Numerical computation of Oscillatory integrals

[muppet]

Hi all,

I have a family of nasty integrals to do, all of the general form

[tex]\int ^{\infty} _0 db b J_0 (b q)(e^{i\chi(b)}-1) [/tex]

where chi is a complicated real function, typically a sum of products of special functions. (q is a real number.) The only thing that the different functions have in common is that they behave like 1/b^n for large values of b.

This is. I think, quite tricky to do numerically, so my supervisor doesn't want to just take Mathematica's answer as final, which gives me the problem of how to do better!

The standard approach in dealing with oscillatory integrals numerically seems to be:
Identify the oscillatory kernel;
Integrate between each zero and the next, generating an alternating series;
Use some kind of transformation on the series to accelerate its convergence.

Problem one is that I have two oscillatory kernels, unless q=0. Because the function chi will eventually decrease to less than Pi, the second of these kernels switches off after a while. Is this likely to upset the convergence of the early terms in the sequence to the final answer?

Problem two is that the best way of accelerating the convergence of the sequence seems to depend on the problem at hand. (For example, I read that the Shanks transformation was one that was commonly employed, but that only works well I believe if the nth partial sum of the series behaves roughly like
[tex]S_n =\text{(Right answer)} + \alpha \epsilon^n[/tex]
and don't see any reason why I should see that behaviour here. Can anyone point me on the right path to working out what the most appropriate transformation is for my series?

Thanks in advance.

 

[jvicens]

Hi there,

It sounds like you have a challenging problem on your hands. Dealing with oscillatory integrals can be tricky, especially when there are multiple oscillatory kernels involved. However, there are some techniques that can help improve the numerical accuracy of your results.

One approach you can try is to use a quadrature method, such as Gaussian quadrature or Clenshaw-Curtis quadrature. These methods are specifically designed to handle oscillatory integrals and can provide more accurate results than standard numerical integration techniques. They also have the advantage of being relatively easy to implement.

Another option is to use a spectral method, such as a Fourier or Chebyshev method. These methods use a series of basis functions to approximate the integral, which can be more accurate than traditional numerical integration methods.

In terms of accelerating the convergence of your alternating series, there are several techniques you can try. As you mentioned, the Shanks transformation is one option, but it may not be the best choice for your particular problem. Other options include the Levin u transformation and the Euler transformation.

Ultimately, the best approach will depend on your specific problem and the behavior of your oscillatory kernels. It may require some trial and error to determine the most effective technique, but don't be discouraged. With persistence and some experimentation, you should be able to find a method that works well for your integrals.

I hope this helps. Best of luck with your integrals!