Computing highly oscillatory integrals. (English) Zbl 1400.65004

Deaño, Alfredo; Huybrechs, Daan; Iserles, Arieh

Other Titles in Applied Mathematics 155. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM) (ISBN 978-1-61197-511-6/pbk; 978-1-61197-512-3/ebook). x, 180 p. (2018).

Highly oscillatory integrals occur in fluid dynamics, acoustic, and electromagnetic scattering. This important monograph presents efficient algorithms for computing highly oscillatory integrals, such as \[ I_{\omega}[f] := \int_{-1}^1 f(x)\,{\mathrm e}^{{\mathrm i}\,\omega \,g(x)}\, {\mathrm d}x \] where \(f\) is a smooth function, \(\omega \gg 1\), and \(g(x) = x\) or \(g(x) = x^2\). In contrast to \(g(x) = x\), the function \(g(x) = x^2\) has a stationary point at \(x=0\).
Chapter 1 has preliminary character. Since efficient numerical methods for oscillatory integrals use the asymptotic behavior of \(I_{\omega}[f]\) for large \(\omega\), Chapter 2 presents an asymptotic theory of highly oscillatory integrals. Chapter 3 handles with the Filon quadrature and Levin-type methods. In the Filon method one calculates \(I_{\omega}[p]\) with \(g(x) = x\), where \(p\) is a polynomial \(p\) interpolating \(p^{(j)}(-1) = f^{(j)}(-1)\) and \(p^{(j)}(1) = f^{(j)}(1)\), \(j=0,\ldots,s\). Chapter 4 is devoted to extended Filon methods, such as Filon-Jacobi quadrature and Filon-Clenshaw-Curtis quadrature. Numerical methods based on steepest descent are discussed in Chapter 5. Complex-valued Gaussian quadrature for oscillatory integrals are presented in Chapter 6. In Chapter 7, the authors compare the various quadrature methods at several test functions. The final Chapter 8 contains some conclusions and further extensions. In an appendix, properties of orthogonal polynomials on \(\mathbb R\) and orthogonal polynomials with complex weight function are sketched.
This well-written monograph is intended for graduate students in applied mathematics, scientists, and engineers who encounter highly oscillatory integrals. The authors consider mainly univariate oscillatory integrals and give some hints for the multivariate case. This book contains numerous examples and instructive figures. Doubtless, this excellent work will be stimulated the further research of computing highly oscillatory integrals.

Reviewer: Manfred Tasche (Rostock)

MSC:

65-02	Research exposition (monographs, survey articles) pertaining to numerical analysis
65D32	Numerical quadrature and cubature formulas
42C05	Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis

Keywords:

quadrature formulas; highly oscillatory integrals; Fourier-type integrals; asymptotic method; Filon quadrature; Levin-type methods; extended Filon methods; error estimates; method of steepest descent; complex-valued Gaussian quadrature