Adding noise in frequency domain while solving an SPDE

Hello,

I’m implementing a numerical solver to a certain kind of a 2D stochastic PDE using Fourier Transforms.
So, this solver is a Galerkin type method, which means that choosing the right basis functions I can apply Fourier Transforms to compute the solution.

Long story short, applying the explicit Euler-Maruyama scheme to my SPDE and discretising the domain with N^2 nodes, the discrete version of the Fourier coefficients of V at time step t_i+1 is given by:


where the hat terms corresponds to the fourier coefficients of the equation, eps is the level of additive noise, W is a complex random matrix with distribution N(0,1) and \lambda is a spatial correlation matrix.
Notice that the noise is being added in the frequency domain!

However, when comparing results with other numerical methods the effect of the noise is not the same, actually I have to scale up eps by a factor.

I’m using the fftw.jl package to compute the Fourier Transforms. Thus, it’s known that fft(V) = N^2*v_mn, i.e. the function fft gives us the fourier coefficients of a certain vector (matrix) scaled by a factor of N (N^2).
So, the stochastic term of my expression showed above has not the same magnitude as the rest of the coefficients. Am I right?

To overcome this issue, I simply have to multiply eps by N^2, is this easy? Or adding noise in the frequency domain is more complicated than this?

Actually, an experimental evidence that the number of nodes influences the noise effect in the solution, is if I increase N I’ll also have to increase eps to continue to have similar behaviours than with less nodes.

Sorry for my question, but I’m a bit newbie in the Fourier world.

Thank you!

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One other factor to consider is whether the noise being added is band-limited or not, i.e., does it extend from 0 to a fraction of Nyquist or does it go up to Nyquist. This would require a normalization factor to preserve the meaning of epsilon in terms of noise variance.

More familiar with noise added in the physical data domain but, just in case, for white noise added in the frequency domain, in addition to the constant magnitude epsilon, one needs to randomize the phase - see stack overflow post here.

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I think you got it right: The standard fft / ifft methods only normalize on the inverse transform (the classical convention; not the quantum convention). So if you want to manipulate data in the Fourier domain you need to scale your terms accordingly with the number of points.

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Hmm, I will study what you said here with more attention. I don’t know exactly if my noise is band-limited or not, I have to check if it extends up to the Nyquist frequency.

Thank you for mentioning this post! Well, the noise I’m working on is additive and I think it’s white noise, since W is a matrix with complex numbers where both real and imaginary parts are generated independently by a distribution N(0,1).
This means that I’m uniformly randomising the phase? No! Right? is being generated by a normal distribution and not by an uniform distribution.

Actually, my data is real, so one of my uncertainties was if I needed to introduce complex noise or not. If not, randomising the phase does not apply to my case.