generalized kahler manifolds commuting complex structures and split tangent bundles we study generalized kahler manifolds for which the corresponding complex structures commute and classify completely the compact four dimensional ones

exact energy and momentum conserving algorithms for general models in nonlinear elasticity implicit time integration schemes that inherit the conservation laws of total energy linear and angular momentum are considered for initial boundary value problems in finite deformation elastodynamics conserving schemes are constructed for general hyperelastic material models both compressible and incompressible and are formulated in a way that is independent of spatial discretization three numerical examples for ogden type material models implemented using a finite element discretization in space are given to illustrate the performance of the proposed schemes these examples show that relative to the standard implicit mid point rule the conserving schemes exhibit superior numerical stability properties without a compromise in accuracy

lms algorithms for tracking slow markov chains with applications to hidden markov estimation and adaptive multiuser detection this paper analyzes the tracking properties of the least mean squares lms algorithm when the underlying parameter evolves according to a finite state markov chain with infrequent jumps first using perturbed liapunov function methods mean square error estimates are obtained for the tracking error then using recent results on two time scale markov chains mean ordinary differential equation and diffusion approximation results are obtained it is shown that a sequence of the centered tracking errors converges to an ordinary differential equation moreover a suitably scaled sequence of the tracking errors converges weakly to a diffusion process it is also shown that iterate averaging of the tracking algorithm results in optimal asymptotic convergence rate in an appropriate sense two application examples analysis of the performance of an adaptive multiuser detection algorithm in a direct sequence code division multiple access ds cdma system and tracking analysis of the state of a hidden markov model hmm with infrequent jumps are presented

weighted means in stochastic approximation of minima weighted averages of kiefer wolfowitz type procedures which are driven by larger step lengths than usual can achieve the optimal rate of convergence a priori knowledge of a lower bound on the smallest eigenvalue of the hessian matrix is avoided the asymptotic mean squared error of the weighted averaging algorithm is the same as would emerge using a newton type adaptive algorithm several different gradient estimates are considered one of them leads to a vanishing asymptotic bias this gradient estimate applied with the weighted averaging algorithm usually yields a better asymptotic mean squared error than applied with the standard algorithm

on almost sure bounds for the lms algorithm almost sure a s bounds for linear constant gain adaptive filtering algorithms are investigated for instance under general pseudo stationarity and dependence conditions on the driving data spl psi k k 1 2 3 y sub k k 0 1 2 a s convergence and rates of a s convergence as the algorithm gain spl epsiv spl rarr 0 are established for the lms algorithm h sub k 1 sup spl epsiv h sub k sup spl epsiv spl epsiv y sub k spl psi sub k 1 y sub k sup t h sub k sup spl epsiv subject to some nonrandom initial condition h sub 0 sup spl epsiv h sub 0 in particular defining g sub k sup spl epsiv sub k 0 sup spl infin by g sub 0 sup spl epsiv h sub 0 and g sub k 1 sup spl epsiv g sub k sup spl epsiv spl epsiv e y sub k spl psi sub k 1 e y sub k y sub k sup t g sub k sup spl epsiv for k 0 1 2 the author shows that for any spl gamma 0 max sub 0 spl les k spl les spl gamma spl epsiv 1 h sub k sup spl epsiv g sub k sup spl epsiv spl rarr 0 as spl epsiv spl rarr 0 a s and under a stronger dependency condition the author shows that for any 0 0 max sub 0 spl les k spl les spl gamma spl epsiv spl zeta h sub k sup spl epsiv g sub k sup spl epsiv converges as spl epsiv spl rarr 0 a s at a rate marginally slower than o spl epsiv sup 2 spl zeta log log spl epsiv sup spl zeta sup 1 2 then under a stronger pseudostationarity assumption it is shown that similar results hold if the sequences g sub k sup spl epsiv sub k 0 sup spl infin spl epsiv 0 in the above results are replaced with the solution g sup 0 spl middot of a nonrandom linear ordinary differential equation i e one has max sub 0 spl les k spl les spl gamma spl epsiv spl zeta h sub k sup spl epsiv g sup 0 spl epsiv k spl rarr 0 as spl epsiv spl rarr 0 a s where one can attach a rate to this convergence under the stronger dependency condition the almost sure bounds contained in the paper complement previously developed weak convergence results in kushner and shwartz ieee trans information theory it 30 2 177 182 1984 and as are near optimal moreover the proofs used to establish these bounds are quite elementary

tracking analysis of an arma parameter estimation algorithm the problem of adaptively estimating parameters of a time varying autoregressive moving average arma process using a constant step size gauss newton algorithm is studied using weak convergence theory and the concept of prescaling it is shown that an ordinary differential equation can be used to describe the mean behavior of the adaptive filter coefficients computer simulations are provided to substantiate the analysis

asymptotic properties of sign algorithms for adaptive filtering this paper develops asymptotic properties of a class of sign error algorithms with expanding truncation bounds for adaptive filtering under merely stationary ergodicity and finite second moments of the reference and output signals and using trajectory subsequence ts method it is proved that the algorithm convergers almost surely then a mean squares estimate is derived for the estimation error and a suitably scaled sequence of the estimation error is shown to converge to a diffusion process the scaling factor together with the stationary covariance gives the rate of convergence result moreover an algorithm under mean squares criterion with expanding truncation bounds is also examined compared with the existing results in the literature sufficient conditions for almost sure convergence are much relaxed a simple example is provided for demonstration purpose

model free control of nonlinear stochastic systems with discrete time measurements consider the problem of developing a controller for general nonlinear and stochastic systems where the equations governing the system are unknown using discrete time measurement this paper presents an approach for estimating a controller without building or assuming a model for the system such an approach has potential advantages in accommodating complex systems with possibly time varying dynamics the controller is constructed through use of a function approximator such as a neural network or polynomial this paper considers the use of the simultaneous perturbation stochastic approximation algorithm which requires only system measurements a convergence result for stochastic approximation algorithms with time varying objective functions and feedback is established it is shown that this algorithm can greatly enhance the efficiency over more standard stochastic approximation algorithms based on finite difference gradient approximations

on surfaces of class vii 0 with numerically anticanonical divisor we consider minimal compact complex surfaces s with betti numbers b 1 and n bi 0 a theorem of donaldson gives n exceptional line bundles we prove that if in a deformation these line bundles have sections s is a degeneration of blown up hopf surfaces besides if there exists an integer m 1 and a flat line bundle f such that h s mk 0 these surfaces admit no nonconstant mero morphic functions the major problem in classification of non kahlerian surfaces is to achieve the classification of surfaces s of class viij all known surfaces of this class contain global spherical shells gss i e admit a biholomorphic map ip u v from a neighbourhood u c c2 0 of the sphere 53 db2 onto an open set v such that i f s3 does not disconnect s are there other surfaces in first section we investigate the general situation a theorem of donaldson 13 gives a z base of 2 s z such that b of s these line bundles form families we propose the following conjecture which can be easily checked for surfaces with gss

asymptotic properties of an adaptive beam former algorithm the asymptotic properties of a recursive adaptive beam former algorithm are studied both decreasing gain and constant gain cases are treated for the case of decreasing gain the mean square convergence result is obtained whereas for constant gain a sharp bound is derived and asymptotic analysis for the normalized error is carried out the analysis provides a clear picture of the local behaviour of the iterates near the optimal value a sequence of scale deviations or normalized errors is shown to converge to a gauss markov diffusion process which satisfies a stochastic differential equation

optimization of the transient and steady state behavior of discrete event systems we present a general framework for applying simulation to optimize the behavior of discrete event systems our approach involves modeling the discrete event system under study as a general state space markov chain whose distribution depends on the decision parameters we then show how simulation and the likelihood ratio method can be used to evaluate the performance measure of interest and its gradient and we present conditions that guarantee that the robbins monro stochastic approximation algorithm will converge almost surely to the optimal values of the decision parameters both transient and steady state performance measures are considered for steady state performance measures we consider both the case when the markov chain of interest is regenerative in the standard sense as well as the case when this markov chain is harris recurrent and thereby regenerative in a wider sense

tracking characteristics of the constrained iir adaptive notch filter the tracking behavior of a constrained iir notch filter the coefficients of which are estimated using a constant step size gauss newton algorithm is studied using weak convergence theory and the concept of prescaling it is shown that the mean behavior of the adaptive filter coefficients can be studied by using an ordinary differential equation ode approximate and simple closed form results are derived for the tracking behavior of a second order notch filter computer simulations are presented to substantiate the analysis

tracking of time varying mobile radio channels 1 the wiener lms algorithm adaptation algorithms with constant gains are designed for tracking smoothly time varying parameters of linear regression models in particular channel models occurring in mobile radio communications in a companion paper an application to channel tracking in the is 136 tdma system is discussed the proposed algorithms are based on two key concepts first the design is transformed into a wiener filtering problem second the parameters are modeled as correlated arima processes with known dynamics this leads to a new framework for systematic and optimal design of simple adaptation laws based on prior information the algorithms can be realized as wiener filters called learning filters or as lms newton updates complemented by filters that provide predictions or smoothing estimates the simplest algorithm named the wiener lms is presented all parameters are here assumed governed by the same dynamics and the covariance matrix of the regressors is assumed known the computational complexity is of the same order of magnitude as that of lms for regressors which are either white or have autoregressive statistics the tracking performance is however substantially improved

stochastic approximation with averaging of the iterates optimal asymptotic rate of convergence for general processes consider the stochastic approximation algorithm x n 1 x n a n g x n xi n in an important paper polyak and juditsky siam j control optim 30 1992 pp 838 855 showed that loosely speaking if the coefficients a n go to zero slower than o 1 n then the averaged sequence sum nolimits i 1 n x i n converged to its limit at an optimum rate for any coefficient sequence the conditions were rather special and direct constructions were used here a rather simple proof is given that results of this type are generic to stochastic approximation and essentially hold any time that the classical asymptotic normality of the normalized and centered iterates holds considerable intuitive insight is provided into the procedure simulations have well borne out the importance of the method

asymptotically optimal smoothing of averaged lms estimates for regression parameter tracking the sequence of estimates formed by the lms algorithm for a standard linear regression estimation problem is considered it is known since earlier that smoothing these estimates by simple averaging will lead to asymptotically the recursive least squares algorithm in this paper it is first shown that smoothing the lms estimates using a matrix updating will lead to smoothed estimates with optimal tracking properties also in case the true parameters are slowly changing as a random walk the choice of smoothing matrix should be tailored to the properties of the random walk second it is shown that the same accuracy can be obtained also for a modified algorithm slams which is based on averages and requires much less computations

stability of numerical methods for jump diffusions and markovian switching jump diffusions this work focuses on stability analysis of numerical solutions to jump diffusions and jump diffusions with markovian switching due to the use of poisson processes using asymptotic expansions as in the usual approach of treating diffusion processes does not work different from the existing treatments of euler maruyama methods for solutions of stochastic differential equations we use techniques from stochastic approximation we analyze the almost sure exponential stability and exponential p stability the benchmark test model in numerical solutions namely one dimensional linear scalar jump diffusion is examined first and easily verifiable conditions are presented then markovian regime switching jump diffusions are dealt with moreover analysis of stability of numerical methods for linearizable and multi dimensional jump diffusions is carried out

least mean square algorithms with markov regime switching limit we analyze the tracking performance of the least mean square lms algorithm for adaptively estimating a time varying parameter that evolves according to a finite state markov chain we assume the markov chain jumps infrequently between the finite states at the same rate of change as the lms algorithm we derive mean square estimation error bounds for the tracking error of the lms algorithm using perturbed lyapunov function methods then combining results in two time scale markov chains with weak convergence methods for stochastic approximation we derive the limit dynamics satisfied by continuous time interpolation of the estimates unlike most previous analyzes of stochastic approximation algorithms the limit we obtain is a system of ordinary differential equations with regime switching controlled by a continuous time markov chain next to analyze the rate of convergence we take a continuous time interpolation of a scaled sequence of the error sequence and derive its diffusion limit somewhat remarkably for correlated regression vectors we obtain a jump markov diffusion finally two novel examples of the analysis are given for state estimation of hidden markov models hmms and adaptive interference suppression in wireless code division multiple access cdma networks

iterate averaging sign algorithms for adaptive filtering with applications to blind multiuser detection motivated by the developments on iterate averaging of recursive stochastic approximation algorithms and asymptotic analysis of sign error algorithms for adaptive filtering this work develops two stage sign algorithms for adaptive filtering the proposed algorithms are based on constructions of a sequence of estimates using large step sizes followed by iterate averaging our main effort is devoted to improving the performance of the algorithms by establishing asymptotic normality of a suitably scaled sequence of the estimation errors the asymptotic covariance is calculated and shown to be the smallest possible hence the asymptotic efficiency or asymptotic optimality is obtained then variants of the algorithm including sign regressor procedures and constant step algorithms are studied the minimal window width of averaging is also dealt with finally iterate averaging algorithms for blind multiuser detection in direct sequence code division multiple access ds cdma systems are proposed and developed and numerical examples are examined

une construction elementaire des surfaces d inoue hirzebruch 0 introduction 663 1 surfaces d inoue hirzebruch et th6or6me principal 664 2 application aux champs de vecteurs globaux et fibr6 canonique 668 3 rev ements et automorphismes des surfaces de kato 673 4 d monstration du th or me principal 681 r6f6rences 682

wiener design of adaptation algorithms with time invariant gains a design method is presented that extends least mean squared lms adaptation of time varying parameters by including general linear time invariant filters that operate on the instantaneous gradient vector the aim is to track time varying parameters of linear regression models in situations where the regressors are stationary or have slowly time varying properties the adaptation law is optimized with respect to the steady state parameter error covariance matrix for time variations modeled as vector arima processes the design method systematically uses prior information about time varying parameters to provide filtering prediction or fixed lag smoothing estimates for arbitrary lags the method is based on a transformation of the adaptation problem into a wiener filter design problem the filter works in open loop for slow parameter variations whereas a time varying closed loop has to be considered for fast variations in the latter case the filter design is performed iteratively the general form of the solution at each iteration is obtained by a bilateral diophantine polynomial matrix equation and a spectral factorization for white gradient noise the diophantine equation has a closed form solution further structural constraints result in very simple design equations under certain model assumptions the wiener designed adaptation laws reduce to lms adaptation compared with kalman estimators the channel tracking performance becomes nearly the same in mobile radio applications whereas the complexity is in general much lower

asymptotic almost sure efficiency of averaged stochastic algorithms first we define the notion of almost sure efficiency for a decreasing stepsize stochastic algorithm and then we show that the averaging method which gives asymptotically efficient algorithms also gives asymptotically almost surely efficient algorithms moreover we prove that the averaged algorithm also satisfies a law of the iterated logarithm as well as an almost sure central limit theorem

how does a stochastic optimization approximation algorithm adapt to a randomly evolving optimum root with jump markov sample paths stochastic optimization approximation algorithms are widely used to recursively estimate the optimum of a suitable function or its root under noisy observations when this optimum or root is a constant or evolves randomly according to slowly time varying continuous sample paths in comparison this paper analyzes the asymptotic properties of stochastic optimization approximation algorithms for recursively estimating the optimum or root when it evolves rapidly with nonsmooth jump changing sample paths the resulting problem falls into the category of regime switching stochastic approximation algorithms with two time scales motivated by emerging applications in wireless communications and system identification we analyze asymptotic behavior of such algorithms our analysis assumes that the noisy observations contain a nonsmooth jump process modeled by a discrete time markov chain whose transition frequency varies much faster than the adaptation rate of the stochastic optimization algorithm using stochastic averaging we prove convergence of the algorithm rate of convergence of the algorithm is obtained via bounds on the estimation errors and diffusion approximations remarks on improving the convergence rates through iterate averaging and limit mean dynamics represented by differential inclusions are also presented

simplified analysis of the lms adaptive filter using a transfer function approximation a simple approximation to the behavior of the lms adaptive filter as a discrete transfer function is developed this representation is a valid description for both deterministic inputs and for the expected results with random inputs including correlated inputs the results are shown to be exact for some classes of input including periodic signals one result of this analysis is the demonstration that the lms filter can produce results which are biased from the least squares solution under the combined conditions of a nonzero mean primary and a correlated reference input

infinite bubbling in non kahlerian geometry in a holomorphic family x b b in b of non kahlerian compact manifolds the holomorphic curves representing a fixed 2 homology class do not form a proper family in general the deep source of this fundamental difficulty in non kahler geometry is the explosion of the area phenomenon the area of a curve c b subset x b in a fixed 2 homology class can diverge as b b0 this phenomenon occurs frequently in the deformation theory of class vii surfaces for instance it is well known that any minimal gss surface x0 is a degeneration of a 1 parameter family of simply blown up primary hopf surfaces x z z in d setminus 0 so one obtains non proper families of exceptional divisors e z subset x z whose area diverge as z 0 our main goal is to study in detail this non properness phenomenon in the case of class vii surfaces we will prove that under certain technical assumptions a lift widetilde e z of ez in the universal cover widetilde x z does converge to an effective divisor widetilde e 0 in widetilde x 0 but this limit divisor is not compact we prove that this limit divisor is always bounded towards the pseudo convex end of widetilde x 0 and that when x0 is a minimal surface with global spherical shell it is given by an infinite series of compact rational curves whose coefficients can be computed explicitly this phenomenon degeneration of a family of compact curves to an infinite union of compact curves should be called infinite bubbling we believe that such a decomposition result holds for any family of class vii surfaces whose generic fiber is a blown up primary hopf surface this statement would have important consequences for the classification of class vii surfaces

asymptotic properties of distributed and communication stochastic approximation algorithms the asymptotic properties of extensions of the type of distributed or decentralized stochastic approximation proposed in 1 are developed such algorithms have numerous potential applications in decentralized estimation detection and adaptive control or in decentralized monte carlo simulation for system optimization where they can exploit the possibilities of parallel processing the structure involves several isolated processors recursive algorithms that communicate to each other asynchronously and at random intervals the asymptotic small gain properties are derived the communication intervals need not be strictly bounded and they and the system noise can depend on the communicating system state state space constraints are also handled in many applications the dynamical terms are merely indicator functions or have other types of discontinuities the typical such case is also treated as is the case where there is noise in the communication the linear stochastic differential equation

conservation properties of a time fe method part i time stepping schemes forn body problems in the present paper one step implicit integration algorithms for the n body problem are developed the time stepping schemes are based on a petrov galerkin finite element method applied to the hamiltonian formulation of the n body problem the approach furnishes algorithmic energy conservation in a natural way the proposed time finite element method facilitates a systematic implementation of a family of time stepping schemes a particular algorithm is specified by the associated quadrature rule employed for the evaluation of time integrals the influence of various standard as well as non standard quadrature formulas on algorithmic energy conservation and conservation of angular momentum is examined in detail for linear and quadratic time elements copyright 2000 john wiley sons ltd

convergence of learning algorithms with constant learning rates the behavior of neural network learning algorithms with a small constant learning rate epsilon in stationary random input environments is investigated it is rigorously established that the sequence of weight estimates can be approximated by a certain ordinary differential equation in the sense of weak convergence of random processes as epsilon tends to zero as applications backpropagation in feedforward architectures and some feature extraction algorithms are studied in more detail

conservation properties of a time fe method part iv higher order energy and momentum conserving schemes in the present paper a systematic development of higher order accurate time stepping schemes which exactly conserve total energy as well as momentum maps of underlying finite dimensional hamiltonian systems with symmetry is shown the result of this development is the enhanced galerkin eg finite element method in time the conservation of the eg method is generally related to its collocation property total energy conservation in particular is obtained by a new projection technique the eg method is moreover based on objective time discretization of the used strain measure this paper is concerned with particle dynamics and semi discrete non linear elastodynamics the related numerical examples show good performance in presence of stiffness as well as for calculating large strain motions copyright 2005 john wiley sons ltd

a structural view of asymptotic convergence speed of adaptive iir filtering algorithms i infinite precision implementation for pt i see ibid vol 41 no 4 p 1493 1517 1993 finite precision fp implementation is the ultimately inevitable reality of all adaptive filters including adaptive infinite impulse response iir filters this paper continues to examine the asymptotic convergence speed of adaptive iir filters of various structures and algorithms including the simple constant gain type and the newton type but under fp implementation a stochastic differential equation sde approach is used in the analysis such an approach not only greatly simplifies the fp analysis which is traditionally very involved algebraically but it also provides valuable information about the first order as well as the second order moments that the latter are not available using the ordinary differential equation ode approach the asymptotic convergence speed as well as the convergent values of the pertinent moments of fp errors are examined in terms of unknown system pole zero locations the adverse effects of lightly damped low frequency ldlf poles resulting from fast sampling on the local transient and convergent behavior of various structures and algorithms are analyzed and compared the new results agree with the existing ones when reduced to the finite impulse response fir case in particular the explosive behavior of pertinent error variances of newton type iir algorithms when the forgetting factor spl lambda 1 is also concluded computer simulation verifies the predicted theoretical results

recursive algorithms for estimation of hidden markov models and autoregressive models with markov regime this paper is concerned with recursive algorithms for the estimation of hidden markov models hmms and autoregressive ar models under the markov regime convergence and rate of convergence results are derived acceleration of convergence by averaging of the iterates and the observations are treated finally constant step size tracking algorithms are presented and examined

averaging procedures in adaptive filtering an efficient approach an averaging procedure for adaptive filtering is developed in contrast to the traditional approach two sequences x sub n and x sub n are constructed where x sub n is the arithmetic average of x sub n the authors show that the algorithm so designed has optimal rate of convergence therefore the averaging approach is asymptotically efficient

stochastic approximation with averaging and feedback rapidly convergent on line algorithms consider the stochastic approximation x sub n 1 x sub n a sub n g x sub n spl xi sub n where 0 0 the averaging method is essentially off line in the sense that the actual sa iterate x sub n is not influenced by the averaging in many applications x sub n itself is of greatest interest since that is the operating parameter this paper deals with the problem of stochastic approximation with averaging and with appropriate feedback of the averages into the original algorithm it is shown both mathematically and via simulation that it works very well and has numerous advantages it is a clear improvement over the system x sub n by itself it is fairly robust and quite often it is much preferable to the use of the above averages without feedback the authors deal in particular with linear algorithms of the type appearing in parameter estimators adaptive noise cancellers channel equalizers adaptive control and similar applications the main development is for the constant parameter case because of its importance in applications but analogous results hold for the case where a sub n spl rarr 0

weak convergence properties of constrained emphatic temporal difference learning with constant and slowly diminishing stepsize we consider the emphatic temporal difference td algorithm etd for learning the value functions of stationary policies in a discounted finite state and action markov decision process the etd algorithm was recently proposed by sutton mahmood and white 2016 to solve a long standing divergence problem of the standard td algorithm when it is applied to off policy training where data from an exploratory policy are used to evaluate other policies of interest the almost sure convergence of etd has been proved in our recent work under general off policy training conditions but for a narrow range of diminishing stepsize in this paper we present convergence results for constrained versions of etd with constant stepsize and with diminishing stepsize from a broad range our results characterize the asymptotic behavior of the trajectory of iterates produced by those algorithms and are derived by combining key properties of etd with powerful convergence theorems from the weak convergence methods in stochastic approximation theory for the case of constant stepsize in addition to analyzing the behavior of the algorithms in the limit as the stepsize parameter approaches zero we also analyze their behavior for a fixed stepsize and bound the deviations of their averaged iterates from the desired solution these results are obtained by exploiting the weak feller property of the markov chains associated with the algorithms and by using ergodic theorems for weak feller markov chains in conjunction with the convergence results we get from the weak convergence methods besides etd our analysis also applies to the off policy td algorithm when the divergence issue is avoided by setting sufficiently large it yields for that case new results on the asymptotic convergence properties of constrained off policy td with constant or slowly diminishing stepsize

the orientation of yang mills moduli spaces and 4 manifold topology si x est une 4 variete lisse orientee fermee dont la forme intersection q h 2 x z torsion z est definie negative alors la forme est equivalente sur les entiers a la forme standard 1 1 1 soit x une 4 variete lisse orientee fermee avec les trois proprietes suivantes 1 h 1 x z n a pas de 2 torsion 2 la forme intersection q sur h 2 x torsion a une partie positive de rang 1 ou 2 3 la forme intersection est paire alors q est equivalente sur les entiers a une des formes 01 10 01 10 01 10

law of the iterated logarithm for a constant gain linear stochastic gradient algorithm we study almost sure limiting properties taken as varepsilon searrow 0 of the finite horizon sequence of random estimates theta 0 varepsilon theta 1 varepsilon theta 2 varepsilon ldots theta left lfloor t varepsilon right rfloor varepsilon for the linear stochastic gradient algorithm theta varepsilon n 1 theta varepsilon n varepsilon left a n 1 theta varepsilon n prime x n 1 right x n 1 theta 0 varepsilon mbox stackrel triangle theta mbox nonrandom noindent where t in 0 infty is an arbitrary constant varepsilon in 0 1 is a small adaptation gain and an and xn are data sequences which drive the algorithm these limiting properties are expressed in the form of a functional law of the iterated logarithm

stochastic approximation methods for systems over an infinitehorizon the paper develops efficient and general stochastic approximation sa methods for improving the operation of parametrized systems of either the continuous or discrete event dynamical systems types and which are of interest over a long time period for example one might wish to optimize or improve the stationary or average cost per unit time performance by adjusting the systems parameters the number of applications and the associated literature are increasing at a rapid rate this is partly due to the increasing activity in computing pathwise derivatives and adapting them to the average cost problem although the original motivation and the examples come from an interest in the infinite horizon problem the techniques and results are of general applicability in sa we present an updating and review of powerful ordinary differential equation type methods in a fairly general context and based on weak convergence ideas the results and proof techniques are applicable to a wide variety of applications exploiting the full potential of these ideas can greatly simplify and extend much current work their breadth as well as the relative ease of using the basic ideas are illustrated in detail via typical examples drawn from discrete event dynamical systems piecewise deterministic dynamical systems and a stochastic differential equations model in these particular illustrations we use either infinitesimal perturbation analysis type estimators mean square derivative type estimators or finite difference type estimators markov and non markov models are discussed the algorithms for distributed asynchronous updating as well as the fully synchronous schemes are developed

general multilevel adaptations for stochastic approximation algorithms of robbins monro and polyak ruppert type in this article we present and analyse new multilevel adaptations of classical stochastic approximation algorithms for the computation of a zero of a function f d rightarrow mathbb r d defined on a convex domain d subset mathbb r d which is given as a parameterised family of expectations the analysis of the error and the computational cost of our method is based on similar assumptions as used in giles oper res 56 3 607 617 2008 for the computation of a single expectation additionally we essentially only require that f satisfies a classical contraction property from stochastic approximation theory under these assumptions we establish error bounds in pth mean for our multilevel robbins monro and polyak ruppert schemes that decay in the computational time as fast as the classical error bounds for multilevel monte carlo approximations of single expectations known from giles oper res 56 3 607 617 2008 our approach is universal in the sense that having multilevel implementations for a particular application at hand it is straightforward to implement the corresponding stochastic approximation algorithm

averaged stochastic gradient algorithms for adaptive blind multiuser detection in ds cdma systems in this paper we present a blind adaptive gradient bag algorithm for code aided suppression of multiple access interference mai and narrow band interference nbi in direct sequence code division multiple access ds cdma systems this bag algorithm is based on the concept of accelerating the convergence of a stochastic gradient algorithm by averaging this ingenious concept of averaging was invented by polyak and juditsky 1992 this paper examines its application to blind multiuser detection and nbi suppression in ds cdma systems we prove that bag has identical convergence and tracking properties to recursive least squares lms but has a computational cost similar to the least mean squares lms algorithm i e an order of magnitude lower computational cost than rls simulations are used to compare our averaged gradient algorithm with the blind lms and lms schemes

conservation properties of a time fe method part ii time stepping schemes for non linear elastodynamics in the present paper one step implicit integration algorithms for non linear elastodynamics are developed the discretization process rests on galerkin methods in space and time in particular the continuous galerkin method applied to the hamiltonian formulation of semidiscrete non linear elastodynamics lies at the heart of the time stepping schemes algorithmic conservation of energy and angular momentum are shown to be closely related to quadrature formulas that are required for the calculation of time integrals we newly introduce the assumed strain method in time which enables the design of energy momentum conserving schemes and which can be interpreted as temporal counterpart of the well established assumed strain method for finite elements in space the numerical examples deal with quasi rigid motion as well as large strain motion copyright 2001 john wiley sons ltd

on projected stochastic gradient descent algorithm with weighted averaging for least squares regression the problem of least squares regression of a d dimensional unknown parameter is considered a stochastic gradient descent based algorithm with weighted iterate averaging that uses a single pass over the data is studied and its convergence rate is analyzed we first consider a bounded constraint set of the unknown parameter under some standard regularity assumptions we provide an explicit o 1 k upper bound on the convergence rate depending on the variance due to the additive noise in the measurements and the size of the constraint set we show that the variance term dominates the error and decreases with rate 1 k while the term that is related to the size of the constraint set decreases with rate log k k 2 we then compare the asymptotic ratio rho between the convergence rate of the proposed scheme and the empirical risk minimizer erm as the number of iterations approaches infinity we show that rho leq 4 for all d geq 1 when the random entries of the sensing vector are uncorrelated and identically distributed we further improve the upper bound by showing that rho leq 4 3 for the case of d 1 and unbounded parameter set when the random sensing entries are equal across time simulation results demonstrate strong performance of the algorithm as compared to existing methods and coincide with rho leq 4 3 even for large d in practice

the pseudo effective cone of a non kahlerian surface and applications we describe the positive cone and the pseudo effective cone of a non kahlerian surface we use these results for two types of applications

analysis of adaptive step size sa algorithms for parameter tracking we present proofs and data for adaptive step size algorithms for tracking time varying parameters when recursive stochastic approximation type algorithms are used a classical problem in adaptive control and communication theory concerns the tracking of the best fit of a given form when the statistics or the parameters change slowly a major and yet unresolved problem has been the choice of the step sizes in the tracking algorithm an algorithm for adapting the step size using the same system measurements which are used for the tracking was suggested by benveniste and various examples worked out by brossier the numerical results were very encouraging but proofs were lacking these proofs are supplied here together with supporting numerical data the proofs are based on recent results in stochastic approximation the adaptive step size technique works very well indeed much supporting analysis is presented particularly concerning the interpretation of certain stationary processes as stationary pathwise derivatives finite difference forms are also treated these are mathematically simpler and can be applied to a wide variety of systems even when the system is not well modeled the data shows that they work well

bihermitian surfaces with odd first betti number compact bihermitian surfaces are considered that is compact oriented conformal four manifolds admitting two distinct compatible complex structures it is shown that if the first betti number is odd then with respect to either complex structure such a manifold belongs to class vii of the enriques kodaira classification moreover it must be either a special hopf or an inoue surface in the strongly bihermitian case or obtained by blowing up a minimal class vii surface with curves in the non strongly bihermitian case

classification of solitons for pluriclosed flow on complex surfaces we give a classification of compact solitons for the pluriclosed flow on complex surfaces first by exploiting results from the kodaira classification of surfaces we show that the complex surface underlying a soliton must be kahler except for the possibility of steady solitons on minimal hopf surfaces then we construct steady solitons on all class 1 hopf surfaces by exploiting a symmetry ansatz

strong points of weak convergence a study using rpa gradient estimation for automatic learning this paper analyzes the behavior of adaptive control schemes for automatic learning estimates of the sensitivities are used in a gradient based stochastic approximation procedure in order to drive the process along the steepest descent trajectory in search for the optimum the learning rates are kept constant for adaptability for such procedures convergence can be established in a weak sense a model problem of a flexible machine is presented for which the control parameter is a probability vector we propose a new sensitivity estimator generalizing the phantom rare perturbation analysis rpa estimator to multi valued decisions from the basic properties of the estimators we build several updating rules based on the weak convergence theory to ensure asymptotic optimality we illustrate the predicted theoretical behavior with computer simulations finally we present the comparison between the behavior of our proposed scheme with a regenerative one for which we can establish strong convergence our results show that weak convergence yields a dramatic improvement in the rate of convergence in addition to the capability of adaptation or tracking

weak convergence and asymptotic properties of adaptive filters with constant gains the basic adaptive filtering algorithm x n 1 epsilon x n epsilon epsilon y n y n x n epsilon psi n is analyzed using the theory of weak convergence apart from some very special cases the analysis is hard when done for each fixed epsilon 0 but the weak convergence techniques are set up to provide much information for small epsilon the relevant facts from the theory are given define x epsilon cdot by x epsilon t x n epsilon on n epsilon n epsilon epsilon then weak distributional convergence of x epsilon cdot and of x epsilon cdot t epsilon is proved under very weak assumptions where t epsilon rightarrow infty as epsilon rightarrow 0 the normalized errors x n epsilon theta sqrt epsilon are analyzed where theta is a stable point for the mean algorithm the asymptotic properties of a projection algorithm are developed where the x n epsilon are truncated at each iteration if they fall outside of a given set

weak convergence and local stability properties of fixed step size recursive algorithms a recursive equation that subsumes several common adaptive filtering algorithms is analyzed for general stochastic inputs and disturbances by relating the motion of the parameter estimate errors to the behavior of an unforced deterministic ordinary differential equation ode the odes describing the motion of several common adaptive filters are examined in some simple settings including the least mean square lms algorithm and all three of its signed variants the signed regressor the signed error and the sign sign algorithms stability and instability results are presented in terms of the eigenvalues of a correlation like matrix this generalizes known results for lms signed regressor lms and signed error lms and gives new stability criteria for the sign sign algorithm the ability of the algorithms to track moving parameterizations can be analyzed in a similar manner by relating the time varying system to a forced ode the asymptotic distribution about the forced ode is an ornstein uhlenbeck process the properties of which can be described in a straightforward manner

acceleration of stochastic approximation by averaging a new recursive algorithm of stochastic approximation type with the averaging of trajectories is investigated convergence with probability one is proved for a variety of classical optimization and identification problems it is also demonstrated for these problems that the proposed algorithm achieves the highest possible rate of convergence

instantons and curves on class vii surfaces we develop a general strategy based on gauge theoretical methods to prove existence of curves on class vii surfaces we prove that for b 2 2 every minimal class vii surface has a cycle of rational curves hence by a result of nakamura is a global deformation of a one parameter family of blown up primary hopf surfaces the case b 2 1 was solved in a previous article the fundamental object intervening in our strategy is the moduli space m pst 0 k of polystable bundles with c 2 0 det k for large b 2 the geometry of this moduli space becomes very complicated the case b 2 2 treated here in detail requires new ideas and difficult techniques of both complex geometric and gauge theoretical nature we explain the substantial obstacles which must be overcome in order to extend our methods to the case b 2 3

stochastic optimization by simulation numerical experiments with the m m 1 queue in steady state this paper gives numerical illustrations of the behavior of stochastic approximation combined with different derivative estimation techniques to optimize a steady state system it is a companion paper to l ecuyer and glynn 1993 which gives convergence proofs for most of the variants experimented here the numerical experiments are made with a simple m m 1 queue which while simple serves to illustrate the basic convergence properties and possible pitfalls of the various techniques

budget dependent convergence rate of stochastic approximation convergence rate results are derived for a stochastic optimization problem where a performance measure is minimized with respect to a vector parameter t assuming that a gradient estimator is available and that both the bias and the variance of the estimator are known functions of the budget devoted to its computation the gradient estimator is employed in conjunction with a stochastic approximation sa algorithm our interest is to figure out how to allocate the total available computational budget to the successive sa iterations the effort is devoted to solving the asymptotic version of this problem by finding the convergence rate of sa toward the optimizer first as a function of the number of iterations and then as a function of the total computational effort as a result the optimal rate of increase of the computational budget per iteration can be found explicit expressions for the case where the computational budget devoted to an iteration is a polynomial in the iteration number and where the bias and variance of the gradient estimator are polynomials of the computational budget are derived applications include the optimization of steady state simulation models with likelihood ratio perturbation analysis or finite difference gradient estimators optimization of infinite horizon models with discounting optimization of functions of several expectations and so on several examples are discussed our results readily generalize to general root finding problems

an invariant measure approach to the convergence of stochastic approximations with state dependent noise a new method is presented for quickly getting the ode ordinary differential equation associated with the asymptotic properties of the stochastic approximation x n 1 x n a n f x n zeta n or the projected algorithm for the constrained problem either a n to 0 or a n can be constant in which case the analysis is on the sequence obtained when a to 0 the method requires that x n zeta n 1 be markov with a feller transition function but little else the simplest result requires that if x n equiv x the corresponding noise process zeta n x n geqq 0 have a unique invariant measure but the nonunique case can also be treated no mixing condition is required nor the construction of averaged test functions and f cdot cdot need not be continuous a detailed analysis of the way that zeta n varies with x n is not required for the class of sequences treated the conditions seem easier to verify than for other methods there

automatic energy momentum conserving time integrators for hyperelastic waves an energy momentum conserving time integrator coupled with an automatic finite element algorithm is developed to study longitudinal wave propagation in hyperelastic layers the murnaghan strain energy function is used to model material nonlinearity and full geometric nonlinearity is considered an automatic assembly algorithm using algorithmic differentiation is developed within a discrete hamiltonian framework to directly formulate the finite element matrices without recourse to an explicit derivation of their algebraic form or the governing equations the algorithm is illustrated with applications to longitudinal wave propagation in a thin hyperelastic layer modeled with a two mode kinematic model solution obtained using a standard nonlinear finite element model with newmark time stepping is provided for comparison

bounded error identification of time varying parameters by rls techniques the performance of the recursive least squares algorithm with constant forgetting factor in the identification of time varying parameters is studied in a stochastic framework it is shown that the mean square tracking error keeps bounded if and only if the so called covariance matrix of the algorithm is l sup 1 bounded then a feasibility range for the forgetting factor is worked out in correspondence of which the covariance matrix and therefore the tracking error keeps bounded

smooth deformations of singular contractions of class vii surfaces we consider normal compact surfaces y obtained from a minimal class vii surface x by contraction of a cycle c of r rational curves with c 2 0 our main result states that if the obtained cusp is smoothable then y is globally smoothable the proof is based on a vanishing theorem for h 2 theta y if r b 2 x any smooth small deformation of y is rational and if r b 2 x i e when x is a half inoue surface any smooth small deformation of y is an enriques surface the condition the cusp is smoothable in our main theorem can be checked in terms of the intersection numbers of the cycle using the looijenga conjecture which has recently become a theorem therefore this is a decidable condition we prove that this condition is always satisfied if r b 2 x leqslant 11 therefore the singular surface y obtained by contracting a cycle c of r rational curves in a minimal class vii surface x with r b 2 x leqslant 11 is always smoothable by rational surfaces the statement holds even for unknown class vii surfaces

continuous time tracking algorithms involving two time scale markov chains this work is concerned with least mean squares lms algorithms in continuous time for tracking a time varying parameter process a distinctive feature is that the true parameter process is changing at a fast pace driven by a finite state markov chain the states of the markov chain are divisible into a number of groups within each group the transitions take place rapidly among different groups the transitions are infrequent introducing a small parameter into the generator of the markov chain leads to a two time scale formulation the tracking objective is difficult to achieve nevertheless a limit result is derived yielding algorithms for limit systems moreover the rates of variation of the tracking error sequence are analyzed under simple conditions it is shown that a scaled sequence of the tracking errors converges weakly to a switching diffusion in addition a numerical example is provided and an adaptive step size algorithm developed

on closed loop adaptive noise cancellation given the mean limit ordinary differential equation for the stochastic approximation defining the adaptive algorithm for a closed loop adaptive noise cancellation we characterize the limit points under appropriate conditions it is shown that as the dimension of the weight vector increases the sequence of corresponding limit points converges in the sense of l sub 2 to the infinite dimensional optimal weight vector also the limit point of the algorithm is nearly optimal if the dimension of the weight vector is large enough the gradient of the mean square error with respect to the weight vector evaluated at the limit goes to zero in l sub 1 and l sub 2 as the dimension increases as does the gradient with respect to the coefficients in the transfer function connecting the reference noise signal with the error output thus the algorithm is nearly a gradient descent algorithm and is error reducing for large enough dimension under broad conditions iterative averaging can be used to get a nearly optimal rate of convergence

a finite element model for a higher order shear deformable beam theory the theory for a higher order shear deformable beam model is first developed it is based on a higher order displacement model and incorporates linear and quadratic variation of transverse normal strain and transverse shearing strain respectively through the beam thickness the effects of the transverse normal and shear stresses are included in the definition of the material s constitutive law the warping of the transverse normal cross section of the beam is automatically incorporated in the mathematical model the question of selecting a shear correction coefficient as in a first order shear deformable timoshenko theory does not arise a linear two noded finite element model of this theory is introduced and developed next both static and free vibration results of this theory are presented and compared with those of euler and timoshenko theories for various boundary and loading conditions

rates of convergence of adaptive step size of stochastic approximation algorithms we propose a new adaptive algorithm with decreasing step size for stochastic approximations the use of adaptive algorithms in various applications is widely spread across fields such as system identification and adaptive control we analyze the rate of convergence of the proposed algorithms an averaging algorithm on its optimality of the rate of convergence is used to control the step sizes our proofs are based on recent results in stochastic approximations and gaussian approximation theorem