a maximum entropy framework for nonexponential distributions probability distributions having power law tails are observed in a broad range of social economic and biological systems we describe here a potentially useful common framework we derive distribution functions for situations in which a joiner particle k pays some form of price to enter a community of size where costs are subject to economies of scale maximizing the boltzmann gibbs shannon entropy subject to this energy like constraint predicts a distribution having a power law tail it reduces to the boltzmann distribution in the absence of economies of scale we show that the predicted function gives excellent fits to 13 different distribution functions ranging from friendship links in social networks to protein protein interactions to the severity of terrorist attacks this approach may give useful insights into when to expect power law distributions in the natural and social sciences

the maximum entropy inference of solutions to pdes we use the maximum entropy variational technique to infer solutions to second order partial differential equations we restrict ourselves to problems with dirichlet boundary conditions first we construct a basis of moment functions in terms of which we set up the constraints for a maximum entropy inversion for the inference of solutions to elliptic equations then we extend the scheme to the inference of time dependence in evolution equations this extension is done by solving a small system of first order initial value differential equations for the moments of the solution vector

support vector machines for 3d object recognition support vector machines svms have been recently proposed as a new technique for pattern recognition intuitively given a set of points which belong to either of two classes a linear svm finds the hyperplane leaving the largest possible fraction of points of the same class on the same side while maximizing the distance of either class from the hyperplane the hyperplane is determined by a subset of the points of the two classes named support vectors and has a number of interesting theoretical properties in this paper we use linear svms for 3d object recognition we illustrate the potential of svms on a database of 7200 images of 100 different objects the proposed system does not require feature extraction and performs recognition on images regarded as points of a space of high dimension without estimating pose the excellent recognition rates achieved in all the performed experiments indicate that svms are well suited for aspect based recognition

a boundary value technique for singular perturbation problems abstract a new way to solve singular perturbation problems is introduced it is designed for the practicing engineer or applied mathematician who needs a practical tool for these problems easy to use modest problem preparation and ready computer implementation as with other methods the original problem is partitioned into inner and outer solution differential equation systems however asymptotic solution techniques are not employed the method is distinguished by the following facts the inner solution problem is solved as a two point boundary problem where the terminal boundary conditions are supplied by the solution of the outer solution problem in turn the outer solution differential equations are solved as an initial value system where the initial conditions are supplied by the inner solution at its terminal point the method is iterative on the terminal point of the inner solution problem three numerical examples are included

solving differential equations by a maximum entropy minimum norm method with applications to fokker planck equations the method of maximum entropy minimum norm is utilized to produce a general method of solving differential equations the technique is a generalization and extension of previous work performed by baker jarvis j math phys 30 302 1989 it is found that introducing an additional constraint on the norm of the solution vector produces a probability distribution that is integrable over the entire real axis a number of simplifications occur in this extended method the lagrange multipliers and solution vector can be solved for explicitly thus eliminating the necessity of solving systems of nonlinear equations for the lagrange multipliers as was required in the previous approach it is shown that the solution obtained is equivalent to a minimum norm approximation the maximum entropy solution of differential equations with fourier moments is shown to be identical to a fourier series solution additionally the new method is applied to solving the random walk and fokker planck equations

approximate solution of fredholm integral equations by the maximum entropy method an approximate means of solving fredholm integral equations by the maximum entropy method is developed the fredholm integral equation is converted to a generalized moment problem whose approximate solution by maximum entropy methods has been successfully implemented in a previous paper by mead and papanicolaou l r mead and n papanicolaou j math phys 25 2404 1984 several explicit examples are given of approximate maximum entropy solutions of fredholm integral equations of the first and second kinds and of the wiener hopf type both the weaknesses and strengths of the method are discussed

erratum a corrected proof of the stochastic verification theorem within the framework of viscosity solutions we correct the proof of theorem 4 1 in gozzi swiech and zhou siam j control optim 43 2005 pp 2009 2019 by imposing additional conditions on the viscosity subsolution u

a novel method for solving a class of singularly perturbed boundary value problems based on reproducing kernel method abstract in this paper a novel method is presented for solving a class of singularly perturbed boundary value problems firstly the original problem is reformulated as a new boundary value problem whose solution does not change rapidly via a proper transformation then the reproducing kernel method is employed to solve the boundary value new problem numerical results show that the present method can provide very accurate analytical approximate solutions

a maximum entropy method for solving frobenius perron operator equations let s 0 1 0 1 be a nonsingular transformation that preserves an absolutely continuous invariant measure with density f in this paper we develop a general algorithm for approximating f using the maximum entropy approach

on the convergence of moment problems we study the problem of estimating a nonnegative density given a finite number of moments such problems arise in numerous practical applications as the number of moments increases the estimates will always converge weak as measures but need not converge weakly in l this is related to the existence of functions on a compact metric space which are not essentially riemann integrable in some suitable sense we characterize the type of weak convergence we can expect in terms of riemann integrability and in some cases give error bounds when the estimates are chosen to minimize an objective function with weakly compact level sets such as the bolzmann shannon entropy they will converge weakly in l when an lp norm 1 p is used as the objective the estimates actually converge in norm these results provide theoretical support to the growing popularity of such methods in practice

a non linear single step explicit scheme for non linear two point singularly perturbed boundary value problems via initial value technique abstract in this paper a method based on initial value technique is proposed for solving non linear two point singularly perturbed boundary value problems for second order ordinary differential equations odes with a boundary layer at one either left or right end the original singularly perturbed boundary value problem is reduced to an initial value problem approximated by its outer solution asymptotic approximation the new initial value problem is solved by proposed non linear single step explicit scheme followed the idea given in f d van niekerk non linear one step methods for initial value problems comput math appl 13 1987 367 371 the proposed scheme has been shown to be of order two to demonstrate the applicability of the proposed scheme several linear and non linear problems have been solved it is observed that the present scheme approximate the exact solution very well

maximum entropy approach with higher moments for solving fokker planck equation abstract the maximum entropy method with higher number of moments is used to solve the fokker planck equation an adopted newton method is used to iterate the maximum entropy set of equations the method is used to calculate the probability density function of the fokker planck equation the calculations are carried out for three examples 1 the bistable systems of double well potential that is used in many problems related to the fluctuation and relaxation processes in far from equilibrium systems 2 the malthus verhulst model which is used to study the evolution of the number of individuals of an ecological species and the evolution of the intensity of the laser light 3 the black scholes equation used in financial market option pricing although the maximum entropy approach has several advantages it is not convergent at large times and so cannot be used to calculate the steady state solution

robust stochastic maximum principle for multi model worst case optimization this paper develops a version of the robust maximum principle applied to the minimax mayer problem formulated for stochastic differential equations with a control dependent diffusion term the parametric families of first and second order adjoint stochastic processes are introduced to construct the corresponding hamiltonian formalism the hamiltonian function used for the construction of the robust optimal control is shown to be equal to the sum of the standard stochastic hamiltonians corresponding to each possible value of the parameter the cost function is defined on a finite horizon and contains the mathematical expectation of a terminal term a terminal condition given by a vector function is also considered the optimal control strategies adapted for available information for the wide class of multi model systems given by a stochastic differential equation with parameters from a given finite set are constructed this problem belongs to the class of minimax stochastic optimization problems the p

fractal based brain tumor detection in multimodal mri in this work we investigate the effectiveness of fusing two novel texture features along with intensity in multimodal magnetic resonance mr images for pediatric brain tumor segmentation and classification one of the two texture features involves our piecewise triangular prism surface area ptpsa algorithm for fractal feature extraction the other texture feature exploits our novel fractional brownian motion fbm framework that combines both fractal and wavelet analyses for fractalwavelet feature extraction we exploit three mr image modalities such as t1 gadolinium enhanced t2 and fluid attenuated inversion recovery flair respectively the extracted features from these multimodality mr images are fused using self organizing map som for a total of 204 t1 contrast enhanced t2 and flair mr images obtained from nine different pediatric patients our successful tumor segmentation is 100 our experimental results suggest that the fusion of fractal fractalwavelet and intensity features in multimodality mr images offers better tumor segmentation results when compared to that of just fractal and intensity features in single modality mr images next we exploit a multi layer feedforward neural network with automated bayesian regularization to classify the tumor regions from non tumor regions the receiver operating characteristic roc curves are obtained to evaluate tumor classification performance the roc suggests that at a threshold value of 0 7 the true positive fraction tpf values range from 75 to 100 for different patients with the average value of 90

an approximate method for solving a class of singular perturbation problems abstract an approximate method for the numerical solution of a class of singularly perturbed two point boundary value problems is presented the given region is divided into inner and outer regions the original second order problem is replaced by an asymptotically equivalent first order problem and solved as an initial value problem in the inner region a terminal boundary condition is then obtained from the solution of the inner region problem in turn an outer region problem is obtained by replacing the second order differential equation by an approximate first order differential equation with a small deviating argument and solved efficiently by employing the trapezoidal formula coupled with a discrete invariant imbedding algorithm the proposed method is iterative on the terminal point of the inner region problem several numerical examples have been solved to demonstrate the applicability of the method

a new image segmentation method based on multifractal detrended moving average analysis abstract in order to segment and delineate some regions of interest in an image we propose a novel algorithm based on the multifractal detrended moving average analysis mf dma in this method the generalized hurst exponent h q is calculated for every pixel firstly and considered as the local feature of a surface and then a multifractal detrended moving average spectrum mf dms d h q is defined by the idea of box counting dimension method therefore we call the new image segmentation method mf dms based algorithm the performance of the mf dms based method is tested by two image segmentation experiments of rapeseed leaf image of potassium deficiency and magnesium deficiency under three cases namely backward 0 centered 0 5 and forward 1 with different q values the comparison experiments are conducted between the mf dms method and other two multifractal segmentation methods namely the popular mfs based and latest mf dfs based methods the results show that our mf dms based method is superior to the latter two methods the best segmentation result for the rapeseed leaf image of potassium deficiency and magnesium deficiency is from the same parameter combination of 0 5 and d h 10 when using the mf dms based method an interesting finding is that the d h 10 outperforms other parameters for both the mf dms based method with centered case and mf dfs based algorithms by comparing the multifractal nature between nutrient deficiency and non nutrient deficiency areas determined by the segmentation results an important finding is that the gray value s fluctuation in nutrient deficiency area is much severer than that in non nutrient deficiency area

initial value technique for singularly perturbed two point boundary value problems using an exponentially fitted finite difference scheme in this paper we present an approximate method initial value technique for the numerical solution of quasilinear singularly perturbed two point boundary value problems in ordinary differential equations having a boundary layer at one end left or right point it is motivated by the asymptotic behavior of singular perturbation problems the original problem is reduced to an asymptotically equivalent first order initial value problem by approximating the zeroth order term by outer solution obtained by asymptotic expansion and then this initial value problem is solved by an exponentially fitted finite difference scheme some numerical examples are given to illustrate the given method it is observed that the presented method approximates the exact solution very well for crude mesh size h

solving singular two point boundary value problem in reproducing kernel space in this paper we present a new method for solving singular two point boundary value problem for certain ordinary differential equation having singular coefficients its exact solution is represented in the form of series in reproducing kernel space in the mean time the n term approximation u n x to the exact solution u x is obtained and is proved to converge to the exact solution some numerical examples are studied to demonstrate the accuracy of the present method results obtained by the method are compared with the exact solution of each example and are found to be in good agreement with each other

on the use of stochastic differential games against nature to ergodic control problems with unknown parameters in this paper we propose an application of the so called games against nature for giving solution to an ergodic control problem governed by a general class of markov diffusion processes whose coefficients depend on an unknown and non observable parameter to this end we assume that the values of the parameter are taken by means of actions made by some opposite player of the controller the nature then the problem reduces to finding optimality for the controller given that the nature has chosen its best strategy such a control is also known as the worst case optimal control our analysis is based on the use of the dynamic programming technique by showing among other facts the existence of classical twice differentiable solutions of the so called hamilton jacobi bellman equation we also provide an example on economic welfare to illustrate our results

reducing degeneracy in maximum entropy models of networks based on jaynes s maximum entropy principle exponential random graphs provide a family of principled models that allow the prediction of network properties as constrained by empirical data observables however their use is often hindered by the degeneracy problem characterized by spontaneous symmetry breaking where predictions fail here we show that degeneracy appears when the corresponding density of states function is not log concave which is typically the consequence of nonlinear relationships between the constraining observables exploiting these nonlinear relationships here we propose a solution to the degeneracy problem for a large class of systems via transformations that render the density of states function log concave the effectiveness of the method is demonstrated on examples

additive logistic regression a statistical view of boosting boosting is one of the most important recent developments in classification methodology boosting works by sequentially applying a classification algorithm to reweighted versions of the training data and then taking a weighted majority vote of the sequence of classifiers thus produced for many classification algorithms this simple strategy results in dramatic improvements in performance we show that this seemingly mysterious phenomenon can be understood in terms of well known statistical principles namely additive modeling and maximum likelihood for the two class problem boosting can be viewed as an approximation to additive modeling on the logistic scale using maximum bernoulli likelihood as a criterion we develop more direct approximations and show that they exhibit nearly identical results to boosting direct multiclass generalizations based on multinomial likelihood are derived that exhibit performance comparable to other recently proposed multiclass generalizations of boosting in most situations and far superior in some we suggest a minor modification to boosting that can reduce computation often by factors of 10 to 50 finally we apply these insights to produce an alternative formulation of boosting decision trees this approach based on best first truncated tree induction often leads to better performance and can provide interpretable descriptions of the aggregate decision rule it is also much faster computationally making it more suitable to large scale data mining applications

robust optimal control for minimax stochastic linear quadratic problem the robust maximum principle applied to the minimax linear quadratic problem is derived for stochastic differential equations containing a control dependent diffusion term the parametric families of the first and second order adjoint stochastic processes are obtained to construct the corresponding hamiltonian formalism the hamiltonian function used for the construction of the robust optimal control is shown to be equal to the sum of the standard stochastic hamiltonians corresponding to each value of the uncertain parameter from a given finite set the cost function is considered on a finite horizon contains the mathematical expectation of both an integral and a terminal term and on an infinite one a time averaged losses function these problems belong to the class of minimax stochastic optimization problems it is shown that the construction of the minimax optimal controller can be reduced to an optimization problem on a finitedimensional simplex and consists in the analysis of the dependence of ric

solution to boundary value problems using the method of maximum entropy the method of maximum entropy is used to solve a class of linear boundary value problems the method is based on using various moments of the differential equation as constraints when maximizing the entropy various examples are presented and compared to exact solutions for varying numbers of moments it is found that the maximum entropy approximation is in many cases better than a fourier series solution for a given number of expansion terms and moments the method is very general and will find applications in many areas of physics a comparison of the amount of work necessary for the maximum entropy solution versus finite difference techniques is presented and it is found that the maximum entropy technique shows promise as an alternative solution technique

local fractal and multifractal features for volumic texture characterization for texture analysis several features such as co occurrence matrices gabor filters and the wavelet transform are used recently fractal geometry appeared to be an effective feature to analyze texture but it is often restricted to 2d images while 3d information can be very important especially in medical image processing moreover applications are limited to the use of fractal dimension this study focuses on the benefits of fractal geometry in a classification method based on volumic texture analysis the proposed methods make use of fractal and multifractal features for a 3d texture analysis of a voxel neighborhood they are validated with synthetic data before being applied on real images their efficiencies are proved by comparison to some other texture features in supervised classification processes adaboost and support vector machine classifiers the results showed that features based on fractal geometry by combining fractal and multifractal features contributed to new texture characterization information on new features was useful and complementary for a classification method this study suggests that fractal geometry can provide a new useful information in 3d texture analysis especially in medical imaging

a modified piecewise linear markov approximation of markov operators in this paper we propose a modified piecewise linear markov method for approximating markov operators defined on l 1 0 1 we present numerical results with the new method as compared to those using the original piecewise linear markov method

multiple resolution texture analysis and classification textures are classified based on the change in their properties with changing resolution the area of the gray level surface is measured at serveral resolutions this area decreases at coarser resolutions since fine details that contribute to the area disappear fractal properties of the picture are computed from the rate of this decrease in area and are used for texture comparison and classification the relation of a texture picture to its negative and directional properties are also discussed

a maximum entropy method for solving the boundary value problem of second order ordinary differential equations we propose a new method to solve the boundary value problem for a class of second order linear ordinary differential equations which has a non negative solution the method applies the maximum entropy principle to approximating the solution numerically the theoretical analysis and numerical examples show that our method is convergent we propose a new method to solve the boundary value problem for a class of second order linear ordinary differential equations which has a non negative solution the method applies the maximum entropy principle to approximating the solution numerically the theoretical analysis and numerical examples show that our method is convergent

maximum entropy in the problem of moments the maximum entropy approach to the solution of underdetermined inverse problems is studied in detail in the context of the classical moment problem in important special cases such as the hausdorff moment problem we establish necessary and sufficient conditions for the existence of a maximum entropy solution and examine the convergence of the resulting sequence of approximations a number of explicit illustrations are presented in addition to some elementary examples we analyze the maximum entropy reconstruction of the density of states in harmonic solids and of dynamic correlation functions in quantum spin systems we also briefly indicate possible applications to the lee yang theory of ising models to the summation of divergent series and so on the general conclusion is that maximum entropy provides a valuable approximation scheme a serious competitor of traditional pade like procedures

asymptotic initial value methods for two parameter singularly perturbed boundary value problems for second order ordinary differential equations in this paper some numerical methods are presented for singularly perturbed two point boundary value problems for second order ordinary differential equations with two small parameters multiplying the derivatives these methods are distinguished by the fact that initial value problems and or terminal value problems are constructed deduced from the given boundary value problem and then solved by a fitted operator method further error estimates are derived the methods suggested are simple to use and easy to implement the important aspect is that some of the methods suggested are well suited for parallel computing examples are provided to illustrate the methods

approximate time dependent solutions of partial differential equations the maxent minimum norm approach a new approach to inverse problems with incomplete data the maxent minimum norm scheme was recently introduced by baker jarvis bj this method is based on the application of jaynes s maxent principle to a probability distribution defined on the functional space of all the possible solutions of the problem in the present work we consider the application of these ideas to non equilibrium time dependent systems in the spirit of jaynes s information theory approach to statistical mechanics we focus on the behaviour of a small number of physically relevant mean values by recourse to the bj procedure the equations of motion of those mean values are closed and an approximate description of the system s evolution is obtained it is also shown that the bj method is closely related to curado s case i e q 2 and standard mean values of tsallis nonextensive thermostatistics

solving fredholm integral equations via a piecewise linear maximum entropy method we propose a piecewise linear approximation method based on the maximum entropy principle to approximate a nonnegative solution of a fredholm integral equation numerically the theoretical analysis and numerical examples show that our method has a convergence rate of order 2 and it can get more accurate approximations with more moments used without ill condition of the classic maximum entropy approach the method can also be applied to solve fredholm integral equations with singular kernels

convergence of best entropy estimates given a finite number of moments of an unknown density on a finite measure space the best entropy estimate that nonnegative density x with the given moments which minimizes the boltzmann shannon entropy i x x log x is considered a direct proof is given that i has the kadec property in l if yn converges weakly to 37 and i yn converges to i 37 then yn converges to 37 in norm as a corollary it is obtained that as the number of given moments increases the best entropy estimates converge in li norm to the best entropy estimate of the limiting problem which is simply in the determined case furthermore for classical moment problems on intervals with strictly positive and sufficiently smooth error bounds and uniform convergence are actually obtained

morphology based multifractal estimation for texture segmentation multifractal analysis is becoming more and more popular in image segmentation community in which the box counting based multifractal dimension estimations are most commonly used however in spite of its computational efficiency the regular partition scheme used by various box counting methods intrinsically produces less accurate results in this paper a novel multifractal estimation algorithm based on mathematical morphology is proposed and a set of new multifractal descriptors namely the local morphological multifractal exponents is defined to characterize the local scaling properties of textures a series of cubic structure elements and an iterative dilation scheme are utilized so that the computational complexity of the morphological operations can be tremendously reduced both the proposed algorithm and the box counting based methods have been applied to the segmentation of texture mosaics and real images the comparison results demonstrate that the morphological multifractal estimation can differentiate texture images more effectively and provide more robust segmentations

computational methods for solving two parameter singularly perturbed boundary value problems for second order ordinary differential equations singularly perturbed two point boundary value problems spbvps for second order ordinary differential equations odes with two small parameters multiplying the derivatives are considered computational methods are suggested in this paper to solve this type of problems one of these methods is fitted operator method fom and the other is bondary value technique bvt in fom we suggest an exponentially fitted finite difference effd scheme throughout the domain of definition of the differential equation and the error estimates are derived in bvt the domain of definition of the differential equation is divided into three subintervals called inner and outer regions then we solve the two parameter spbvp tpspbvp over these regions as two point boundary value problems an exponentially fitted finite difference scheme is used in the inner regions and a classical finite difference scheme in the outer region the data for boundary conditions at the transition points are obtained using zero order asymptotic expansion approximation of the solution of the problem the bvt is well suited for parallel computing numerical examples are given to illustrate the method

wavelet analysis and synthesis of fractional brownian motion fractional brownian motion fbm offers a convenient modeling for nonstationary stochastic processes with long term dependencies and 1 f type spectral behavior over wide ranges of frequencies statistical self similarity is an essential feature of fbm and makes natural the use of wavelets for both its analysis and its synthesis a detailed second order analysis is carried out for wavelet coefficients of fbm it reveals a stationary structure at each scale and a power law behavior of the coefficients variance from which the fractal dimension of fbm can be estimated conditions for using orthonormal wavelet decompositions as approximate whitening filters are discussed consequences of discretization are considered and some connections between the wavelet point of view and previous approaches based on length measurements analysis or dyadic interpolation synthesis are briefly pointed out

design and implementation of an estimator of fractal dimension using fuzzy techniques abstract this paper presents a new method for estimating the fractal dimension of one dimensional profiles in this approach the real fractal dimension d is considered as an implicit continuous function of the estimated fractal dimension de the resolution and several other elements by approximating this function from a number of experimental data we can obtain more precise estimates of the fractal dimension d this approximation is done using a fuzzy logic controller and an averaging procedure permitting to respectively decrease two kinds of estimation errors 1 systematic errors which are associated with values of d resolution trends of profiles and etc 2 stochastic errors which are mainly caused by the choice of the sequence ek representing the sizes of structuring elements corresponding to different scales the effectiveness of this method is shown by estimating fractal dimensions for two sample functions and a number of natural and synthetic fibers

a maximum entropy method based on piecewise linear functions for the recovery of a stationary density of interval mappings let s 0 1 0 1 be a nonsingular transformation such that the corresponding frobenius perron operator p s l 1 0 1 l 1 0 1 has a stationary density f we propose a maximum entropy method based on piecewise linear functions for the numerical recovery of f an advantage of this new approximation approach over the maximum entropy method based on polynomial basis functions is that the system of nonlinear equations can be solved efficiently because when we apply newton s method the jacobian matrices are positive definite and tri diagonal the numerical experiments show that the new maximum entropy method is more accurate than the markov finite approximation method which also uses piecewise linear functions provided that the involved moments are known this is supported by the convergence rate analysis of the method

accurate solution of the thomasfermi equation using the fractional order of rational chebyshev functions in this paper the nonlinear singular thomasfermi differential equation for neutral atoms is solved using the fractional order of rational chebyshev orthogonal functions frcs of the first kind ftn t l on a semi infinite domain where l is an arbitrary numerical parameter first using the quasilinearization method the equation be converted into a sequence of linear ordinary differential equations ldes and then these ldes are solved using the frcs collocation method using 300 collocation points we have obtained a very good approximation solution and the value of the initial slope y 0 1 5880710226113753127186845094239501095 highly accurate to 37 decimal places

modeling and control of stochastic systems with poorly known dynamics this paper is concerned with controlling poorly known systems for which only a simplified and rough model is available for control design there are many systems that cannot be reasonably probed for the sake of identification yet they are important for areas such as economy populations or medicine the ideas are developed around an alternative way to account for the bare modeling in a stochastic based setting and to heighten the control features for such a modified model the mathematical framework for the optimal control reveals important features such as the raising of a precautionary feedback policy of keep the action unchanged inaction for short on a certain state space region this feature is not seen in the robust approach but has been pointed out and permeates part of the economics literature the control problem relies on the viscosity solution for the hamilton jacobi bellman equation and the value of the problem is shown to be convex when specialized to the quadratic problem with discounted cost the exact solution inside the inaction region is given by a lyapunov type of equation and asymptotically for large state values by a riccati like equation this scenario bridges to the stochastic stability analysis for the controlled model the single control input is developed in full part analytically part numerically for the scalar case and an approximation is tested for the multidimensional case the advantage of the precautionary policy is substantial in some situations

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