Fourier spectral discretization

Finite difference formulas create dispersion effects not found in original PDE. Similar effects seen in crystals, which are discrete by nature.

One way to avoid these, is to let the order of the finite difference formula tend to infinity. We then get spectral methods. Simplest favours are:

• Periodic domains: Fourier spectral methods.
• Non-periodic domains: Chebysev spectral methods.

In the limit of infinite order, those finite differences approach the infinite Laurent matrix (or Laurent operator).

Suppose we have the values of the solution function $v$ on our discrete periodic grid. Spectral approximations to $v'$, $w$ is given by:

$w=Dv$

where D here is the spectral differentiation matrix.

The fundamental idea of spectral collocation methods is :

1. Interpolate the data by a global interpolant (for example, a periodic trigonometric polynomial):

$p(x) = \sum_{j=-N/2}^{N/2} a_j e^{ijx}$

2. Differentiate $p(x)$ and evaluate at the grid points.

From properties of exponential, another way to compute the 2nd Fourier spectral derivative is:

1. Given $u$, compute its DFT (discrete Fourier transform) U = fft(u) (using MATLAB notation for fast Fourier transform (FFT), an efficient algorithm to compute DFT).

2. Multiply by $-j^2$: $W(j) = -j^2 U_j$.

3. Take the inverse transform w = ifft(u).

Similar ideas lead to the one-way wave equation.

Fill details below from lecture 10, when it's published (https://www0.maths.ox.ac.uk/courses/course/28839, and vid).

Fourier series

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Quadrature: trapezoidal rule $\Leftrightarrow$ integrating the interpolant

Rootfinding: via eigenvalues of companion matrix

Laurent series

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Chebysev series

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Relations to Grid cells and Spatial navigation (see my work on the second rotation of the SysBio DTC)

Spectral Inference Networks: Unifying Deep and Spectral Learning