotics on the corresponding singularity formation that occurs through the concentration of a universal bubble of energy at the speed predicted by van den berg hulshof and king our approach lies in the continuation of the study of the 1 equivariant energy critical wave map and schrodinger map with 2 target by merle raphael and rodnianski 2012 wiley periodicals inc

a parallel vlasov solver based on local cubic spline interpolation on patches a method for computing the numerical solution of vlasov type equations on massively parallel computers is presented in contrast with particle in cell methods which are known to be noisy the method is based on a semi lagrangian algorithm that approaches the vlasov equation on a grid of phase space as this kind of method requires a huge computational effort the simulations are carried out on parallel machines to that purpose we present a local cubic splines interpolation method based on a domain decomposition e g devoted to a processor hermite boundary conditions between the domains using ad hoc reconstruction of the derivatives provide a good approximation of the global solution the method is applied on various physical configurations which show the ability of the numerical scheme

winding behaviour of finite time singularities of the harmonic map heat flow we settle a number of questions about the possible behaviour of the harmonic map heat flow at finite time singularities in particular we show that a type of nonuniqueness of bubbles can occur at finite time we show that the weak limit of the flow at the singular time can be discontinuous we determine exactly the polynomial rate of blow up in one particular example and we show that winding behaviour of the flow can lead to an unexpected failure of convergence when the flow is locally lifted to the universal cover of the target manifold

distributed consensus algorithms in sensor networks quantized data and random link failures the paper studies the problem of distributed average consensus in sensor networks with quantized data and random link failures to achieve consensus dither small noise is added to the sensor states before quantization when the quantizer range is unbounded countable number of quantizer levels stochastic approximation shows that consensus is asymptotically achieved with probability one and in mean square to a finite random variable we show that the mean squared error mse can be made arbitrarily small by tuning the link weight sequence at a cost of the convergence rate of the algorithm to study dithered consensus with random links when the range of the quantizer is bounded we establish uniform boundedness of the sample paths of the unbounded quantizer this requires characterization of the statistical properties of the supremum taken over the sample paths of the state of the quantizer this is accomplished by splitting the state vector of the quantizer in two components one along the consensus subspace and the other along the subspace orthogonal to the consensus subspace the proofs use maximal inequalities for submartingale and supermartingale sequences from these we derive probability bounds on the excursions of the two subsequences from which probability bounds on the excursions of the quantizer state vector follow the paper shows how to use these probability bounds to design the quantizer parameters and to explore tradeoffs among the number of quantizer levels the size of the quantization steps the desired probability of saturation and the desired level of accuracy away from consensus finally the paper illustrates the quantizer design with a numerical study

stochastic approximation based consensus dynamics over markovian networks this paper considers consensus problems with random networks a key object of our analysis is a sequence of stochastic matrices which involve markovian switches and decreasing step sizes we establish ergodicity of the backward products of these stochastic matrices the basic technique is to consider the second moment dynamics of an associated markovian jump linear system and exploit its two scale interaction property resulting from the decreasing step sizes the mean square convergence rate of the backward products is also obtained the ergodicity results are used to prove mean square consensus of stochastic approximation algorithms where agents collect noisy information the approach is further applied to a token scheduled averaging model

lp decoding corrects a constant fraction of errors we show that for low density parity check ldpc codes whose tanner graphs have sufficient expansion the linear programming lp decoder of feldman karger and wainwright can correct a constant fraction of errors a random graph will have sufficient expansion with high probability and recent work shows that such graphs can be constructed efficiently a key element of our method is the use of a dual witness a zero valued dual solution to the decoding linear program whose existence proves decoding success we show that as long as no more than a certain constant fraction of the bits are flipped by the channel we can find a dual witness this new method can be used for proving bounds on the performance of any lp decoder even in a probabilistic setting our result implies that the word error rate of the lp decoder decreases exponentially in the code length under the binary symmetric channel bsc this is the first such error bound for ldpc codes using an analysis based on pseudocodewords recent work by koetter and vontobel shows that lp decoding and min sum decoding of ldpc codes are closely related by the graph cover structure of their pseudocodewords in their terminology our result implies that that there exist families of ldpc codes where the minimum bsc pseudoweight grows linearly in the block length

flux corrected transport this paper describes a class of explicit eulerian finite difference algorithms for solving the continuity equation which are built around a technique called flux correction these flux corrected transport fct algorithms are of indeterminate order but yield realistic accurate results in addition to the mass conserving property of most conventional algorithms the fct algorithms strictly maintain the positivity of actual mass densities so steep gradients and inviscid shocks are handled particularly well this first paper concentrates on a simple one dimensional version of fct utilizing shasta a new transport algorithm for the continuity equation which is described in detail

coding with side information for rate constrained consensus average consensus algorithms are protocols to compute the average value of all sensor measurements via near neighbors communications they offer a natural tradeoff between the number of messages exchanged among terminals and the accuracy in the computation most of the models adopted for the message exchange in the literature however neither include explicit rate constraints nor explore the rate distortion tradeoff associated with the algorithm the contribution of our work is in examining the impact of such constraints and in finding strategies to minimize the communication cost in terms of rate the main motivation behind the proposed coding strategies is the observation that consensus algorithms offer the perfect example of a network communication problem where there is an increasing correlation between the data exchanged as the algorithm iterates henceforth it is possible to utilize previously exchanged data and current side information to significantly reduce the demands of quantization bit rate for a certain precision we analyze the case of a network where the links are assumed to be reliable at a constant bit rate we explore the conditions on the quantization noise which lead to a consensus value whose mean squared distance from the initial average is bounded in the case of infinite length vector coding with gaussian states we show that our proposed schemes achieve bounded convergence with vanishing rates as the iteration index tends to infinity

convergence of classes of high order semi lagrangian schemes for the vlasov poisson system abstract in this paper we present some classes of high order semi lagran gian schemes for solving the periodic one dimensional vlasov poisson system in phase space on uniform grids we prove that the distribution function f t x v and the electric field e t x converge in the l 2 norm with a rate of displaystyle mathcal o left delta t 2 h m 1 frac h m 1 delta t right where m is the degree of the polynomial reconstruction and delta t and h are respectively the time and the phase space discretization parameters

repulsion and quantization in almost harmonic maps and asymptotics of the harmonic map flow we present an analysis of bounded energy low tension maps between 2 spheres by deriving sharp estimates for the ratio of length scales on which bubbles of opposite orientation develop we show that we can establish a quantization estimate which constrains the energy of the map to lie near to a discrete energy spectrum one application is to the asymptotics of the harmonic map flow we find uniform exponential convergence in time in the case under consideration

consensus in ad hoc wsns with noisy links part i distributed estimation of deterministic signals we deal with distributed estimation of deterministic vector parameters using ad hoc wireless sensor networks wsns we cast the decentralized estimation problem as the solution of multiple constrained convex optimization subproblems using the method of multipliers in conjunction with a block coordinate descent approach we demonstrate how the resultant algorithm can be decomposed into a set of simpler tasks suitable for distributed implementation different from existing alternatives our approach does not require the centralized estimator to be expressible in a separable closed form in terms of averages thus allowing for decentralized computation even of nonlinear estimators including maximum likelihood estimators mle in nonlinear and non gaussian data models we prove that these algorithms have guaranteed convergence to the desired estimator when the sensor links are assumed ideal furthermore our decentralized algorithms exhibit resilience in the presence of receiver and or quantization noise in particular we introduce a decentralized scheme for least squares and best linear unbiased estimation blue and establish its convergence in the presence of communication noise our algorithms also exhibit potential for higher convergence rate with respect to existing schemes corroborating simulations demonstrate the merits of the novel distributed estimation algorithms