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.
Audience: 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 A: Orthogonal polynomials.
Alfredo Deaño is a Lecturer in Mathematics at the School of Mathematics, Statistics and Actuarial Science, University of Kent (UK). His main research interests include the theory of classical special functions, orthogonal polynomials, and Painleve equations.
Daan Huybrechs is a Professor at KU Leuven, Belgium, in the section on numerical analysis and applied mathematics (NUMA) in the Department of Computer Science. He is an associate editor of the IMA Journal of Numerical Analysis. His main research interests include oscillatory integrals, approximation theory.
Arieh Iserles recently retired from the Chair in Numerical Analysis of Differential Equations at University of Cambridge. He is the managing editor of Acta Numerica, Editor-in-Chief of IMA Journal of Numerical Analysis and of Transactions of Mathematics and its Applications, and an editor of many other journals and book series. His research interests comprise geometric numerical integration, the computation of highly oscillatory phenomena, computations in quantum mechanics, approximation theory, and the theory of orthogonal polynomials.
