Computing Highly Oscillatory Integrals

by Alfredo Deaño, Daan Huybrechs, Arieh Iserles

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. The text is intended for advanced undergraduate and graduate students, as well as applied mathematicians, scientists, and engineers who encounter highly oscillatory integrals as a critical difficulty in their computations. Contents : Chapter 1: Introduction; Chapter 2: Asymptotic theory of highly oscillatory integrals; Chapter 3: Filon and Levin methods; Chapter 4: Extended Filon method; Chapter 5: Numerical steepest descent; Chapter 6: Complex-valued Gaussian quadrature; Chapter 7: A highly oscillatory olympics; Chapter 8: Variations on the highly oscillatory theme; Appendix Orthogonal polynomials.

190 pages, Paperback

Published December 27, 2017

About the author

Filósofo y lógico español, que ejerció como profesor en la Universidad Autónoma de Madrid.

Se interesó profundamente por las relaciones entre la lógica formal y disciplinas como la lingüística, la psicología o la propia filosofía. Su postura se aleja tanto del ya casi universalmente abandonado (pero aún latente) psicologismo de la lógica, como de posturas puristas que entiendan la lógica únicamente como un cálculo algebraico cerrado y ajeno al lenguaje natural.

Entre sus obras destaca la Introducción a la lógica formal (1974), con la que logra una introducción amena y con sentido del humor a los conceptos generales de lógica, a la lógica de enunciados y la lógica de predicados de primer orden, así como esboza también algunas otras lógicas (clásicas y "no-clásicas"), y plantea algunas de las cuestiones lógico-filosóficas más importantes.

Deaño falleció inesperadamente de un infarto en Madrid en 1978, a los 33 años de edad. En 1980 se publicó su obra póstuma Las concepciones de la lógica.