Computing highly oscillatory integrals

by Deaño, Alfredo

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Title Statements

Full title :Computing highly oscillatory integrals / Alfredo Deaño, Daan Huybrechs, Arieh Iserles.

Summary

Highly oscillatory phenomena range across numerous areas in science and engineering and their computation represents a difficult challenge. A case in point is integrals of rapidly oscillating functions in one or more variables. The quadrature of such integrals has been historically considered very demanding. Research in the past 15 years (in which the authors played a major role) resulted in a range of very effective and affordable algorithms for highly oscillatory quadrature. This is the only monograph bringing together the new body of ideas in this area in its entirety. The starting point is that approximations need to be analyzed using asymptotic methods rather than by more standard polynomial expansions. As often happens in computational mathematics, once a phenomenon is understood from a mathematical standpoint, effective algorithms follow. As reviewed in this monograph, we now have at our disposal a number of very effective quadrature methods for highly oscillatory integrals--Filon-type and Levin-type methods, methods based on steepest descent, and complex-valued Gaussian quadrature. Their understanding calls for a fairly varied mathematical toolbox--from classical numerical analysis, approximation theory, and theory of orthogonal polynomials all the way to asymptotic analysis--yet this understanding is the cornerstone of efficient algorithms.

Edition Statements

Publication Date:2018

Publication Statements

Publisher(s):SIAM, Society for Industrial and Applied Mathematics
Place of Publication:Philadelphia
Language:eng
Publication Date:2018

The Physical Object

Description:1 online resource

Edition Identifiers

ISBN:9781611975123, 1611975123
oclc:on1013497641
LC Classification:QA308
Dewey Decimal:515/.43

1. Introduction -- 2. Asymptotic theory of highly oscillatory integrals -- 3. Filon and Levin methods -- 4. Extended Filon method -- 5. Numerical steepest descent -- 6. Complex-valued Gaussian quadrature -- 7. A highly oscillatory olympics -- 8. Variations on the highly oscillatory theme -- Appendix A. Orthogonal polynomials.

Subjects

Asymptotic analysis Bifurcation theory Calcul intégral Calculus, Integral High oscillation Integrales Integrals Orthogonal polynomials Quadrature Steepest descent Théorie de la bifurcation