BandlimitedOperators.jl

A simple package providing objects that implicitly represent the action of a bandlimited kernel matrix on a vector. This action is computed at the cost of two NUFFTs, which can often turn O(n^2) work into O(n \log n). The implementation here is based on the approach of the fast sinc transform. It uses FINUFFT internally, and so transforms are available in one, two, and three dimensions.

A quick demo:

```julia using BandlimitedOperators

pts1 = sort(rand(1000)).3 pts2 = sort(rand(1000)).3

The matrix we want to approximate, which corresponds to a kernel matrix whose

kernel function's Fourier transform is supported on [-18.1, 18.1].

M = [sinc(218.1(xj-xk)) for xj in pts1, xk in pts2]

We approximate it with a FastBandlimited object. Note the Fourier transform of

this kernel function is precisely ω ↦ 2*χ(|ω| ≤ 18.1)/18.1, so that is the

third argument here.

fastM = FastBandlimited(pts1, pts2, x->inv(2*18.1), 18.1)

x = randn(1000) @show maximum(abs, Mx - fastMx) # ~1e-13 ```

Non-smooth points in your kernel's Fourier transform

If you have a kernel function whose Fourier transform has non-smooth points, please pass those in with the roughpoints API. For example, here is how you would do a fast sinc squared transform: ```julia pts1 = sort(rand(1000)).3 pts2 = sort(rand(1000)).3

Note that the Fourier transform of sinc(x)^2 is a triangle function. That is

still an easy numerical integral...if you use a panel quadrature rule

because the function is not smooth at the origin. So we provide that location

for splitting panels with roughpoints=(0.0,) (that arg should be an iterable):

triangle(x, bw) = max(0.0, 1-abs(x)/bw) fastM = FastBandlimited(pts1, pts2, x->triangle(x, 18.1)/18.1, 18.1; roughpoints=(0.0,))

x = randn(length(pts1))

You can also use the in-place mul! of course:

buf = zeros(length(x)) mul!(buf, fastM, x)

M = [sinc(18.1(xj-xk))^2 for xj in pts1, xk in pts2] @show maximum(abs, Mx - buf) # ~1e-13 ```

NOTE: Even if the Fourier transform of your kernel is smooth at the origin, if it has anything resembling a sharp peak there (or at any other location), it is likely advantageous for you to provide the locations for those sharp peaks as roughpoints. The default roughpoints argument is now (as of v0.1.7) the origin (except in the case of polar=true, see below).

Applying to multiple vectors at a time

By default, FastBandlimited uses planned "guru-mode" NUFFTs, which can provide a big speedup. But guru mode plans require committing to a certain number of vectors to apply to at once, so applying to a different number means looping over columns or other fallback strategies that are less efficient. If you plan to apply your operator to multiple vectors at a time, it can sometimes be faster to instead use the non-guru mode and exploit the multiple-column speedup that FINUFFT offers. The downside is simply that the mul! command will now make allocations. To enable this, all you need to do is provide the allocating_mul=trud kwarg: julia fastM = FastBandlimited(pts1, pts2, fn, bandlimit; allocating_mul=true, kwargs...)

Experimental: Bandlimited operators with support on a disk

Some functions have a Fourier transform that is supported on regions other than rectangles. This package now offers the option to specify the support of your kernel FT as being a disk of radius bandwidth in two dimensions only like so: ```julia pts1 = [...] # ::Vector{SVector{2,Float64}} pts2 = [...] # ::Vector{SVector{2,Float64}}

the kernel ft fn should still be in cartesian coordinates!

fastM = FastBandlimited(pts1, pts2, fn, bandwidth; polar=true) ``` Note: As of now, the quadrature rule being used in the Fourier domain is a hybrid trapezoidal and Gauss-Legendre. Panel support is not currently implemented, so your kernel FT should be smooth. Secondly, the size of the quadrature rule is not very optimally designed and probably has more oversampling than is necessary.

Please see the files in ./example for various other demonstrations.

Roadmap (PRs welcome!)

• polar=true in 3D
• More rigorous quadrature rule design for polar=true option
• Supporting ellipsoidal support regions for kernel FTs in 2 and 3 dimensions