optimal gaussian solutions of nonlinear stochastic partial differential equations we present a linearization procedure of a stochastic partial differential equation for a vector field x i t x t 0 x r d i l n t x i t x b i x t x d x i t x i f i t x here is the laplace beltrami operator inr d and f i t x is a gaussian random field with f i t x f j t x ij t t x x the procedure is a natural extension of the equivalent linearization for stochastic ordinary differential equations the linearized solution is optimal in the sense that the distance between true and approximate solutions is minimal when it is measured by the kullback leibler entropy the procedure is applied to the scalar valued ginzburg landau model in r1 withb 1 z z vz 3 stationary values of mean variance and correlation length are calculated they almost agree with exact ones if 1 24 2 1 4 d 1 1 3 c when c there appear quasistationary states fluctuating around one of the bottoms of the potentialu z b 1 z dz the second moment at the quasistationary states almost agrees with the exact one transient phenomena are also discussed half width at half maximum of a structure function decays liket 1 2 for small t the diffusion term 2 x accelerates the relaxation from the neighborhood of an unstable initial statex 0 x 0

comments on estimation of frequencies and damping factors by two dimensional esprit type methods for original paper see rouquette and najim ieee trans signal process vol 49 no 1 p 237 45 2001 the original authors present two high resolution approaches to estimate two dimensional 2 d signal parameters the first approach decomposes the 2 d estimation problem into two one dimensional 1 d ones estimates the 1 d frequencies with the memp method and then forms correct pairs using a new pairing scheme introduced in vanpoucke et al 1994 the second method the 2 d esprit method directly estimates the 2 d frequencies it is shown that both methods are more accurate than the related methods the present authors note that the original paper appears to have a key error in the permutation matrix specification of the first method in this comment they derive the correct one with the performance simulations

a comparative study of numerical solutions of a class of kdv equation abstract in this work we present a comparative study of meshless method modified bernstein polynomials bp and b spline finite element method bs fem for the numerical solution of two different models of korteweg de vries kdv equation the multiquadric mq and gaussian ga radial basis functions rbfs are used due to exponential convergence rate excellent agreement is found between exact and rbfs solutions and same accuracy is obtained as bernstein polynomials and b spline finite element method

numerical solution of kdv equation using modified bernstein polynomials abstract here we present an algorithm for approximating numerical solution of korteweg de vries kdv equation in a modified b polynomial basis a set of continuous polynomials over the spatial domain is used to expand the desired solution requiring discretization with only the time variable galerkin method is used to determine the expansion coefficients to construct initial trial functions for the time variable the system of equations is solved using fourth order runge kutta method the accuracy of the solutions is dependent on the size of the b polynomial basis set we have presented our numerical result with an exact analytical result excellent agreement is found between exact and approximate solutions this procedure has a potential to be used in more complex system of differential equations where no exact solution is available

statistical replacement for systems with delta correlated fluctuations nonlinear systems with stochastic parameters are approximated by simpler systems using a method that we call statistical replacement this method is an extension of the previously developed agree which was restricted to systems with additive fluctuations statistical replacement incorporates the distinctions between globally stable thermodynamically closed systems and thermodynamically open systems that can be unstable

a variational approach to stochastic nonlinear problems a variational principle is formulated which enables the mean value and higher moments of the solution of a stochastic nonlinear differential equation to be expressed as stationary values of certain quantities approximations are generated by using suitable trial functions in this variational principle and some of these are investigated numerically for the case of a bernoulli oscillator driven by white noise comparison with exact data available for this system shows that the variational approach to such problems can be quite effective

piecewise optimal linearization method for nonlinear stochastic differential equations a method for estimating the dynamical statistical properties of the solutions of nonlinear langevin type stochastic differential equations is presented the non linear equation is linearized within a small interval of the independent variable and statistical properties are expressed analytically within the interval the linearization procedure is optimal in the sense of the chebyshev inequality long term behavior of the solution process is obtained by appropriately matching the approximate solutions at the boundaries between intervals the method is applied to a model nonlinear equation for which the exact time dependent moments can be obtained by numerical methods the calculations demonstrate that the method represents a significant improvement over the method of statistical linearization in time regimes far from equilibrium

quasilinear approximations for the propagator of the fokker planck equation abstract four novel classes of the approximate fokker planck propagators are developed by means of a decoupling operator treatment for a small time increment t these are correct at least through second order t the propagators have an ornstein uhlenbeck form thus leading to novel discrete path integrals with a linear reference system their ambiguities are characterized by an arbitrary function of the time increment and by a matrix and a vector field which may be both time and state dependent the efficiency of the treatment used is very sensitive to the choice of these functions several forms of which are proposed the power of the treatment is illustrated by a physically meaningful fokker planck equation relevant to laser physics being used as the global approximations the approximate propagators obtained are shown to provide a correct description of a bistable system in the entire time domain a scaling theory of suzuki including its unified version emerges from them in a very natural way another attractive feature of our general treatment when compared to the others known in the literature is that it allows for equations with singular diffusion matrices two of which a kramers equation and a colored noise problem are considered

estimation of frequencies and damping factors by two dimensional esprit type methods in this paper we consider two dimensional 2 d signals modeled by the sum of damped cisoids we propose two high resolution approaches to estimate their frequencies and damping factors both high resolution methods are based on the shift invariance structure of the signal subspace related to each dimension the first one estimates the frequency components in both dimensions as in the matrix enhancement and matrix pencil memp method before pairing them with a new algorithm the second one consists of the direct estimation of the signal frequency pairs without an additional step to pair the frequencies related to each dimension we show how these methods can estimate the scattering points of radar images

approximate gaussian representation of evolution equations i single degree of freedom nonlinear equations we have developed a generalization of the method of statistical linearization to enable us to describe transient and other nonstationary phenomena obeying stochastic nonlinear differential equations this approximation technique provides an optimal gaussian representation with time dependent parameters the algorithm specifies a set of ordinary differential equations for the gaussian parameters in terms of the time dependent average nonlinearities we apply the general formalism developed herein for single degree of freedom dissipative systems to a particular example

diffusion in a bistable potential a comparative study of different methods of solution the problem of diffusion in a bistable potential is studied by considering the associated nonlinear langevin equation and its equivalent fokker planck equation two numerically exact methods of solution namely the monte carlo solution of the nonlinear langevin equation and the solution of the fokker planck equation via the finite difference technique are considered the latter method has the advantage that it directly gives the evolution of the probability distribution function approximate analyses of the fluctuations using the system size expansion the gaussian decoupling procedure and the scaling approach are also carried out these investigations are performed on a representative problem for two specific cases 1 evolution from intrinsically unstable states and 2 evolution from extensive regime the fluctuations obtained using these approximate methods are compared with those obtained via the numerically exact methods the study brings out the advantages and limitations of each of the methods considered

fundamentals of statistical signal processing estimation theory minimum variance unbiased estimation cramer rao lower bound linear models general minimum variance unbiased estimation best linear unbiased estimators maximum likelihood estimation least squares method of moments the bayesian philosophy general bayesian estimators linear bayesian estimators kalman filters summary of estimators extension for complex data and parameters

unstable state dynamics treatment of colored noise the diffusion of a particle set near an unstable point in a bistable potential is considered the scaling theory of fluctuations proposed originally for onedimensional systems driven by gaussian white noise is extended to arbitrary dimensions the merits and drawbacks of the scaling theory are discussed by taking a model problem in one dimension it is shown in passing that the saddle point approximation enables one to get analytic expressions for various moments of the stochastic process the two different methods to include asymptotic fluctuations which are absent in the usual scaling solution are shown to be equivalent an alternate way of including asymptotic fluctuations is attempted by solving the associated fokker planck equation using the fer formula the reason for the failure of this method is traced after this it is argued that the unified scaling theory should be applicable for treatment of colored noise as well for the scaling assumption is independent of the statistical property of the driving noise explicit monte carlo simulation of a model onedimensional system driven by exponentially correlated gaussian noise is performed and compared with the scaling solution to bolster this point the agreement is very good

on the linearization of nonlinear langevin type stochastic differential equations we show that a logical extension of the piecewise optimal linearization procedure leads to the gaussian decoupling scheme where no iteration is required the scheme is equivalent to solving a few coupled equations the method is applied to models which represent a a single steady state b passage from an initial unstable state to a final preferred stable state by virtue of a finite displacement from the unstable state and c a bivariate case of passage from an unstable state to a final stable state the results are shown to be in very good agreement with the monte carlo calculations carried out for these cases the method should be of much value in multidimensional cases

the validity of nonlinear langevin equations in the presence of internal noise the variables describing a system are intrinsically stochastic if they constitute a markov process the expansion enables one to extract a deterministic macroscopic equation and to compute the fluctuations in successive approximations in the lowest or linear noise approximation the fluctuations can be represented by a langevin equation provided it is handled appropriately higher orders cannot be described by any white noise langevin equation the question whether the equation has to be interpreted according to ito or stratonovich concerns these higher orders for which the equation is not valid anyway

universal trellis codes for diagonally layered space time systems foschini s diagonally layered space time transmission system known as d blast is an architecture designed for a rayleigh fading environment using multiple element antenna arrays at both the transmit and receive sites to achieve very high spectral efficiencies in this paper we propose a simple coding technique for d blast that uses a single trellis code with finite traceback viterbi decoding we examine the performance of universal trellis codes that are designed to have a distance structure that is matched to the periodic signal to noise ratio snr variation of the channel created by d blast under the assumption that the channel is static during one burst but may change from burst to burst we show that a universal 64 state trellis code on a 2 spl times 2 d blast system with long enough block lengths displays universal behavior working on almost every 2 spl times 2 channel with at least the mutual information required by a standard 64 state awgn trellis code the only 2 spl times 2 channel where more mutual information is required is a certain rotation of the zero eigenvalue channel we also present 4 spl times 4 and 8 spl times 8 examples

estimating two dimensional frequencies by matrix enhancement and matrix pencil a new method called the matrix enhancement and matrix pencil memp method is presented for estimating two dimensional 2 d frequencies in the memp method an enhanced matrix is constructed from the data samples and then the matrix pencil approach is used to extract out the 2 d sinusoids from the principal eigenvectors of the enhanced matrix the memp method yields the estimates of the 2 d frequencies efficiently without solving the roots of a 2 d polynomial or searching in a 2 d space it is shown that the memp method can be faster than a 2 d fft method if the number of the 2 d sinusoids is much smaller than the data set simulation results are provided to show that the accuracy of the memp method can be very close to the cramer rao lower bound

the parameter r2 in multiquadric interpolation the multiquadric mq method is an effective bivariate interpolant to three dimensional data xi yi zi where the xi yi are arbitrarily located in the plane the accuracy of the mq method is dependent on a user defined parameter r2 and most practitioners select r2 based upon the number of data points and the locations of the xi yi in the plane we observe that the optimal value of r2 is a strong function of the zi and that it is essentially independent of both the number and the locationsof the xi yi points contrary to some opinions we observe that the mq method can effectively interpolate to track data that is data sampled densely along tracks in the plane together with several other observations we present an algorithm that generally yields an effective value for r2

numerical solution of korteweg de vries equation by galerkin b spline finite element method a finite element solution of the kdv equation is presented to demonstrate the efficiency of the method two test problems are considered the numerical solutions of the kdv equation are compared with both the exact solutions and other numerical solutions in the literature the numerical solutions are found to be in good agreement with the exact solutions

analytical and numerical aspects of certain nonlinear evolution equations 1v numerical modified korteweg de vries equation various numerical methods are used in order to approximate the korteweg de vries equation namely i zabusky kruskal scheme ii hopscotch method iii a scheme due to goda iv a proposed local scheme v a proposed global scheme vi a scheme suggested by kruskal vii split step fourier method by tappert viii an improved split step fourier method and ix pseudospectral method by fornberg and whitham comparisons between our proposed scheme which is developed using notions of the inverse scattering transform and the other utilized schemes are obtained