Nonlinear Waves Seminar - Gregory Beylkin

ACCURATE EVALUATION OF OSCILLATORY INTEGRALS

Ubiquitous in a variety of applications, oscillatory integrals are typically evaluated via asymptotic
expansions rather than via quadratures. The obvious reason is that for such integrals the standard
quadrature rules are highly inefficient since the cost of evaluation grows proportionally to the number
of oscillations of the integrand. For example, consider the Fourier-type integral

I (ω) = ∫ ¹₋₁ ƒ (χ) e^iωg(χ)dχ, ω > 0,

where we assume that the real-valued functions ƒ and g, usually referred to as the amplitude and the
phase, are smooth and only mildly oscillatory. The integrand becomes highly oscillatory for ω ≫ 1
and, in order to avoid quadratures, the classical approach is to evaluate I (ω) by constructing its
asymptotic expansion with respect to inverse powers of ω.

Recently we developed a new method for functional representation of oscillatory integrals within
any user-supplied accuracy. Our approach is based on robust methods for nonlinear approximation
of functions via exponentials. The resulting complexity of evaluation of functional representations
of the oscillatory integrals no longer depends or depends only mildly on the size of the parameter
responsible for the oscillatory behavior (e.g., O(1) or O(log ω) for the integral above). In the talk
I will describe our approach. This is a joint work with Lucas Monzón.