Estimation of the probability density function from the statistical power moments presents a challenging nonlinear numerical problem posed by unbalanced nonlinearities, numerical instability and a lack of convergence, especially for larger numbers of moments. Despite many numerical improvements over the past two decades, the classical moment problem of maximum entropy (MaxEnt) is still a very demanding numerical and statistical task. Among others, it was presented how Fup basis functions with compact support can significantly improve the convergence properties of the mentioned nonlinear algorithm, but still, there is a lot of obstacles to an efficient pdf solution in different applied examples. Therefore, besides the mentioned classical nonlinear Algorithm 1, in this paper, we present a linear approximation of the MaxEnt moment problem as Algorithm 2 using exponential Fup basis functions. Algorithm 2 solves the linear problem, satisfying only the proposed moments, using an optimal exponential tension parameter that maximizes Shannon entropy. Algorithm 2 is very efficient for larger numbers of moments and especially for skewed pdfs. Since both Algorithms have pros and cons, a hybrid strategy is proposed to combine their best approximation properties.
SažetakAtomske bazne funkcije (ABF) posjeduju svojstvo univerzalnosti vektorskog prostora kao klasične bazne funkcije i svojstvo finitnosti (konačna duljina nosača) kao splineovi te na taj način popunjavaju skup elementarnih funkcija. U ovom radu ukratko su opisana svojstva eksponencijalnih ABF EFup n (x, w), koji za razliku od algebarskih ABF EFup n (x), sadrže parametar ili frekvenciju ω koja im omogućava dodatna aproksimacijska svojstva. Problem izbora vrijednosti parametra ω u numeričkoj analizi riješen je primjenom tzv. trostruke baze. Primjena atomskih baznih funkcija EFup n (x, w) ilustrirana je na primjeru rješavanja singularno perturbiranog rubnog problema (SPRP) i to korištenjem trostruke baze u metodi kolokacije uz primjenu multirezolucijskog postupka. Ključne riječi: ABF eksponencijalnog tipa, frekvencija, trostruka baza, SPRP Solving the SPRP by using a triple base ABF AbstractAtomic basis functions (ABF) possess the property of universality of the vector space as a classical basis functions and finiteness as splines, and thus filling a set of elementary functions. In this paper basic properties of exponential ABF EFup n (x, w) are described briefly, which, unlike the algebraic ABF EFup n (x), contain a parameter or frequency ω which gives them additional approximation properties. The problem of selecting parameter value ω in numerical analysis has been solved using the so-called triple base. The application of the basis function EFup n (x, w) is illustrated in the example of solving the singularly perturbated boundary problem (SPBP) by using a triple base in the collocation method using the multi-resolution procedure.
In this work basic properties of algebraic Atomic Basis Functions (ABF) are systematized and, using analogous approach, ABF of exponential type, so far known only at the basic level, are developed. For the first time the properties of exponential ABFs are thoroughly investigated and expressions for calculating the values and all the necessary derivatives of the functions in an arbitrary points of the domain are developed as well as some special features required for their practical application in a form suitable for numerical analysis. A software module for calculating all necessary values of the exponential ABFs, including its own graphics support, is created within this work. Thus, the exponential ABF is prepared to use as users or compiler function. The presented 1D verification examples of the function approximation and the examples of solving differential equations illustrate and confirm the practical advantage of the ABFs of the exponential type in relation to the, so far mostly used, algebraic functions, especially for describing expressed fronts and/or waves contained in the numerical solutions of various technical tasks.
The purpose of this paper is to present the class of atomic basis functions (ABFs) which are of exponential type and are denoted by EFup n (x, ω). While ABFs of the algebraic type are already represented in the numerical modeling of various problems in mathematical physics and computational mechanics, ABFs of the exponential type have not yet been sufficiently researched. These functions, unlike the ABFs of the algebraic type Fup n (x), contain the tension parameter ω, which gives them additional approximation properties. Exponential monomials up to the nth degree can be described exactly by the linear combination of the functions EFup n (x, ω). The function EFup n for n = 0 is called the "mother" ABF of the exponential type, i.e., EFup 0 (x, ω) ≡ Eup(x, ω). In other words, the functions EFup n (x, ω) are elements of the linear vector space EUP n and retain all the properties of their "mother" function Eup(x, ω). Thus, this paper, in terms of its content and purpose, can be understood as a sequel of the article by Brajčić Kurbaša et al., which shows the basic properties and application of the basis function Eup (x, ω). This paper presents, in an analogous way, the development and application of the exponential basis functions EFup n (x, ω). Here, for the first time, expressions for calculating the values of the functions EFup n (x, ω) and their derivatives are given in a form suitable for application in numerical analyses, which is shown in the verification examples of the approximations of known functions.
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