domain of definition of levermore s five moment system the simplest system in levermore s moment hierarchy involving moments higher than second order is the five moment closure it is obtained by taking velocity moments of the one dimensional boltzmann equation under the assumption that the velocity distribution is a maximum entropy function the moment vectors for which a maximum entropy function exists consequently make up the domain of definition of the system the aim of this article is a complete characterization of the structure of the domain of definition and the connected maximum entropy problem the space homogeneous case of the equation and numerical aspects are also addressed

dynamic spin pair correlations in a heisenberg chain at infinite temperature based on an extended short time expansion exact coefficients of the short time expansion of the spin pair correlations are computed up to order t30 in the one dimensional spin 1 2 heisenberg model with isotropic nearest neighbour interaction at infinite temperature the key points of the computational algorithm are sketched the newly derived coefficients are used to obtain lower and upper bounds on the pair correlations which are graphically indistinguishable up to time nearly twice as large as those up to which previous bounds are reasonably close together in view of the spin diffusion hypothesis the spatial spread of the pair correlations is studied as a function of time but no indication for a diffusive long time behaviour is found in the extended time region where the new bounds provide control

an improved algorithm for the multidimensional moment constrained maximum entropy problem the maximum entropy principle is a versatile tool for evaluating smooth approximations of probability density functions with the least bias beyond specified constraints in the recent paper we introduced new computational framework for the moment constrained maximum entropy problem in a multidimensional domain and developed a simple numerical algorithm capable of computing maximum entropy problem in a two dimensional domain with moment constraints of order up to 4 here we design an improved numerical algorithm for computing the maximum entropy problem in a two and higher dimensional domain with higher order moment constraints the algorithm features multidimensional orthogonal polynomial basis in the dual space of lagrange multipliers to achieve numerical stability and rapid convergence of newton iterations the new algorithm is found to be capable of solving the maximum entropy problem in the two dimensional domain with moment constraints of order up to 8 in the three dimensional domain with moment constraints of order up to 6 and in the four dimensional domain with moment constraints of order up to 4 corresponding to the total number of moment constraints of 44 83 and 69 respectively the two and higher dimensional maximum entropy test problems in the current work are based upon long term statistics of numerical simulation of the real world geophysical model for wind stress driven oceanic currents such as the gulf stream and the kuroshio

moment methods for decision analysis decision models involving continuous probability distributions almost always require some form of approximation the usual approach to evaluating these kinds of models is to construct a discrete approximation for each continuous distribution and compute value lotteries and certain equivalents using these discrete approximations although decision analysts are quite comfortable with this approach there has been relatively little consideration of how these discrete approximations affect the results of the analysis in the first part of this paper we review three common methods of constructing discrete approximations and compare their performance in a simple example the results of the example suggest a different approach that offers potential improvements in accuracy and efficiency over the usual approach the basic idea is that given discrete approximations that accurately represent the moments of assessed input distributions we may easily and accurately compute the moments of the output distribution or value lotteries these moments then summarize what we know about the value lottery and certain equivalent and provide the basis for computing approximate value lotteries and certain equivalents in this paper we discuss the methods supporting this moment approach and evaluate their performance in the context of the example

hausdorff moment problem and maximum entropy a unified approach hausdorff moment problem in the approach of maximum entropy is reconsidered some aspects as existence condition entropy convergence entropy decreasing stability are studied through a unified approach results known in literature and some new ones are illustrated

lyapunov exponents and the natural invariant density determination of chaotic maps an iterative maximum entropy ansatz we apply the maximum entropy principle to construct the natural invariant density and the lyapunov exponent of one dimensional chaotic maps using a novel function reconstruction technique that is based on the solution of the hausdorff moment problem via maximizing shannon entropy we estimate the invariant density and the lyapunov exponent of nonlinear maps in one dimension from a knowledge of finite number of moments the accuracy and the stability of the algorithm are illustrated by comparing our results to a number of nonlinear maps for which the exact analytical results are available furthermore we also consider a very complex example for which no exact analytical result for the invariant density is available a comparison of our results to those available in the literature is also discussed

a piecewise quadratic maximum entropy method for the statistical study of chaos abstract let the frobenius perron operator p s l 1 0 1 l 1 0 1 related to a nonsingular transformation s 0 1 0 1 have an invariant density f we propose a piecewise quadratic maximum entropy method for the numerical approximation of f the role of the partition of unity property of the quadratic functions for the numerical recovery of f has been depicted the proposed algorithm overcomes the ill conditioning shortage of the traditional maximum entropy method which only employs polynomials so that any number of moments can be used to increase the accuracy of the computed invariant density the convergence rate of the method is shown to be of order 3 numerical results are presented to justify the theoretical analysis of the method

derivation of operational matrices of differentiation for orthogonal polynomials abstract a direct method for deriving an operational matrix of differentiation for legendre polynomials is presented the general case is also considered the presented formulas are formally proved results are compared with the other known method the new method is accurate for all matrix dimensions while the other method may fail for higher dimensions

a new concept of relative suitability of moment function sets in modeling statistical data by using entropy optimization distributions the choice of a suitable moment function set and various combinations of their elements are very important it is also important to have criteria to compare various moment sets for suitability considering this in the present study by choosing special moment function sets we have obtained maxmaxent minmaxent distributions and shown that the minmaxent distribution more exactly represents the given statistical data moreover in order to compare various moment function sets for suitability we have suggested a new concept of relative suitability of moment function sets for given statistical data and the importance of this concept has been presented in some applications using real data

the hausdorff moments in statistical mechanics a new method for solving the hausdorff moment problem is presented which makes use of pollaczek polynomials this problem is severely ill posed a regularized solution is obtained without any use of prior knowledge when the problem is treated in the l 2 space and the moments are finite in number and affected by noise or round off errors the approximation converges asymptotically in the l 2 norm the method is applied to various questions of statistical mechanics and in particular to the determination of the density of states concerning this latter problem the method is extended to include distribution valued densities computing the laplace transform of the expansion a new series representation of the partition function z 1 k bt is obtained which coincides with a watson resummation of the high temperature series for z

application of general orthogonal polynomials to the optimal control of time varying linear systems the operational matrices for forward or backward integration of general orthogonal polynomials are derived the tensor expression for the generalized orthogonal polynomial approximation of any two arbitrary functions is also introduced it is shown that the approximation solution obtained using the chebyshev polynomial is readily obtainable as a special case of the results derived thus the present results include results presented by shih in 1983 and chou and horng in 1984 a linear time varying optimal control system with a quadratic performance measure is solved by using the generalized orthogonal polynomials comparison of the results with those obtained using several different classical orthogonal polynomial approximations is also 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

a new approach for parameter identification of time varying systems via generalized orthogonal polynomials abstract a very effective method of using the generalized orthogonal polynomials gop for identifying the parameters of a process whose behaviour can be modelled by a linear differential equation with time varying coefficients in the form of finite order polynomials is presented it is based on the differentiation operational matrix of the gop which can represent all kinds of individual orthogonal polynomials the main advantage of using the differentiation operational matrix is that parameter estimation can be made starting at any time of interest i e without the restriction of starting at zero time in addition the present computation algorithm is simpler than that of the integration operational matrix using the concept of gop expansion for state and control functions the differential input output equation is converted into a set of linear algebraic equations the unknown parameters are evaluated by a weighted least squares estimation method very satisfactory results for illustrative example are

moment information for probability distributions without solving the moment problem ii main mass tails and shape approximation how much information does a small number of moments carry about the unknown distribution function is it possible to explicitly obtain from these moments some useful information e g about the support the modality the general shape or the tails of a distribution without going into a detailed numerical solution of the moment problem in this previous and subsequent papers clear and easy to implement answers will be given to some questions of this type first the question of how to distinguish between the main mass interval and the tail regions in the case we know only a number of moments of the target distribution function will be addressed the answer to this question is based on a version of the chebyshev stieltjes markov inequality which provides us with upper and lower moment based bounds for the target distribution then exploiting existing asymptotic results in the main mass region an explicit moment based approximation of the target probability density function is provided although the latter cannot be considered in general as a satisfactory solution it can always serve as an initial approximation in any iterative scheme for the numerical solution of the moment problem numerical results illustrating all the theoretical statements are also presented

high order entropy based closures for linear transport in slab geometry ii a computational study of the optimization problem we present a numerical algorithm to implement entropy based m n moment models in the context of a simple linear kinetic equation for particles moving through a material slab the closure for these models as is the case for all entropy based models is derived through the solution of a constrained convex optimization problem the algorithm has two components the first component is a discretization of the moment equations which preserves the set of realizable moments thereby ensuring that the optimization problem has a solution in exact arithmetic the discretization is a second order kinetic scheme which uses muscl type limiting in space and a strong stability preserving runge kutta time integrator the second component of the algorithm is a newton based solver for the dual optimization problem which uses an adaptive quadrature to evaluate integrals in the dual objective and its derivatives the accuracy of the numerical solution to the dual problem plays a key role in the time step restric

a model free non parametric method for density determination with application to asset returns abstract the distribution of the returns of an asset is still an open problem despite the variety of models and methods devised to deal with it in this note we propose a model free nonparametric method for the estimation of asset returns which is applicable also to a wider set of similar problems the laplace transform of a shifted rate of return or other percent change of a positive random variables can be transformed into a fractional moment problem which can then be solved by the method of maximum entropy this is a very robust method and requires only a few values of the laplace transform to provide good reconstructions this method can be applied in multiple fields where model free density estimation of functionals like percent changes of random variables is of interest

partially finite programming in l_1 and the existence of maximum entropy estimates best entropy estimation is a technique that has been widely applied in many areas of science it consists of estimating an unknown density from some of its moments by maximizing some measure of the entropy of the estimate this problem can be modelled as a partially finite convex program with an integrable function as the variable a complete duality and existence theory is developed for this problem and for an associated extended problem which allows singular measure theoretic solutions this theory explains the appearance of singular components observed in the literature when the burg entropy is used it also provides a unified treatment of existence conditions when the burg boltzmann shannon or some other entropy is used as the objective some examples are discussed

function reconstruction as a classical moment problem a maximum entropy approach we present a systematic study of the reconstruction of non negative functions via maximum entropy approach utilizing the information contained in a finite number of moments of the functions for testing the efficacy of the approach we reconstruct a set of functions using an iterative entropy optimization scheme and study the convergence profile as the number of moments is increased a wide variety of functions are considered that include a distribution with a sharp discontinuity an oscillatory function a distribution with singularities and finally a distribution with several spikes and fine structure the last example is important in the context of the determination of the natural density of the logistic map the convergence of the method is studied by comparing the moments of the approximated functions with the exact ones furthermore by varying the number of moments and iterations we examine to what extent the features of the functions such as the divergence behavior at singular points within the interval is reproduced the proximity of the reconstructed maximum entropy solution to the exact solution is examined via kullback leibler divergence and variation measures for different number of moments

gaussian process regression for maximum entropy distribution abstract maximum entropy distributions offer an attractive family of probability densities suitable for moment closure problems yet finding the lagrange multipliers which parametrize these distributions turns out to be a computational bottleneck for practical closure settings motivated by recent success of gaussian processes we investigate the suitability of gaussian priors to approximate the lagrange multipliers as a map of a given set of moments examining various kernel functions the hyperparameters are optimized by maximizing the log likelihood the performance of the devised data driven maximum entropy closure is studied for couple of test cases including relaxation of non equilibrium distributions governed by bhatnagar gross krook and boltzmann kinetic equations

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

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

a comparative study of the operational matrices of integration and differentiation for orthogonal polynomial series in the paper a comparison is made between two approximate methods of system analysis based on the use of the operational matrices of integration and differentiation both methods applied to model problems involving linear time invariant differential equations yielded results of equal order of accuracy for the same number of orthogonal polynomial terms however the operational matrix of differentiation method is not limited to initial value problems but may easily be applied to boundary value problems in addition due to the exact definition of this matrix transformations from an orthogonal polynomial base to another may easily be performed with the use of the operational matrix of polynomial series transformation the operational matrix of differentiation approach may be easily applied to the analysis of other types of systems such as time varying and distributed parameter systems and can be easily extended for other types of orthogonal differentiable base functions such as the fourier trigonometric

on the application of maximum entropy to the moments problem the maximum entropy approach to the solution of classical inverse problem of moments in which one seeks to reconstruct a function p x where x 0 from the values of a finite set n 1 of its moments is studied it is shown that for n 4 such a function always exists while for n 2 and n 3 the acceptable values of the moments are singled out analytically the paper extends to the general case where the results were previously bounded to the case n 2

some physical features of the burr type xii distribution aspects related to driven nonequilibrium systems considered in beck cohen superstatistics bcs formalism and in addition the unbiased maximum entropy density estimation show the pertinence of considering the three parameter burr type xii distribution as a suitable way to describe stationary states of complex and nonequilibrium systems the above is shown following a variant to the procedure presented in the formulation of bcs which consists of the incorporation of an expanded boltzmann factor through the mittag leffler function on the other hand maximization of shannon boltzmann gibbs entropy and other generalized forms of entropy show that the burr type xii distribution may emerge as the least informative distribution for the mentioned systems

new developments of the extended quadrature method of moments to solve population balance equations abstract population balance models have a wide range of applications in many industrial fields as they allow accounting for heterogeneity among properties which are crucial for some system modelling they actually describe the evolution of a number density function ndf using a population balance equation pbe for instance they are applied to gas liquid columns or stirred reactors aerosol technology crystallisation processes fine particles or biological systems there is a significant interest for fast stable and accurate numerical methods in order to solve for pbes a class of such methods actually does not solve directly the ndf but resolves their moments these methods of moments and in particular quadrature based methods of moments have been successfully applied to a variety of systems point wise values of the ndf are sometimes required but are not directly accessible from the moments to address these issues the extended quadrature method of moments eqmom has been developed in the past few years and approximates the ndf from its moments as a convex mixture of kernel density functions kdfs of the same parametric family in the present work eqmom is further developed on two aspects the main one is a significant improvement of the core iterative procedure of that method the corresponding reduction of its computational cost is estimated to range from 60 up to 95 the second aspect is an extension of eqmom to two new kdfs used for the approximation the weibull and the laplace kernels all matlab source codes used for this article are provided with this article

a sufficient condition for the existence of bound states for scalar spherically symmetric potentials we consider the class of three dimensional attractive finite range or similar potentials depending on a strength constant beyond a critical value the potential has at least one bound state for the s wave we propose simple bounds for obtained by formally solving the schrodinger equation for the zero energy bound state the various bounds are compared with the exact value for a set of usual potentials they are also compared with bounds derived in earlier works by glaser et al and by calogero we show that the case is solved by equations very similar to ones

a high order moment method simulating evaporation and advection of a polydisperse liquid spray in this paper we tackle the modeling and numerical simulation of sprays and aerosols that is dilute gas droplet flows for which polydispersity description is of paramount importance starting from a kinetic description for point particles experiencing transport either at the carrier phase velocity for aerosols or at their own velocity for sprays as well as evaporation we focus on an eulerian high order moment method in size and consider a system of partial differential equations pdes on a vector of successive integer size moments of order 0 to n n 2 over a compact size interval there exists a stumbling block for the usual approaches using high order moment methods resolved with high order finite volume methods the transport algorithm does not preserve the moment space indeed reconstruction of moments by polynomials inside computational cells coupled to the evolution algorithm can create n dimensional vectors which fail to be moment vectors it is impossible to find a size distribution for which there are the moments we thus propose a new approach as well as an algorithm which is second order in space and time with very limited numerical diffusion and allows to accurately describe the advection process and naturally preserves the moment space the algorithm also leads to a natural coupling with a recently designed algorithm for evaporation which also preserves the moment space thus polydispersity is accounted for in the evaporation and advection process very accurately and at a very reasonable computational cost these modeling and algorithmic tools are referred to as the eulerian multi size moment emsm model we show that such an approach is very competitive compared to multi fluid approaches where the size phase space is discretized into several sections and low order moment methods are used in each section as well as with other existing high order moment methods an accuracy study assesses the order of the method as well as the low level of numerical diffusion on structured meshes whereas the extension to unstructured meshes is provided we focus in this paper on cartesian meshes and two 2d test cases are presented taylor green vortices and turbulent free jets where the accuracy and efficiency of the approach are assessed

size velocity correlations in hybrid high order moment multi fluid methods for polydisperse evaporating sprays modeling and numerical issues kah et al 2010 30 33 recently developed the eulerian multi size moment model emsm which tackles the modeling and numerical simulation of polydisperse multiphase flows using a high order moment method in a compact interval they suggested to reconstruct the number density function ndf by entropy maximization which leads to a unique and realizable ndf potentially in several size intervals thus leading to an hybrid method between multifluid and high order this reconstruction is used to simulate the evaporation process by an evaluation of the flux of droplet disappearance at zero size the fluxes of droplets between size intervals and an accurate description of the size shift induced by evaporation massot et al 2010 15 although this method demonstrated its potential for evaporating polydisperse flows two issues remain to be addressed first the emsm only considers one velocity for all droplets thus decoupling size from velocity which is too restrictive for distributions with a large size spectrum in most applications size conditioned dynamics have to be accounted for second the possibility to have separated dynamics for each size can lead to quasi monodisperse distributions which corresponds to a hard limiting case for the em algorithm so the behavior of the algorithm needs to be investigated in order to reproduce the entire moment space with a reasonable accuracy the aim of this paper is thus twofold the em and its related algorithm are enhanced by using a more accurate integration method in order to handle ndf close to the frontier of the moment space associated with an adaptive number of parameters to reconstruct the ndf accurately and efficiently as well as tabulated initial guess to optimize the computational time then a new model called csvm coupled size velocity moments model is introduced size velocity correlations are addressed either in the evaporation and drag processes or in the convective transport to reach this goal a velocity reconstruction for each size is suggested using only one additional moment per dimension and which can be directly applied to several size intervals thus this method is a direct generalization of emsm to handle the convective transport a flux splitting scheme is proposed based on the underlying kinetic description of the disperse phase comparing to existing approaches a main novelty of the csvm is that our kinetic approach ensures built in realizability conditions no additional corrections of the moments being needed at each time step the full strategy is first evaluated in 0d and 1d cases which either demonstrates the ability to reproduce both evaporation drag force and convection with size velocity correlations or the possible extension to several size intervals finally the method is applied on 2d cases with only one section showing the ability of the csvm and its related algorithms to capture the main physics of polydisperse evaporating sprays with a minimal number of moments

fourth order pseudo maximum likelihood methods we extend pml theory to account for information on the conditional moments up to order four but without assuming a parametric model to avoid a risk of misspecification of the conditional distribution the key statistical tool is the quartic exponential family which allows us to generalize the pml2 and qgpml1 methods proposed in gourieroux et al 1984 to pml4 and qgpml2 methods respectively an asymptotic theory is developed the key numerical tool that we use is the gauss freud integration scheme that solves a computational problem that has previously been raised in several fields simulation exercises demonstrate the feasibility and robustness of the methods

single site monthly streamflow simulation using entropy theory 1 using entropy theory a new method for single site monthly streamflow simulation is developed which is capable of preserving mean standard deviation skewness and lag one correlation the method entails deriving joint and conditional probability density functions using the entropy theory determining the lagrange multipliers using the information obtained from the historical record and then simulating streamflow by sequential sampling from the conditional distribution the advantage of the entropy based method is that it does not make any assumptions about the probability distribution of the streamflow data it can also preserve the cross correlation between streamflow of the last month of the previous year and the first month of the current year and avoid the generation of negative values the method can also be extended to incorporate higher order moments and more lag correlations if needed the disadvantage is that the method will be cumbersome when more statistics need to be preserved and the bimodality that may exist in the empirical probability density function cannot yet be resolved application to the colorado river basin shows that the entropy based method satisfactorily simulates monthly streamflow

hamburger moment problem and maximum entropy on the existence conditions the existence of maximum entropy solution for the reduced hamburger moment problem is reconsidered existence conditions previously appeared in literature are revisited allowing an easy way to identifying the existence of maximum entropy solution the obtained results suggest that except for special sequences of moments unknown a priori the maximum entropy solution for the non symmetric reduced hamburger moment problem exists for practical purposes the replacing of the support r with a large enough finite interval finds a theoretical warranty the symmetric case may be formulated as follows once assigned the first 2m moments if maxent density does not exist conclusion drawn uniquely from numerical evidence maxent density with the first 2m 2 moments exists in such a case even if the first 2m moments are known we have to settle for a density which carries less information theoretical results are illustrated through some numerical examples

probability density methods for smooth function approximation and learning in populations of tuned spiking neurons this article proposes a new method for interpreting computations performed by populations of spiking neurons neural firing is modeled as a rate modulated random process for which the behavior of a neuron in response to external input can be completely described by its tuning function i show that under certain conditions cells with any desired tuning functions can be approximated using only spike coincidence detectors and linear operations on the spike output of existing cells i show examples of adaptive algorithms based on only spike data that cause the underlying cell tuning curves to converge according to standard supervised and unsupervised learning algorithms unsupervised learning based on principal components analysis leads to independent cell spike trains these results suggest a duality relationship between the random discrete behavior of spiking cells and the deterministic smooth behavior of their tuning functions classical neural network approximation methods and learning algorithms based on continuous variables can thus be implemented within networks of spiking neurons without the need to make numerical estimates of the intermediate cell firing rates

critical behaviour of weakly bound systems we consider the class of three dimensional finite range or similar potentials depending on a strength constant we study the behaviour of the eigenvalue e as a function of where is the critical value at the transition from 0 1 bound state for the case we find whereas the relationship is linear for treating as a continuous parameter in the radial schrodinger equation we give the evolution of the power law between and besides spherically symmetric scalar potentials we also discuss the case of a repulsive scalar potential combined with a spin orbit component of the thomas form

a practical computational framework for the multidimensional moment constrained maximum entropy principle the maximum entropy principle is a versatile tool for evaluating smooth approximations of probability density functions with a least bias beyond given constraints in particular the moment based constraints are often a common prior information about a statistical state in various areas of science including that of a forecast ensemble or a climate in atmospheric science with that in mind here we present a unified computational framework for an arbitrary number of phase space dimensions and moment constraints for both shannon and relative entropies together with a practical usable convex optimization algorithm based on the newton method with an additional preconditioning and robust numerical integration routine this optimization algorithm has already been used in three studies of predictability and so far was found to be capable of producing reliable results in one and two dimensional phase spaces with moment constraints of up to order 4 the current work extensively references those earlier studies as practical examples of the applicability of the algorithm developed below

maximum entropy algorithm with inexact upper entropy bound based on fup basis functions with compact support the maximum entropy maxent principle is a versatile tool for statistical inference of the probability density function pdf from its moments as a least biased estimation among all other possible pdf s it maximizes shannon entropy satisfying the moment constraints thus the maxent algorithm transforms the original constrained optimization problem to the unconstrained dual optimization problem using lagrangian multipliers the classic moment problem cmp uses algebraic power moments causing typical conventional numerical methods to fail for higher order moments m 5 10 due to different sensitivities of lagrangian multipliers and unbalanced nonlinearities classic maxent algorithms overcome these difficulties by using orthogonal polynomials which enable roughly the same sensitivity for all lagrangian multipliers in this paper we employ an idea based on different principles using fup n basis functions with compact support which can exactly describe algebraic polynomials but only if the fup order n is greater than or equal to the polynomial s order our algorithm solves the cmp with respect to the moments of only low order fup 2 basis functions finding a fup 2 optimal pdf with better balanced lagrangian multipliers the algorithm is numerically very efficient due to localized properties of fup 2 basis functions implying a weaker dependence between lagrangian multipliers and faster convergence only consequences are an iterative scheme of the algorithm where power moments are a sum of fup 2 and residual moments and an inexact entropy upper bound however due to small residual moments the algorithm converges very quickly as demonstrated on two continuous pdf examples the beta distribution and a bi modal pdf and two discontinuous pdf examples the step and double dirac pdf finally these pdf examples present that fup maxent algorithm yields smaller entropy value than classic maxent algorithm but differences are very small for all practical engineering purposes

log density estimation in linear inverse problems we estimate a probability density function p which is related by a linear operator k to a density function q in sequences of regular exponential families based on a random sample from q in this paper deconvolution and positron emission tomography are considered the logarithm of the density function is approximated by basis functions consisting of singular functions of k while direct maximum likelihood or minimum kullback leibler density estimation in exponential families selects the parameters to match the moments of the basis functions to the sample moments in the inverse problem the moment of each singular function is related to a corresponding moment of the direct problem by a factor given by a singular value a of k thus an appropriate analogue of the maximum likelihood estimate is obtained by matching moments with respect to p to 1 a times the empirical moments associated with the sample from q bounds on the kullback leibler distance between the true density and the estimators are obtained and rates of convergence are established for log density functions having a measure of smoothness the density estimator converges to the unknown density in the kullback leibler sense and in the l2 sense at a rate determined not only by the order of smoothness of the log density and the dimension of data but also by the decay rate of the singular values of the operator a minimax lower bound for deconvolution is provided under certain conditions numerical examples using simulated data are provided to illustrate the finite sample performance of the proposed method for deconvolution and positron emission tomography

on the recovery of joint distributions from limited information the paper considers the role of entropy and other information theoretic concepts in the description and formation of joint distributions for m random variables the discussion includes a review of methods for constructing discrete and continuous joint distributions from the component marginal distributions we then propose a minimum cross entropy approach that recovers continuous joint distributions from the joint and marginal moments and the marginal densities the large sample properties of the associated estimator are outlined and a simple demonstration problem that highlights the advantages and drawbacks of the proposed method is presented

coarse grained computation for particle coagulation and sintering processes by linking quadrature method of moments with monte carlo the study of particle coagulation and sintering processes is important in a variety of research studies ranging from cell fusion and dust motion to aerosol formation applications these processes are traditionally simulated using either monte carlo methods or integro differential equations for particle number density functions in this paper we present a computational technique for cases where we believe that accurate closed evolution equations for a finite number of moments of the density function exist in principle but are not explicitly available the so called equation free computational framework is then employed to numerically obtain the solution of these unavailable closed moment equations by exploiting through intelligent design of computational experiments the corresponding fine scale here monte carlo simulation we illustrate the use of this method by accelerating the computation of evolving moments of uni and bivariate particle coagulation and sintering through short simulation bursts of a constant number monte carlo scheme

random walk statistics in moment based o n tight binding and applications in carbon nanotubes a computational framework for a moment based o n tight binding atomistic method is presented analyzed and applied to the problem of electronic properties of deformed carbon nanotubes where n is the number of atoms in the system the moment based approach is based on the maximum entropy and kernel polynomial methods for constructing the electronic density of states from local statistical information about the environment around individual atoms random walk statistics are formally presented as the basis for several methods to collect the moments of the density of states in a computationally efficient manner the computational complexity and accuracy of these methods are systematically analyzed using these methods for the problem of deformed carbon nanotubes it is shown that the computational cost for some cases per atom scales as efficiently as o m phantom rule 0 2em 0ex mathrm log phantom rule 0 2em 0ex m where m is the desired number of moments in the expansion of the density of states these methods are compared to other methods such as direct diagonalization and a green s function approach

maximum entropy approach to series extrapolation and analytic continuation the authors use information theory the principle of maximum entropy to develop an approach to the problem of extrapolating power series they suggest a well defined way to map the extrapolation problem onto a moment problem and show that the use of additional information about the function being extrapolated such as asymptotic behaviour for large arguments is important to obtaining accurate extrapolations they apply the method to the virial expansion for the classical hard sphere equation of state the quantum harmonic oscillator with octic perturbation and the symmetric anderson model of relevance to magnetic impurities in metals in each case the method yields excellent pointwise estimates of the function being extrapolated

entropy copula method for single site monthly streamflow simulation 1 an entropy copula method is proposed for single site monthly streamflow simulation in this method the joint distribution of adjacent monthly streamflows is constructed using the copula method whereas the marginal distribution of streamflow for each month is derived using the entropy method then the conditional distribution is derived from which monthly streamflow is generated moment statistics of monthly streamflow such as mean standard deviation and skewness can be modeled by the entropy based marginal distribution while the dependence structure between streamflows of adjacent months can be modeled by the copula based joint distribution the proposed entropy copula method can be extended by incorporating an aggregate variable for modeling the interannual dependence application to the simulation of monthly streamflow from the colorado river illustrates the effectiveness of the proposed method

on the application of maximum entropy to the moments problem the use of the maximum entropy condition to obtain a solution to the moments problem in which one seeks to reconstruct a function from the values of a finite set of its moments is of interest in a variety of physical applications unfortunately the standard analysis which is available and on which these applications are based takes as its starting point precise values of the moments whereas the physical applications involve imprecise data in this paper it will be shown how to solve the moments problem using the maximum entropy condition starting from imprecise data values with errors specified by a 2 function

construction of probability distributions in high dimension using the maximum entropy principle applications to stochastic processes random fields and random matrices the construction of probabilistic models in computational mechanics requires the effective construction of probability distributions of random variables in high dimension this paper deals with the effective construction of the probability distribution in high dimension of a vector valued random variable using the maximum entropy principle the integrals in high dimension are then calculated in constructing the stationary solution of an ito stochastic differential equation associated with its invariant measure a random generator of independent realizations is explicitly constructed in this paper three fundamental applications are presented the first one is a new formulation of the stochastic inverse problem related to the construction of the probability distribution in high dimension of an unknown non stationary random time series random accelerograms for which the velocity response spectrum is given the second one is also a new formulation related to the construction of the probability distribution of positive definite band random matrices finally we present an extension of the theory when the support of the probability distribution is not all the space but is any part of the space the third application is then a new formulation related to the construction of the probability distribution of the karhunen loeve expansion of non gaussian positive valued random fields

maximum entropy formalism and analytic extrapolation the maximum entropy principle is used to obtain a new analytic extrapolation method just complementary to the pade type method which leads to rigorous upper and lower bounds on the extrapolated function on the cut complex plane among the large class of functions that could equally well represent an analytical function in the experimental region a choice is made of the unique function that maximizes the entropy functional associated to this set of functions the result is the least biased function compatible with the actual experimental data this extrapolation method is applied to kaon nucleon experimental data in order to obtain the most reasonable values for the kny coupling constants compatible with the available experimental data and analytical constraints

the operational matrices of integration and differentiation for the fourier sine cosine and exponential series for the fourier sine cosine series basis vector varphi t and the fourier exponential series basis vector psi t a linear nonsingular transfor marion t is determined such that psi t t varphi t this result is then used to show that the operational matrices of integration p and q for varphi t and psi t respectively are related by the expression tp qt analogous results are derived for the corresponding operational matrices of differentiation d and r general expressions are derived for t p q d and r

maximum entropy autoregressive conditional heteroskedasticity model abstract in many applications it has been found that the autoregressive conditional heteroskedasticity arch model under the conditional normal or student s t distributions are not general enough to account for the excess kurtosis in the data moreover asymmetry in the financial data is rarely modeled in a systematic way in this paper we suggest a general density function based on the maximum entropy me approach that takes account of asymmetry excess kurtosis and also of high peakedness the me principle is based on the efficient use of available information and as is well known many of the standard family of distributions can be derived from the me approach we demonstrate how we can extract information functional from the data in the form of moment functions we also propose a test procedure for selecting appropriate moment functions our procedure is illustrated with an application to the nyse stock returns the empirical results reveal that the me approach with a fewer moment functions leads to a model that captures the stylized facts quite effectively

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

existence conditions and spreading properties of extreme entropy d dimensional distributions abstract the extremization of the information theoretic measures fisher information shannon entropy tsallis entropy which complementary describe the spreading of the physical states of natural systems gives rise to fundamental equations of motion and or conservation laws at times the associated extreme entropy distributions are known for some given constraints usually moments or radial expectation values in this work first we give the existence conditions of the maxent probability distributions in a d dimensional scenario where two moments not necessarily of consecutive order are known then we find general relations which involve four elements the extremized entropy the other two information theoretic measures and the variance of the extremum density in scenarios with different dimensionalities and moment constraints

maximum entropy approach to classical hard sphere and hard disk equations of state the maximum entropy procedure for extrapolating series is applied to sum the known virial series for the hard disk and hard sphere fluids this procedure is found to produce numerical results more accurate than pade approximants and it agrees with monte carlo simulations of these systems the maximum entropy results extrapolate the series smoothly to densities arbitrarily near to maximum close packing and predict no singularities whatsoever

two dimensional distribution of streamwise velocity in open channel flow using maximum entropy principle incorporation of additional constraints based on conservation laws abstract this study derived the vertical and transverse distribution of streamwise velocity in open channels using the information theoretic concept involving the maximum entropy maxent principle moment constraints expressed by the conservation of mass momentum and energy and a hypothesis invoking the connection between probability and space domains through a generalized coordinate system explicit analytical solutions were obtained for velocity profiles using a non perturbation approach along with pade approximation the convergence of the series solution was shown both theoretically and numerically the lagrange multipliers arising through the application of maxent principle were obtained by minimizing a convex potential and the quasi newton method along with bfgs scheme was applied for solving the unconstrained optimization problem for applying the conservation laws the momentum and energy coefficients were found to influence the velocity profile in a channel cross section with the expressions of these coefficients the derived velocity equations were validated for a wide range of experimental and field data

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

realizable second order finite volume schemes for the advection of moment sets of the particle size distribution the accurate description and robust simulation at relatively low cost of a size polydisperse population of fine particles in a carrier fluid is still a major challenge for many applications for this purpose moment methods derived from a population balance equation represent a very interesting strategy however one of the major issues of such methods is the realizability the numerical schemes have to ensure that the moment sets stay realizable i e that an underlying distribution exists this issue is all the more crucial that some moment vectors can be at the boundary of the moment space for practical applications corresponding to a population of particles with only one or a few sizes it is then investigated here for the advection operator for which it is particularly significant then second order realizable kinetic finite volume schemes are designed with two strategies for the fluxes evaluation based on the work of kah et al 1 and of vikas et al 2 which are here completely revisited extended to take into account the boundary of the moment space and any number of moments analyzed and compared in a cartesian mesh context for a potential easiest generalization to unstructured meshes simplified but still realizable versions of these schemes are also developed the high accuracy of all the schemes is then numerically checked on 1d and 2d test cases with cartesian meshes and their robustness is shown even when some moment vectors are at the boundary of the moment space

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

maximum entropy approximation for lyapunov exponents of chaotic maps let s 0 1 0 1 be a nonsingular chaotic map that preserves an integrable density f that describes the statistics of the orbits in this article we use the maximum entropy approach to approximate the density f and the corresponding lyapunov exponent

calculation of maximum entropy densities with application to income distribution the maximum entropy approach is a flexible and powerful tool for density approximation this paper proposes a sequential updating method to calculate the maximum entropy density subject to known moment constraints instead of imposing the moment constraints simultaneously the sequential updating method incorporates the moment constraints into the calculation from lower to higher moments and updates the density estimates sequentially the proposed method is employed to approximate the size distribution of u s family income empirical evidence demonstrates the efficiency of this method

information theoretic measures of the income distribution in food demand a new method to nest estimate and test the rank and functional form of the income terms in an incomplete system of demand equations is developed information theory is employed to infer the u s income distribution from data on quintile and top five percentile income ranges and intra quintile and top five percentile mean incomes maximum entropy income distributions are combined with data on the u s demand for 21 food items to estimate u s food demand over the period 1919 1995 excluding 1942 1946

efficient numerical approximation of maximum entropy estimates a class of algorithms for approximation of the maximum entropy estimate of probability density functions on the basis of a finite number of sampled data is introduced the algorithms are presented as a finite sequence in order of increasing accuracy and decreasing computational efficiency in which the last element of the sequence is the exact maximum entropy estimate numerical and applicative examples are reported

observer participant models of neural processing a model is proposed in which the neuron serves as an information channel channel distortion occurs through the channel since the mapping from input boolean codes to output codes are many to one in that neuron outputs consist of just two distinguished states within the described model the neuron performs a decision making function decisions are made regarding the validity of a question passively posed by the neuron this question becomes defined through learning hence learning is viewed as the process of determining an appropriate question based on supplied input ensembles an application of the shannon information measures of entropy and mutual information taken together in the context of the proposed model lead to the hopfield neuron model with conditionalized hebbian learning rules neural decisions are shown to be based on a sigmoidal transfer characteristic or in the limit as computational temperature tends to zero a maximum likelihood decision rule the described work is contrasted with the information theoretic approach of linsker

exponential functionals and means of neutral to the right priors summary the mean of a random distribution chosen from a neutral to the right prior can be represented as the exponential functional of an increasing additive process this fact is exploited in order to give sufficient conditions for the existence of the mean of a neutralto the right prior and for the absolute continuity of its probability distribution moreover expressions for its moments of any order are provided for illustrative purposes we consider a generalisation of the neutral to the right prior based on the gamma process and the beta stacy process finally by resorting to the maximum entropy algorithm we obtain an approximation to the probability density function of the mean of a neutral tothe right prior the arguments are easily extended to examine means of posterior quantities the numerical results obtained are compared to those yielded by the application of some well established simulation algorithms

discretizing distributions with exact moments error estimate and convergence analysis the maximum entropy principle is a powerful tool for solving underdetermined inverse problems this paper considers the problem of discretizing a continuous distribution which arises in various applied fields we obtain the approximating distribution by minimizing the kullback leibler information relative entropy of the unknown discrete distribution relative to an initial discretization based on a quadrature formula subject to some moment constraints we study the theoretical error bound and the convergence of this approximation method as the number of discrete points increases we prove that i the theoretical error bound of the approximate expectation of any bounded continuous function has at most the same order as the quadrature formula we start with and ii the approximate discrete distribution weakly converges to the given continuous distribution moreover we present some numerical examples that show the advantage of the method and apply to numerically solving an optimal portfolio problem

entropic quadrature for moment approximations of the boltzmann bgk equation abstract using the method of moments to approximate the solution of kinetic equations has become a standard technique especially in the context of solving the boltzmann equation to model rarefied gases one attractive example is the maximum entropy closure which however is computationally barely affordable in this paper we will combine the maximum entropy approach with the quadrature method of moments qmom two methods which have in some sense diametric properties thereby introducing the entropic quadrature eq closure eq calculates a sparse quadrature based reconstruction of the unknown velocity distribution like the qmom where the physical meaningful maximum entropy principle is employed as a criterion for selecting a quadrature among different quadratures fitting the given moments we will discuss the construction of the closure and give several numerical examples of its performance in one and two dimensions

higher order moments and the maximum entropy inference the thermodynamic limit approach according to jaynes 1963 statistical mechanics can be interpreted as a special type of statistical inference based on the principle of maximum entropy the result of such an inference depends on the available information about a given physical system but the principle itself does not decide what kind of information is essential and what is not the well known gibbs canonical state results from the principle when the statistical mean value of energy is supposed to be known the maximum entropy inference employing higher order moments of energy is examined from the point of view of the thermodynamic limit it is shown that the contribution to the total entropy arising from the extra information corresponding to the higher order moments is o n when n to infinity n v constant n is the number of particles and v is the volume consequently from a purely thermodynamic point of view this extra information is non essential and can be neglected in the maximum entropy procedure these conclusions also characterise the maximum entropy inference employing higher order moments of almost arbitrary extensive quantities one can say that the maximum entropy inference has a certain stability property with respect to information corresponding to higher order moments of extensive physical quantities

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

first order continuous and discontinuous galerkin moment models for a linear kinetic equation model derivation and realizability theory abstract we provide two new classes of moment models for linear kinetic equations in slab and three dimensional geometry they are based on classical finite elements and low order discontinuous galerkin approximations on the unit sphere we investigate their realizability conditions and other basic properties numerical tests show that these models are more efficient than classical full moment models in a space homogeneous test when the analytical solution is not smooth

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

bertlmann martin inequality for a weakly bound particle the bertlmann martin bm inequality for the rms radius is checked against the exact value in the case of a particle in a weak central potential having a single bound state a number of usual potentials are considered in one and three dimensions the bm inequality is poorly saturated but the correction factor needed to recover the exact rms radius is weakly depending on the shape of the potential except for the hulth n potential for potentials having a hard core component we found that which depends on the bound state energy tends to a universal curve as the hard core radius increases

high order moment model for polydisperse evaporating sprays towards interfacial geometry description in this paper we propose a new eulerian modeling and related accurate and robust numerical methods describing polydisperse evaporating sprays based on high order moment methods in size the main novelty of this model is its capacity to describe some geometrical variables of the droplet gas interface by analogy with the liquid gas interface in interfacial flows for this purpose we use fractional size moments where the size variable is taken as the droplet surface in order to evaluate the evaporation of the polydisperse spray we use a smooth reconstruction which maximizes the shannon entropy however the use of fractional moments introduces some theoretical and numerical difficulties which need to be tackled first relying on a study of the moment space we extend the maximum entropy me reconstruction of the size distribution to the case of fractional moments then we propose a new accurate and realizable algorithm to solve the moment evolution due to evaporation which preserves the structure of the moment space this algorithm is based on a mathematical analysis of the kinetic evolution due to evaporation where it shown that the evolution of some negative order fractional moments have to be properly predicted a peculiarity related to the use of fractional moments the present model and numerical schemes yield an accurate and stable evaluation of the moment dynamics with minimal number of variables as well as a minimal computational cost as with the emsm model but with the very interesting additional capacity of coupling with diffuse interface model and transport equation of averaged geometrical interface variables which are essential in oder to describe atomization

the maximum entropy method applied to stationary density computation abstract the maximum entropy method maxent is widely used in the context of the moment problem which appears naturally in many branches of physics and engineering it is used to numerically recover the density with least bias from finitely many known moments we introduce the basic idea behind this method and apply this method to approximating fixed densities of markov operators in stochastic analysis and frobenius perron operators in ergodic theory of chaotic dynamics

quantifying predictability through information theory small sample estimation in a non gaussian framework many situations in complex systems require quantitative estimates of the lack of information in one probability distribution relative to another in short term climate and weather prediction examples of these issues might involve the lack of information in the historical climate record compared with an ensemble prediction or the lack of information in a particular gaussian ensemble prediction strategy involving the first and second moments compared with the non gaussian ensemble itself the relative entropy is a natural way to quantify the predictive utility in this information and recently a systematic computationally feasible hierarchical framework has been developed in practical systems with many degrees of freedom computational overhead limits ensemble predictions to relatively small sample sizes here the notion of predictive utility in a relative entropy framework is extended to small random samples by the definition of a sample utility a measure of the unlikeliness that a random sample was produced by a given prediction strategy the sample utility is the minimum predictability with a statistical level of confidence which is implied by the data two practical algorithms for measuring such a sample utility are developed here the first technique is based on the statistical method of null hypothesis testing while the second is based upon a central limit theorem for the relative entropy of moment based probability densities these techniques are tested on known probability densities with parameterized bimodality and skewness and then applied to the lorenz 96 model a recently developed toy climate model with chaotic dynamics mimicking the atmosphere the results show a detection of non gaussian tendencies of prediction densities at small ensemble sizes with between 50 and 100 members with a 95 confidence level

entropy densities with an application to autoregressive conditional skewness and kurtosis abstract the entropy principle yields for a given set of moments a density that involves the smallest amount of prior information we first show how entropy densities may be constructed in a numerically efficient way as the minimization of a potential next for the case where the first four moments are given we characterize the skewness kurtosis domain for which densities are defined this domain is found to be much larger than for hermite or edgeworth expansions last we show how this technique can be used to estimate a garch model where skewness and kurtosis are time varying we find that there is little predictability of skewness and kurtosis for weekly data

treatment of uncertain material interfaces in compressible flows abstract to treat uncertain interface position is an important issue for complex applications in this paper we address the characterization of randomly perturbed interfaces between fluids thanks to stochastic modeling and uncertainty quantification through the 2d euler system the perturbed interface is modeled as a random field and represented by a karhunen loeve expansion the stochastic 2d euler system is solved applying polynomial chaos theory through the intrusive polynomial moment method ipmm this stochastic resolution method is fully explained and studied theoretically and numerically stochastic richtmyer meshkov unstable flows are solved and presented for several configurations of the uncertain interface different rugosities between the fluids the probability density functions of the mass density of the fluid in the vicinity of the interface are computed built and compared for the different simulations the system exhibits strong sensitivity with respect to the stochastic initially leading modes