The nonuniform fast Fourier transform (NUFFT) generalizes the FFT to off-grid data. Its many applications include image reconstruction, data analysis, and the numerical solution of differential equations. We present FINUFFT, an efficient parallel library for type 1 (nonuniform to uniform), type 2 (uniform to nonuniform), or type 3 (nonuniform to nonuniform) transforms, in dimensions 1, 2, or 3. It uses minimal RAM, requires no precomputation or plan steps, and has a simple interface to several languages. We perform the expensive spreading/interpolation between nonuniform points and the fine grid via a simple new kernel-the "exponential of semicircle" e β √ 1−x 2 in x ∈ [−1, 1]-in a cache-aware load-balanced multithreaded implementation. The deconvolution step requires the Fourier transform of the kernel, for which we propose efficient numerical quadrature. For types 1 and 2, rigorous error bounds asymptotic in the kernel width approach the fastest known exponential rate, namely that of the Kaiser-Bessel kernel. We benchmark against several popular CPU-based libraries, showing favorable speed and memory footprint, especially in three dimensions when high accuracy and/or clustered point distributions are desired.

We present a spectrally accurate method for the rapid evaluation of free-space Stokes potentials, i.e., sums involving a large number of free space Green's functions. We consider sums involving stokeslets, stresslets and rotlets that appear in boundary integral methods and potential methods for solving Stokes equations. The method combines the framework of the Spectral Ewald method for periodic problems (Lindbo and Tornberg in J Comput Phys 229(23): 8994-9010, 2010. doi:10.1016/j.jcp.2010.08. 026), with a very recent approach to solving the free-space harmonic and biharmonic equations using fast Fourier transforms (FFTs) on a uniform grid (Vico et al. in J Comput Phys 323:191-203, 2016. doi:10.1016/j.jcp.2016. Convolution with a truncated Gaussian function is used to place point sources on a grid. With precomputation of a scalar grid quantity that does not depend on these sources, the amount of oversampling of the grids with Gaussians can be kept at a factor of two, the minimum for aperiodic convolutions by FFTs. The resulting algorithm has a computational complexity of O(N log N) for problems with N sources and targets. Comparison is made with a fast multipole method to show that the performance of the new method is competitive. BackgroundIn this paper, we consider the evaluation of free-space potentials of Stokes flow, i.e., vector fields defined by sums involving a large number of free space Green's functions such as the so-called stokeslet, stresslet or rotlet. The stokeslet is the free space Green's function for velocity and is given bywith r = |r| and where δ jl is the Kronecker delta. The stresslet and rotlet will be introduced in the following. The discrete sums are on the form

We present a numerical method for suspensions of spheroids of arbitrary aspect ratio which sediment under gravity. The method is based on a periodized boundary integral formulation using the Stokes double layer potential. The resulting discrete system is solved iteratively using GMRES accelerated by the spectral Ewald (SE) method, which reduces the computational complexity to O(N log N ), where N is the number of points used to discretize the particle surfaces. We develop predictive error estimates, which can be used to optimize the choice of parameters in the Ewald summation. Numerical tests show that the method is well conditioned and provides good accuracy when validated against reference solutions.

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