We introduce a general purpose algorithm for rapidly computing certain types of oscillatory integrals which frequently arise in problems connected to wave propagation and general hyperbolic equations. The problem is to evaluate numerically a so-called Fourier integral operator (FIO) of the form @math at points given on a Cartesian grid. Here, @math is a frequency variable, @math is the Fourier transform of the input @math , @math is an amplitude and @math is a phase function, which is typically as large as @math ; hence the integral is highly oscillatory at high frequencies. Because an FIO is a dense matrix, a naive matrix vector product with an input given on a Cartesian grid of size @math by @math would require @math operations. This paper develops a new numerical algorithm which requires @math operations, and as low as @math in storage space. It operates by localizing the integral over polar wedges with small angular aperture in the frequency plane. On each wedge, the algorithm factorizes the kernel @math into two components: 1) a diffeomorphism which is handled by means of a nonuniform FFT and 2) a residual factor which is handled by numerical separation of the spatial and frequency variables. The key to the complexity and accuracy estimates is that the separation rank of the residual kernel is . Several numerical examples demonstrate the efficiency and accuracy of the proposed methodology. We also discuss the potential of our ideas for various applications such as reflection seismology.
We would also like to acknowledge the line of research related to Filon-type quadratures for oscillatory integrals @cite_14 . When the integrand is of the form @math with @math smooth and @math large, it is not always necessary to sample the integrand at the Nyquist rate. For instance, integration of a polynomial interpolant of @math (Filon quadrature) provides an accurate approximation to @math using fewer and fewer evaluations of the function @math as @math . While these ideas are important, they are not directly applicable in the case of FIOs. The reasons are threefold. First, we make no notable assumption on the support of the function to which the operator is applied, meaning that the oscillations of @math may be on the same scale as those of the exponential @math . Second the phase does not in general have a simple formula that would lend itself to precomputations. And third, Filon-type quadratures do not address the problem of simplifying computations of such oscillatory integrals at once (i.e. computing a family of integrals indexed by @math in the case of FIOs).
@cite_14
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Highly-oscillatory integrals are allegedly difficult to calculate. The main assertion of this paper is that that impression is incorrect. As long as appropriate quadrature methods are used, their accuracy increases when oscillation becomes faster and suitable choice of quadrature points renders this welcome phenomenon more pronounced. We focus our analysis on Filon-type quadrature and analyse its behaviour in a range of frequency regimes for integrals of the form ∫ h 0 f(x)e iωx w(x)dx, where h > 0 is small and |ω| large. Our analysis is applied to modified Magnus methods for highly-oscillatory ordinary differential equations.