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

a family of fifth order runge kutta pairs the construction of a runge kutta pair of order 5 4 with the minimal number of stages requires the solution of a nonlinear system of 25 order conditions in 27 unknowns we define a new family of pairs which includes pairs using 6 function evaluations per integration step as well as pairs which additionally use the first function evaluation from the next step this is achieved by making use of kutta s simplifying assumption on the original system of the order conditions i e that all the internal nodes of a method contributing to the estimation of the endpoint solution provide at these nodes cost free second order approximations to the true solution of any differential equation in both cases the solution of the resulting system of nonlinear equations is completely classified and described in terms of five free parameters optimal runge kutta pairs with respect to minimized truncation error coefficients maximal phase lag order and various stability characteristics are presented these pairs were selected under the assumption that they are used in local extrapolation mode the propagated solution of a problem is the one provided by the fifth order formula of the pair numerical results obtained by testing the new pairs over a standard set of test problems suggest a significant improvement in efficiency when using a specific pair of the new family with minimized truncation error coefficients instead of some other existing pairs

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 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

on anomalous diffusion and the fractional generalized langevin equation for a harmonic oscillator the fractional generalized langevin equation fgle is proposed to discuss the anomalous diffusive behavior of a harmonic oscillator driven by a two parameter mittag leffler noise the solution of this fgle is discussed by means of the laplace transform methodology and the kernels are presented in terms of the three parameter mittag leffler functions recent results associated with a generalized langevin equation are recovered

time fractional wave diffusion equation in an inhomogeneous half space we consider the fundamental solution of the time fractional wave diffusion equation in a three dimensional half space medium which contains an inhomogeneity in form of a plane parallel layer the corresponding green s function which is derived by means of the fourier and laplace transforms can be accurately and efficiently evaluated without recourse to the mittag leffler or the fox h function moreover it is shown that in the one dimensional case the fundamental solution in an inhomogeneous half space is no longer a probability density function in addition we consider the advection equation for the fractional laplacian and the caputo time fractional derivative of orders on a bounded domain simple algorithms for accurate evaluation of the m wright function and the mittag leffler function are enclosed at the end of this article

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

similarity solution to fractional nonlinear space time diffusion wave equation in this article the so called fractional nonlinear space time wave diffusion equation is presented and discussed this equation is solved by the similarity method using fractional derivatives in the caputo riesz feller and riesz senses some particular cases are presented and the corresponding solutions are shown by means of 2 d and 3 d plots

characterizations and simulations of a class of stochastic processes to model anomalous diffusion in this paper we study a parametric class of stochastic processes to model both fast and slow anomalous diffusions this class called generalized grey brownian motion ggbm is made up of self similar with stationary increments processes h sssi and depends on two real parameters 0 2 and 0 1 it includes fractional brownian motion when 0 2 and 1 and time fractional diffusion stochastic processes when 0 1 the latter have a marginal probability density function governed by timefractional diffusion equations of order the ggbm is defined through the explicit construction of the underlying probability space however in this paper we show that it is possible to define it in an unspecified probability space for this purpose we write down explicitly all the finite dimensional probability density functions moreover we provide different ggbm characterizations the role of them wright function which is related to the fundamental solution of the time fractional diffusion equation emerges as a natural generalization of the gaussian distribution furthermore we show that the ggbm can be represented in terms of the product of a random variable which is related to the m wright function and an independent fractional brownian motion this representation highlights the h sssi nature of the ggbm and provides a way to study and simulate the trajectories for this purpose we developed a random walk model based on a finite difference approximation of a partial integro differential equation of a fractional type

differentiation to fractional orders and the fractional telegraph equation using methods of differential and integral calculus we present and discuss the calculation of a fractional green function associated with the one dimensional case of the so called general fractional telegraph equation with one space variable this is a fractional partial differential equation with constant coefficients the equation is solved by means of juxtaposition of transforms i e we introduce the laplace transform in the time variable and the fourier transform in the space variable several particular cases are discussed in terms of the parameters some known results are recovered as a by product of our main result we obtain two new relations involving the two parameter mittag leffler function

the multidimensional moment constrained maximum entropy problem a bfgs algorithm with constraint scaling in a recent paper we developed a new algorithm for the moment constrained maximum entropy problem in a multidimensional setting using a multidimensional orthogonal polynomial basis in the dual space of lagrange multipliers to achieve numerical stability and rapid convergence of the newton iterations here we introduce two new improvements for the existing algorithm adding significant computational speedup in situations with many moment constraints where the original algorithm is known to converge slowly the first improvement is the use of the bfgs iterations to progress between successive polynomial reorthogonalizations rather than single newton steps typically reducing the total number of computationally expensive polynomial reorthogonalizations for the same maximum entropy problem the second improvement is a constraint rescaling aimed to reduce relative difference in the order of magnitude between different moment constraints improving numerical stability of iterations due to reduced sensitivity of different constraints to changes in lagrange multipliers we observe that these two improvements can yield an average wall clock time speedup of 5 6 times compared to the original algorithm

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

response characteristics of a fractional oscillator abstract the integral equation of motion of a driven fractional oscillator is obtained by generalizing the corresponding equation of motion of a driven harmonic oscillator to include integrals of arbitrary order according to the methods of fractional calculus the green s function solution for the fractional oscillator is obtained in terms of mittag leffler functions using laplace transforms the response and resonance characteristics of the fractional oscillator are studied for several cases of forcing function

chaotic systems and maximum entropy formalism we show how the maximum entropy formalism can be applied in nonlinear chaotic systems in particular we show how one can use a few moments of the time evolution of a dynamical variable to infer the probability density and hence the lyapunov exponent

on some fractional green s functions in this paper we discuss some fractional green s functions associated with the fractional differential equations which appear in several fields of science more precisely the so called wave reaction diffusion equation and some of its particular cases the methodology presented is the juxtaposition of integral transforms in particular the laplace and the fourier integral transforms some recent results involving the reaction diffusion equation are pointed out

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

adaptive change of basis in entropy based moment closures for linear kinetic equations entropy based m n moment closures for kinetic equations are defined by a constrained optimization problem that must be solved at every point in a space time mesh making it important to solve these optimization problems accurately and efficiently we present a complete and practical numerical algorithm for solving the dual problem in one dimensional slab geometries the closure is only well defined on the set of moments that are realizable from a positive underlying distribution and as the boundary of the realizable set is approached the dual problem becomes increasingly difficult to solve due to ill conditioning of the hessian matrix to improve the condition number of the hessian we advocate the use of a change of polynomial basis defined using a cholesky factorization of the hessian that permits solution of problems nearer to the boundary of the realizable set we also advocate a fixed quadrature scheme rather than adaptive quadrature since the latter introduces unnecessary expense and changes the computationally realizable set as the quadrature changes for very ill conditioned problems we use regularization to make the optimization algorithm robust we design a manufactured solution and demonstrate that the adaptive basis optimization algorithm reduces the need for regularization this is important since we also show that regularization slows and even stalls convergence of the numerical simulation when refining the space time mesh we also simulate two well known benchmark problems there we find that our adaptive basis fixed quadrature algorithm uses less regularization than alternatives although differences in the resulting numerical simulations are more sensitive to the regularization strategy than to the choice of basis

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

fractional wave diffusion equation with periodic conditions we study a time space fractional wave diffusion equation with periodic conditions using laplace transforms and fourier series and presenting its solution in terms of three parameter mittag leffler functions as a particular case we recover a recent result we also present some graphics associated with particular values of the parameters

maximum entropy approach for linear scaling in the electronic structure problem we present a method for obtaining the band structure energy and density of states for large sparse hamiltonian matrices as occur in electronic structure problems the computation scales linearly with the dimension of the matrix we use a statistical approach using random vectors along with the principle of maximum entropy to obtain highly accurate estimates for integrals over the density of states we offer several tests of the approach using tight binding hamiltonians and apply the scheme to large icosahedral fullerenes with 720 and 2160 atoms

a general family of explicit runge kutta pairs of orders 6 5 explicit runge kutta formula pairs of different orders of accuracy form a class of efficient algorithms for treating nonstiff ordinary differential equations so far several runge kutta pairs of order 6 5 have appeared in the literature these pairs use 8 function evaluations per step and belong to certain families of solutions of a set of 54 nonlinear algebraic equations in 44 or 45 coefficients depending on the use of the fsal first stage as last device these equations form a set of necessary and sufficient conditions that a 6 5 runge kutta pair must satisfy the solution of the latter is achieved by employing various types of simplifying assumptions in this paper we make use of the fact that all these families of pairs satisfy a common set of simplifying assumptions using only these simplifying assumptions we define a new family of 6 5 runge kutta pairs its main characteristic which is also a property that no other known family shares is that all of its nodes except the last one which eq

time and space fractional partial differential equations the fundamental solution for time and space fractional partial differential operator dt a2 2 0 is given in terms of the fox s h function here the time fractional derivative in the sense of generalized functions distributions dt is defined by the convolution dt f t t f t where t t 1 and f t 0 as t 0 and the fractional n dimensional laplace operator 2 is defined by its fourier transform with respect to spatial variable f 2g x f g x the solutions for initial value problems for time and space fractional partial differential equation in the sense of caputo and riemann liouville time fractional derivatives respectively are obtained by the fundamental solution

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

the fractional schrodinger equation for delta potentials the fractional schrodinger equation is solved for the delta potential and the double delta potential for all energies the solutions are given in terms of fox s h function

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 and the problem of moments a stable algorithm we present a technique for entropy optimization to calculate a distribution from its moments the technique is based upon maximizing a discretized form of the shannon entropy functional by mapping the problem onto a dual space where an optimal solution can be constructed iteratively we demonstrate the performance and stability of our algorithm with several tests on numerically difficult functions we then consider an electronic structure application the electronic density of states of amorphous silica and study the convergence of the fermi level with increasing number of moments

significance of higher moments for complete characterization of the travel time probability density function in heterogeneous porous media using the maximum entropy principle 1 the travel time formulation of advective transport in heterogeneous porous media is of interest both conceptually e g for incorporating retention processes and in applications where typically the travel time peak early and late arrivals of contaminants are of major concern in a regulatory or remediation context furthermore the travel time moments are of interest for quantifying uncertainty in advective transport of tracers released from point sources in heterogeneous aquifers in view of this interest the travel time distribution has been studied in the literature however the link to the hydraulic conductivity statistics has been typically restricted to the first two moments here we investigate the influence of higher travel time moments on the travel time probability density function pdf in heterogeneous porous media combining monte carlo simulations with the maximum entropy principle the monte carlo experimental pdf is obtained by the adaptive fup monte carlo method afmcm for advective transport characterized by a multi gaussian structure with exponential covariance considering two injection modes in flux and resident and lnk variance up to 8 a maximum entropy maxent algorithm based on fup basis functions is used for the complete characterization of the travel time pdf all travel time moments become linear with distance initial nonlinearity is found mainly for the resident injection mode which exhibits a strong nonlinearity within first 5iy for high heterogeneity for the resident injection mode the form of variance and all higher moments changes from the familiar concave form predicted by the first order theory to a convex form for the in flux mode linearity is preserved even for high heterogeneity the number of moments sufficient for a complete characterization of the travel time pdf mainly depends on the heterogeneity level mean and variance completely describe travel time pdf for low and mild heterogeneity skewness is dominant for lnk variance around 4 while kurtosis and fifth moment are required for lnk variance higher than 4 including skewness seems sufficient for describing the peak and late arrivals linearity of travel time moments enables the prediction of asymptotic behavior of the travel time pdf which in the limit converges to a symmetric distribution and fickian transport however higher order travel time moments may be important for most practical purposes and in particular for advective transport in highly heterogeneous porous media for a long distance from the source

homogeneous fractional embeddings fractional equations appear in the description of the dynamics of various physical systems for lagrangian systems the embedding theory developed by cresson fractional embedding of differential operators and lagrangian systems j math phys 48 033504 2007 provides a univocal way to obtain such equations stemming from a least action principle however no matter how equations are obtained the dimension of the fractional derivative differs from the classical one and may induce problems of temporal homogeneity in fractional objects in this paper we show that it is necessary to introduce an extrinsic constant of time then we use it to construct two equivalent fractional embeddings which retains homogeneity the notion of fractional constant is also discussed through this formalism finally an illustration is given with natural lagrangian systems and the case of the harmonic oscillator is entirely treated

maximum entropy summation of divergent perturbation series in this paper the principle of maximum entropy is used to predict the sum of a divergent perturbation series from the first few expansion coefficients the perturbation expansion for the ground state energy e g of the octic oscillator defined by h p2 2 x2 2 gx8 is a series of the form e g 1 2 1 n 1 angn this series is terribly divergent because for large n the perturbation coefficients an grow like 3n this growth is so rapid that the solution to the moment problem is not unique and ordinary pade summation of the divergent series fails a completely different kind of procedure based on the principle of maximum entropy for reconstructing the function e g from its perturbation coefficients is presented very good numerical results are obtained

solution of the fractional langevin equation and the mittag leffler functions we introduce the fractional generalized langevin equation in the absence of a deterministic field with two deterministic conditions for a particle with unitary mass i e an initial condition and an initial velocity are considered for a particular correlation function that characterizes the physical process and using the methodology of the laplace transform we obtain the solution in terms of the three parameter mittag leffler function as particular cases some recent results are also presented

fractional variational calculus in terms of riesz fractional derivatives this paper presents extensions of traditional calculus of variations for systems containing riesz fractional derivatives rfds specifically we present generalized euler lagrange equations and the transversality conditions for fractional variational problems fvps defined in terms of rfds we consider two problems a simple fvp and an fvp of lagrange results of the first problem are extended to problems containing multiple fractional derivatives functions and parameters and to unspecified boundary conditions for the second problem we present lagrange type multiplier rules for both problems we develop the euler lagrange type necessary conditions which must be satisfied for the given functional to be extremum problems are considered to demonstrate applications of the formulations explicitly we introduce fractional momenta fractional hamiltonian fractional hamilton equations of motion fractional field theory and fractional optimal control the formulations presented and the resulting equations are similar to the formulations for fvps given in agrawal 2002 j math anal appl 272 368 2006 j phys a math gen 39 10375 and to those that appear in the field of classical calculus of variations these formulations are simple and can be extended to other problems in the field of fractional calculus of variations

some physical applications of fractional schrodinger equation the fractional schrodinger equation is solved for a free particle and for an infinite square potential well the fundamental solution of the cauchy problem for a free particle the energy levels and the normalized wave functions of a particle in a potential well are obtained in the barrier penetration problem the reflection coefficient and transmission coefficient of a particle from a rectangular potential wall is determined in the quantum scattering problem according to the fractional schrodinger equation the green s function of the lippmann schwinger integral equation is given

calculation of densities of states and spectral functions by chebyshev recursion and maximum entropy we present an efficient algorithm for calculating spectral properties of large sparse hamiltonian matrices such as densities of states and spectral functions the combination of chebyshev recursion and maximum entropy achieves high energy resolution without significant roundoff error machine precision or numerical instability limitations if controlled statistical or systematic errors are acceptable cpu and memory requirements scale linearly in the number of states the inference of spectral properties from moments is much better conditioned for chebyshev moments than for power moments we adapt concepts from the kernel polynomial method a linear chebyshev approximation with optimized gibbs damping to control the accuracy of fourier integrals of positive nonanalytic functions we compare the performance of kernel polynomial and maximum entropy algorithms for an electronic structure example

fundamental solution of the tempered fractional diffusion equation in this paper we consider the space time fractional diffusion equation dt u x t k dx u x t 0 x r t 0 with the tempered riemann liouville derivative of order 0 1 in space and the caputo derivative of order 0 1 in time the fundamental solution which turns out to be a spatial probability density function is given in computable series form as well as in integral representation the spatial moments of the probability density function are determined explicitly for an arbitrary order n 0 moreover green s function of the untempered neutral fractional diffusion equation is analyzed in view of absolute and relative extreme points at the end of this article we point out a remarkably and important integral representation for accurate evaluation of the m wright mainardi function m x of order 0 1 and arguments x r0