# Flatiron Institute Nonuniform Fast Fourier Transform¶

FINUFFT is a set of libraries to compute efficiently three types of nonuniform fast Fourier transform (NUFFT) to a specified precision, in one, two, or three dimensions, on a multi-core shared-memory machine. The library has a very simple interface, does not need any precomputation step, is written in C++ (using OpenMP and FFTW), and has wrappers to C, fortran, MATLAB, octave, and python. As an example, given $$M$$ arbitrary real numbers $$x_j$$ and complex numbers $$c_j$$, with $$j=1,\dots,M$$, and a requested integer number of modes $$N$$, the 1D type-1 (aka “adjoint”) transform evaluates the $$N$$ numbers

(1)$f_k = \sum_{j=1}^M c_j e^{ik x_j}~, \qquad \mbox{ for } \; k\in\mathbb{Z}, \quad -N/2 \le k \le N/2-1 ~.$

The $$x_j$$ can be interpreted as nonuniform source locations, $$c_j$$ as source strengths, and $$f_k$$ then as the $$k$$th Fourier series coefficient of the distribution $$f(x) = \sum_{j=1}^M c_j \delta(x-x_j)$$. Such exponential sums are needed in many applications in science and engineering, including signal processing, imaging, diffraction, and numerical partial differential equations. The naive CPU effort to evaluate (1) is $$O(NM)$$. The library approximates (1) to a requested relative precision $$\epsilon$$ with nearly linear effort $$O(M \log (1/\epsilon) + N \log N)$$. Thus the speedup over the naive cost is similar to that achieved by the FFT. This is achieved by spreading onto a regular grid using a carefully chosen kernel, followed by an upsampled FFT, then a division (deconvolution) step. For the 2D and 3D definitions, and other types of transform, see below.

The FINUFFT library achieves its speed via several innovations including:

1. The use of a new spreading kernel that is provably close to optimal, yet faster to evaluate than the Kaiser-Bessel kernel
2. Quadrature approximation for the Fourier transform of the spreading kernel
3. Load-balanced multithreading of the type-1 spreading operation

For the same accuracy in 3D, the library is 3-50 times faster on a single core than the single-threaded fast Gaussian gridding CMCL libraries of Greengard-Lee, and in the multi-core setting for spreading-dominated problems is faster than the Chemnitz NFFT3 library even when the latter is allowed a RAM-intensive full precomputation of the kernel. This is especially true for highly non-uniform point distributions and/or high precision. Our library does not require precomputation, and uses minimal RAM.

For the case of small problems where repeated NUFFTs are needed with a fixed set of nonuniform points, we have started to build advanced interfaces for this case. These are a factor of 2 or more faster than repeated calls to the plain interface, since certain costs such as FFTW setup and sorting are performed only once.

Note

For very small repeated problems (less than 10000 input and output points), users should also consider a dense matrix-matrix multiplication against the NUDFT matrix using BLAS3 (eg ZGEMM). Since we did not want BLAS to be a dependency, we have not yet included this option.