Computing Highly Oscillatory Integrals

This is a chatty, informal compendium about the numerical estimation of highly-oscillatory integrals. There have been a lot of developments in this field in the past fifteen years, and this book attempts to cover them all.

The most familiar highly-oscillatory integral is the Fourier transform, \[\int_{-\infty}^\infty f(t) e^{-i \omega t} \, dt.\]The book however deals only lightly with this integral. Instead the two model problems are \[ \int_{-1}^1 f(x) e^{i \omega x} \, dx \qquad \text{ and } \qquad\int_{-1}^1 f(x) e^{i \omega x^2} \, dx\] where the function \(f\) is smooth. The difference is that the second integral has a stationary point (at 0), so it’s not highly-oscillatory over the whole range. The interest here is in applied problems, where \(\omega\) is the frequency in Hertz and is on the order of 50 to 100, and we need a few decimals of accuracy.

The traditional quadrature approach, of dividing the interval of integration into many small pieces, approximating the integrand by a polynomial, then integrating the polynomial, works poorly for these integrals. The integrand does not look at all like a polynomial, even over short intervals. Even the obvious improvement of approximating only \(f\) and explicitly integrating the exponential part doesn’t work well. So current work uses a different approach inspired by the asymptotics of the problem.

It is possible to develop an asymptotic series in \(\omega^{-1}\) for these integrals, and for large \(\omega\) this usually gives good enough approximations, so the interest in the present book is in intermediate values of \(\omega\). The idea is to use the asymptotic series as a guide. One key point from asymptotics is that the series coefficients depend on the values and derivatives of \(f\) at the endpoints and not in the middle, so one approach (Filon’s) is to approximate \(f\) with a polynomial that matches its values and derivatives as the endpoints. Levin’s method attempts to transform the problem from a highly-oscillatory one to a slowly-oscillatory one. The methods of numerical steepest descent and complex Gaussian quadrature both transform the integral by changing it to a contour integral in the complex plane. All these methods are inspired by asymptotic analysis.

The book has good coverage of both univariate and multivariate integrals. It’s skimpy on numerical examples, but it does has lots of graphs showing the error between the true values and quadrature values based on various parameters. The book is very concerned with the number of function evaluations needed to get a good result.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is allenstenger.com. His mathematical interests are number theory and classical analysis.