Nonlinear Waves Seminar - Gregory Beylkin
| Event Description: Gregory Beylkin, Department of Applied Mathematics, °µÍø½ûÇø 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 (Ó¬) =Ìý∫ 1−1ÌýÆ’ (χ) e¾±Ó¬²µ(χ)dχ, Ó¬ > 0, where we assume that the real-valued functionsÌýÆ’ and g, usually referred to as the amplitude and the phase, are smooth and only mildly oscillatory. The integrand becomes highly oscillatory forÌýÓ¬ ≫ 1 and, in order to avoid quadratures, the classical approach is to evaluate I (Ó¬) by constructing its asymptotic expansion with respect to inverse powers of Ó¬. 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 Ó¬) for the integral above). In the talk I will describe our approach. This is a joint work with Lucas Monzón. |
| Location Information: ÌýÌý() 1111 Engineering DR Boulder, CO Room:Ìý226: Applied Math Conference Room |
| Contact Information: Name: Ian Cunningham Phone: 303-492-4668 Email: amassist@colorado.edu |