Stieltjes moment problem and fractional moments

Stieltjes moment problem is considered and a solution, consisting of the use of fractional moments, is proposed. More precisely, a determinate Stieltjes moment problem, whose corresponding Hamburger moment problem is determinate too, is investigated in the setup of Maximum Entropy. Condition number in entropy calculation is provided endowing both Stieltjes moment problem existence conditions and Hamburger moment problem determinacy conditions by a geometric meaning. Then the resorting to fractional moments is considered; numerical aspects are investigated and a stable algorithm for calculating fractional moments from integer moments is proposed.     

TWO-VARIABLE JACOBI POLYNOMIALS FOR SOLVING SOME FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS

Two-variable Jacobi polynomials, as a two-dimensional basis, are applied to solve a class of temporal fractional partial differential equations. The fractional derivative operators are in the Caputo sense. The operational matrices of the integration of integer and fractional orders are presented. Using these matrices together with the Tau Jacobi method converts the main problem into the corresponding system of algebraic equations. An error bound is obtained in a two-dimensional Jacobi-weighted Sobolev space. Finally, the efficiency of the proposed method is demonstrated by implementing the algorithm to several illustrative examples. Results will be compared with those obtained from some existing methods.     

On the solution of the finite moment problem

In this work we propose a method to obtain the normal solution of the finite moment problem both in the absence and in the presence of linear boundary constraints. The method gives the normal solution as a linear combination of Jacobi polynomials and furnishes its coefficients in terms of the moments. A number of examples are given to illustrate the strength of the method.     

An effective approach for probabilistic lifetime modelling based on the principle of maximum entropy with fractional moments

The paper presents a fractional moment method for probabilistic lifetime modelling of uncertain engineering systems. A novel feature of the method is the use of fractional moments, as opposed to integer moments commonly used so far in the structural reliability literature. The fractional moments are calculated from a small, simulated sample of remaining useful life of the system. And the fractional exponents that are used to model the system lifetime distribution are determined through the entropy maximization process, rather than assigned by an analyst in prior. Together with the theory of copula, the efficiency and accuracy of the proposed method are illustrated by the probabilistic lifetime modelling of several dynamical and discontinuous stochastic systems.     

The discrete moment problem with fractional moments

Discrete moment problems (DMP) with integer moments were first introduced by Prékopa to provide sharp lower and upper bounds for functions of discrete random variables. Prékopa also developed fast and stable dual type linear programming methods for the numerical solutions of the problem. In this paper, we assume that some fractional moments are also available and propose basic theory and a solution method for the bounding problems. Numerical experiments show significant improvement in the tightness of the bounds.     

Local moment problem

The work is devoted to the local moment problem, which consists in finding of non-decreasing functions on the real axis having given first 2n + 1, n ≥ 0, power moments on the whole axis and also 2m + 1 first power moments on a certain finite axis interval. Considering the local moment problem as a combination of the Hausdorff and Hamburger truncated moment problems we obtain the conditions of its solvability and describe the class of its solutions with minimal number of growth points if the problem is solvable. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Existence and stability of a discrete probability distribution by maximum entropy approach

The discrete maximum entropy (ME) probability distribution which can take on a finite number of values and whose first moments are assigned, is considered. The necessary and sufficient conditions for the existence of a maximum entropy solution are identical to the general ones for the finite moment problem. The entropy decreasing by adding one more moment is studied. Unstability of the distribution recovering is proved when an increasing number of moments is used.     

Multigrid algorithms for a vertex–centered covolume method for elliptic problems

Summary. We analyze V–cycle multigrid algorithms for a class of perturbed problems whose perturbation in the bilinear form preserves the convergence properties of the multigrid algorithm of the original problem. As an application, we study the convergence of multigrid algorithms for a covolume method or a vertex–centered finite volume element method for variable coefficient elliptic problems on polygonal domains. As in standard finite element methods, the V–cycle algorithm with one pre-smoothing converges with a rate independent of the number of levels. Various types of smoothers including point or line Jacobi, and Gauss-Seidel relaxation are considered. Received August 19, 1999 / Revised version received July 10, 2000 / Published online June 7, 2001     

A fractional‐order Jacobi Tau method for a class of time‐fractional PDEs with variable coefficients

This paper presents a shifted fractional‐order Jacobi orthogonal function (SFJF) based on the definition of the classical Jacobi polynomial. A new fractional integral operational matrix of the SFJF is presented and derived. We propose the spectral Tau method, in conjunction with the operational matrices of the Riemann–Liouville fractional integral for SFJF and derivative for Jacobi polynomial, to solve a class of time‐fractional partial differential equations with variable coefficients. In this algorithm, the approximate solution is expanded by means of both SFJFs for temporal discretization and Jacobi polynomials for spatial discretization. The proposed tau scheme, both in temporal and spatial discretizations, successfully reduced such problem into a system of algebraic equations, which is far easier to be solved. Numerical results are provided to demonstrate the high accuracy and superiority of the proposed algorithm over existing ones. Copyright © 2015 John Wiley & Sons, Ltd.     

A general finite element formulation for fractional variational problems

This paper presents a general finite element formulation for a class of Fractional Variational Problems (FVPs). The fractional derivative is defined in the Riemann-Liouville sense. For FVPs the Euler-Lagrange and the transversality conditions are developed. In the Fractional Finite Element Formulation (FFEF) presented here, the domain of the equations is divided into several elements, and the functional is approximated in terms of nodal variables. Minimization of this functional leads to a set of algebraic equations which are solved using a numerical scheme. Three examples are considered to show the performance of the algorithm. Results show that as the number of discretization is increased, the numerical solutions approach the analytical solutions, and as the order of the derivative approaches an integer value, the solution for the integer order system is recovered. For unspecified boundary conditions, the numerical solutions satisfy the transversality conditions. This indicates that for the class of problems considered, the numerical solutions can be obtained directly from the functional, and there is no need to solve the fractional Euler-Lagrange equations. Thus, the formulation extends the traditional finite element approach to FVPs.     

Subcritical catalytic branching random walk with finite or infinite variance of offspring number

Subcritical catalytic branching random walk on the d-dimensional integer lattice is studied. New theorems concerning the asymptotic behavior of distributions of local particle numbers are established. To prove the results, different approaches are used, including the connection between fractional moments of random variables and fractional derivatives of their Laplace transforms. In the previous papers on this subject only supercritical and critical regimes were investigated under the assumptions of finiteness of the first moment of offspring number and finiteness of the variance of offspring number, respectively. In the present paper, for the offspring number in the subcritical regime, the finiteness of the moment of order 1 + δ is required where δ is some positive number.     

Parametric integer linear programming: A synthesis of branch and bound with cutting planes

A branch and bound algorithm is designed to solve the general integer linear programming problem with parametric right-hand sides. The right-hand sides have the form b + θd where b and d are comformable vectors, d consists of nonnegative constants, and θ varies from zero to one.The method consists of first determining all possible right-hand side integer constants and appending this set of integer constants to the initial tableau to form an expanded problem with a finite number of family members. The implicit enumeration method gives a lower bound on the integer solutions. The branch and bound method is used with fathoming tests which allow one family member possibly to fathom other family members. A cutting plane option applies a finite number of cuts to each node before branching. In addition, the cutting plane method is invoked whenever some members are feasible at a node and others are infeasible. The branching and cutting process is repeated until the entire family of problem has been solved.     

Power moments of negative order for the principal spectral function of a string

We established necessary and sufficient conditions for the existence of finite power moments of all integer negative orders for the principal spectral function of a string. The necesity of this problem is explained by its relation to the so-called strong Stieltjes moment problem. Odessa Academy of Food Technologies, Odessa. Translated from Ukrainskii Matematischeskii Zhurnal, Vol. 48, No. 9, pp. 1209–1222, September, 1996.     

Fractional knapsack problems

The fractional knapsack problem to obtain an integer solution that maximizes a linear fractional objective function under the constraint of one linear inequality is considered. A modification of the Dinkelbach's algorithm [3] is proposed to exploit the fact that good feasible solutions are easily obtained for both the fractional knapsack problem and the ordinary knapsack problem. An upper bound of the number of iterations is derived. In particular it is clarified how optimal solutions depend on the right hand side of the constraint; a fractional knapsack problem reduces to an ordinary knapsack problem if the right hand side exceeds a certain bound.     

Hausdorff moment problem and Maximum Entropy: On the existence conditions

Different existence conditions of the Maximum Entropy solution to finite Hausdorff moment problem have been formulated in literature. Through a counterexample we prove that the most cited one is uncorrect. We do not bound ourselves to a crude counterexample, as we think that a detailed explanation is of interest by itself. It clarifies the difference existing between the finite and infinite Hausdorff moment problem existence conditions.     

Hausdorff moment problem and fractional moments: A simplified procedure

The purpose of this paper is the recovering of a probability density function with support [0, 1] from the knowledge of its sequence of moments, i.e. the classical Hausdorff moment problem. To avoid the well-known ill-conditioning, firstly the moment curve is calculated from the assigned sequence of moments; next the unknown density is approximated by Maximum Entropy (MaxEnt) technique selecting some proper points on the moment curve. Exploiting convergence in entropy, a simplified quick procedure is suggested to recover the approximate density. An application to Laplace Transform inversion is illustrated.     

The modified moment problem in the presence of noise

The modified moment problem is little studied in the literature with respect to the classical moment problem due to the lacking of experiments measuring modified moments. The modified moment problem is analyzed in this paper, when noise affects the modified moments themselves. A numerical method for solving the problem, based on regularization, is given, together with a full theoretical analysis, convergence results and an optimized algorithm. The modified moment problem reveals strongly superior to the classical moment problem in terms of the amplification of the error, conditioning of the matrices involved and ease of computation.     

Optimal control problems for linear fractional-order systems defined by equations with Hadamard derivative

Two optimal control problems are studied for linear stationary systems of fractional order with lumped variables whose dynamics is described by equations with Hadamard derivative, a minimum-norm control problem and a time-optimal problem with a constraint on the norm of the control. The setting of the problem with nonlocal initial conditions is considered. Admissible controls are sought in the class of functions p-integrable on an interval for some p. The main approach to the study is based on the moment method. The well-posedness and solvability of the moment problem are substantiated. For several special cases, the optimal control problems under consideration are solved analytically. An analogy between the obtained results and known results for systems of integer and fractional order described by equations with Caputo and Riemann–Liouville derivatives is specified.

     

Some remarks on the moment problem and its relation to the theory of extrapolation of spaces

It is well known that that the coincidence of integer moments (nth-power moments, where n is an integer) of two nonnegative random variables does not imply the coincidence of their distributions. Moreover, we show that, given coinciding integer moments, the ratio of half-integer moments may tend to infinity arbitrarily fast. Also, in this paper, we give a new proof of uniqueness in the continuous moment problem and show that, in that problem, it is impossible to replace the condition of coincidence of all moments by a two-sided inequality between them, while preserving the inequality between the distributions. In conclusion, we study the relationship with the theory of extrapolation of spaces.     

The truncated Stieltjes moment problem solved by using kernel density functions

In this work the problem of the approximate numerical determination of a semi-infinite supported, continuous probability density function (pdf) from a finite number of its moments is addressed. The target space is carefully defined and an approximation theorem is proved, establishing that the set of all convex superpositions of appropriate Kernel Density Functions (KDFs) is dense in this space. A solution algorithm is provided, based on the established approximate representation of the target pdf and the exploitation of some theoretical results concerning moment sequence asymptotics. The solution algorithm also permits us to recover the tail behavior of the target pdf and incorporate this information in our solution. A parsimonious formulation of the proposed solution procedure, based on a novel sequentially adaptive scheme is developed, enabling a very efficient moment data inversion. The whole methodology is fully illustrated by numerical examples.