Memory issues in numerical integration of oscillatory function

[cyberpotato]

TL;DR

Memory challenges in numerically integrating oscillatory functions using FFT in Python. Starting this thread here as a related thread was discussed earlier: https://www.physicsforums.com/threads/numerical-integration-fourier-transform-or-brute-force.283916/#post-2030214

Hello!

I need to numerically integrate a frequently oscillating, decaying complex function over the interval from 0 to infinity, which is continuous. For brevity, I provide the general integral view
$$\int_{0}^{\infty} A(t)e^{e^{iw't}}dt$$.
I'm using Python libraries for this task. Initially, I tried the quadrature method, but encountered convergence issues when integrating over a large interval. To address this, I started subdividing the integration interval into subintervals, integrating them separately, and summing the results until convergence. However, this process was time-consuming.
To speed up the integration, I switched to using the fast Fourier transform (FFT) from scipy.fft. While this approach improved integration speed, a new issue arose. When passing a vector of function values to scipy.fft, I faced memory constraints for very large integration intervals with small step sizes. To solve this, I considered subdividing the large integration interval, performing FFT on each subinterval, and summing the results until convergence.

Is it correct to solve this issue by breaking down the large integration interval into subintervals, performing FFT on each subinterval, and summing until convergence? Additionally, are there alternative methods for integrating such functions that I should consider?

 

[pasmith]

It may be worth setting $$I_n = \int_{0}^{2\pi/\omega'} A\left(\frac{2n\pi}{\omega'} + t\right)e^{e^{i\omega't}}\,dt, \qquad n \geq 0$$ and seeing how fast $|I_n|$ decays with $n$. Your integral can then be approximated as $\sum_{n=0}^N I_n$ for some $N$.

 

[Fred Wright]

I think what you are asking is that you have,
$$
\int_0^{\infty} A(t)[1 + e^{i\omega t}+ \frac{e^{2i\omega t}}{2!} +\frac{e^{3i\omega t}}{3!} + ...]dt
$$
and wish to take the FFT of your time series against each succeeding term in the expansion. Firstly, you must apply the scaling theorem for FFTs to each term (see Scaling Theorem )which will probably result in interpolation issues which you must resolve. Secondly, by breaking up your time series into segments and taking the FFT of each segment Gibbs oscillations raise their ugly heads. Therefore you must window each segment using a Hamming window or whatnot.

 

[cyberpotato]

It may be worth setting $$I_n = \int_{0}^{2\pi/\omega'} A\left(\frac{2n\pi}{\omega'} + t\right)e^{e^{i\omega't}}\,dt, \qquad n \geq 0$$ and seeing how fast $|I_n|$ decays with $n$. Your integral can then be approximated as $\sum_{n=0}^N I_n$ for some $N$.

Your approach looks very interesting, thank you. I'm trying to understand it. Are you using the periodicity property of the exp(iωt) function? In the case A(t) = 1, the integral will have the form:
$$\int_{0}^{\infty }e^{e^{i\omega t}}=\sum_{n=0}^N \int_{0}^{2\pi/\omega}e^{e^{i\omega t}}$$
Is that right?