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Numerical integration of highly oscillatory functions based on analytic continuation

Presented by: 
S Vandewalle [Leuven]
Date: 
Thursday 5th July 2007 - 09:00 to 10:00
Venue: 
INI Seminar Room 1
Abstract: 

We consider the integration of highly oscillatory functions. Based on analytic continuation, rapidly converging quadrature rules are derived for a fairly general class of oscillatory integrals with an analytic integrand. The accuracy of the quadrature increases both for the case of a fixed number of points and increasing frequency, and for the case of an increasing number of points and fixed frequency. These results are then used to obtain quadrature rules for more general oscillatory integrals, i.e., for functions that exhibit some smoothness but that are not analytic.

The approach described in this paper is related to the steepest descent or saddle point method from complex analysis. However, it does not employ asymptotic expansions. It can be used for small or moderate frequencies as well as for very high frequencies. We consider both the one-dimensional case, and the multi-dimensional case. Finally, we briefly elaborate on the use of the new integration rules in the context of solving highly oscillatory integral equations.

University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons