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Title: Resurgent extrapolation: rebuilding a function from asymptotic data. Painlevé I

Abstract

Extrapolation is a generic problem in physics and mathematics: how to use asymptotic data in one parametric regime to learn about the behavior of a function in another parametric regime. For example: extending weak coupling expansions to strong coupling, or high temperature expansions to low temperature, or vice versa. Such extrapolations are particularly interesting in systems possessing dualities. Here we study numerical procedures for performing such an extrapolation, combining ideas from resurgent asymptotics with well-known techniques of Borel summation, Padé approximants and conformal mapping. We illustrate the method with the concrete example of the Painlevé I equation, which has applications in many branches of physics and mathematics. Starting solely with a finite number of coefficients from asymptotic data at infinity on the positive real line, we obtain a high precision extrapolation of the function throughout the complex plane, even across the phase transition into the pole region. The precision far exceeds that of state-of-the-art numerical integration methods along the real axis. Furthermore, the methods used are both elementary and general, not relying on Painlevé integrability properties, and so are applicable to a wide class of extrapolation problems.

Authors:
ORCiD logo [1];  [1]
  1. Univ. of Connecticut, Storrs, CT (United States)
Publication Date:
Research Org.:
Univ. of Connecticut, Storrs, CT (United States)
Sponsoring Org.:
USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25); National Science Foundation (NSF)
OSTI Identifier:
1594817
Grant/Contract Number:  
SC0010339; DMS 1515755; NSF PHY-1125915
Resource Type:
Accepted Manuscript
Journal Name:
Journal of Physics. A, Mathematical and Theoretical
Additional Journal Information:
Journal Volume: 52; Journal Issue: 44; Journal ID: ISSN 1751-8113
Publisher:
IOP Publishing
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; resurgence; Painlevé; extrapolation

Citation Formats

Dunne, Gerald V., and Costin, Ovidiu. Resurgent extrapolation: rebuilding a function from asymptotic data. Painlevé I. United States: N. p., 2019. Web. doi:10.1088/1751-8121/ab477b.
Dunne, Gerald V., & Costin, Ovidiu. Resurgent extrapolation: rebuilding a function from asymptotic data. Painlevé I. United States. doi:10.1088/1751-8121/ab477b.
Dunne, Gerald V., and Costin, Ovidiu. Thu . "Resurgent extrapolation: rebuilding a function from asymptotic data. Painlevé I". United States. doi:10.1088/1751-8121/ab477b.
@article{osti_1594817,
title = {Resurgent extrapolation: rebuilding a function from asymptotic data. Painlevé I},
author = {Dunne, Gerald V. and Costin, Ovidiu},
abstractNote = {Extrapolation is a generic problem in physics and mathematics: how to use asymptotic data in one parametric regime to learn about the behavior of a function in another parametric regime. For example: extending weak coupling expansions to strong coupling, or high temperature expansions to low temperature, or vice versa. Such extrapolations are particularly interesting in systems possessing dualities. Here we study numerical procedures for performing such an extrapolation, combining ideas from resurgent asymptotics with well-known techniques of Borel summation, Padé approximants and conformal mapping. We illustrate the method with the concrete example of the Painlevé I equation, which has applications in many branches of physics and mathematics. Starting solely with a finite number of coefficients from asymptotic data at infinity on the positive real line, we obtain a high precision extrapolation of the function throughout the complex plane, even across the phase transition into the pole region. The precision far exceeds that of state-of-the-art numerical integration methods along the real axis. Furthermore, the methods used are both elementary and general, not relying on Painlevé integrability properties, and so are applicable to a wide class of extrapolation problems.},
doi = {10.1088/1751-8121/ab477b},
journal = {Journal of Physics. A, Mathematical and Theoretical},
number = 44,
volume = 52,
place = {United States},
year = {2019},
month = {10}
}

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