scholarly journals Extrapolation of perturbation-theory expansions by self-similar approximants

2014 ◽  
Vol 25 (5) ◽  
pp. 595-628 ◽  
Author(s):  
S. GLUZMAN ◽  
V.I. YUKALOV
Keyword(s):  
Perturbation Theory ◽  
Approximation Theory ◽  
Asymptotic Expansions ◽  
Applied Mathematics ◽  
Self Similar ◽  
Similar Approximation ◽  
Asymptotic Perturbation

The problem of extrapolating asymptotic perturbation-theory expansions in powers of a small variable to large values of the variable tending to infinity is investigated. The analysis is based on self-similar approximation theory. Several types of self-similar approximants are considered and their use in different problems of applied mathematics is illustrated. Self-similar approximants are shown to constitute a powerful tool for extrapolating asymptotic expansions of different natures.

scholarly journals From Asymptotic Series to Self-Similar Approximants

Physics ◽  
2021 ◽  
Vol 3 (4) ◽  
pp. 829-878
Author(s):  
Vyacheslav I. Yukalov ◽  
Elizaveta P. Yukalova
Keyword(s):  
Perturbation Theory ◽  
Approximation Theory ◽  
Good Accuracy ◽  
Asymptotic Series ◽  
Approximate Solutions ◽  
Self Similar ◽  
Similar Approximation ◽  
Physics Problems

The review presents the development of an approach of constructing approximate solutions to complicated physics problems, starting from asymptotic series, through optimized perturbation theory, to self-similar approximation theory. The close interrelation of underlying ideas of these theories is emphasized. Applications of the developed approach are illustrated by typical examples demonstrating that it combines simplicity with good accuracy.

scholarly journals SELF-SIMILAR EXTRAPOLATION OF ASYMPTOTIC SERIES AND FORECASTING FOR TIME SERIES

2000 ◽  
Vol 14 (22n23) ◽  
pp. 791-800 ◽  
Author(s):  
V. I. YUKALOV
Keyword(s):  
Time Series ◽  
Probability Measure ◽  
Approximation Theory ◽  
Asymptotic Series ◽  
The Self ◽  
Complete Family ◽  
Self Similar ◽  
Similar Approximation

The method of extrapolating asymptotic series, based on the self-similar approximation theory, is developed. Several important questions are answered, which makes the foundation of the method unambiguous and its application straightforward. It is shown how the extrapolation of asymptotic series can be reformulated as forecasting for time series. The probability measure is introduced characterizing the ensemble of forecasted scenarios. The method of choosing the complete family of databases is put forward.

scholarly journals Convergent perturbation theory for lattice models with fermions

2016 ◽  
Vol 31 (13) ◽  
pp. 1650072 ◽  
Author(s):  
V. K. Sazonov
Keyword(s):  
Perturbation Theory ◽  
Asymptotic Expansions ◽  
Lattice Model ◽  
Lattice Models ◽  
Convergent Series ◽  
Standard Perturbation Theory

The standard perturbation theory in QFT and lattice models leads to the asymptotic expansions. However, an appropriate regularization of the path or lattice integrals allows one to construct convergent series with an infinite radius of the convergence. In the earlier studies, this approach was applied to the purely bosonic systems. Here, using bosonization, we develop the convergent perturbation theory for a toy lattice model with interacting fermionic and bosonic fields.

scholarly journals Self-similar extrapolation of nonlinear problems from small-variable to large-variable limit

2020 ◽  
Vol 34 (21) ◽  
pp. 2050208
Author(s):  
V. I. Yukalov ◽  
E. P. Yukalova
Keyword(s):  
Asymptotic Series ◽  
Nonlinear Problems ◽  
Numerical Convergence ◽  
Physical Problems ◽  
Variable Limit ◽  
Small Parameters ◽  
Self Similar ◽  
Similar Approximation ◽  
Similar Factor ◽  
Class Of Functions

Complicated physical problems are usually solved by resorting to perturbation theory leading to solutions in the form of asymptotic series in powers of small parameters. However, finite, and even large values of the parameters, are often of main physical interest. A method is described for predicting the large-variable behavior of solutions to nonlinear problems from the knowledge of only their small-variable expansions. The method is based on self-similar approximation theory resulting in self-similar factor approximants. The latter can well approximate a large class of functions, rational, irrational, and transcendental. The method is illustrated by several examples from statistical and condensed matter physics, where the self-similar predictions can be compared with the available large-variable behavior. It is shown that the method allows for finding the behavior of solutions at large variables when knowing just a few terms of small-variable expansions. Numerical convergence of approximants is demonstrated.

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