Fast Error-bounded Surfaces and Derivatives Computation for Volumetric Particle Data

Volumetric smooth particle data arise as atomic coordinates with electron density kernels for molecular structures, as well as fluid particle coordinates with a smoothing kernel in hydrodynamic flow simulations. In each case there is the need for efficiently computing approximations of relevant surfaces (molecular surfaces, material interfaces, shock waves, etc), along with surface and volume derivatives (normals, curvatures, etc.), from the irregularly spaced smooth particles. Additionally, molecular properties (charge density, polar potentials), as well as field variables from numerical simulations are often evaluated on these computed surfaces. In this paper we show how all the above problems can be reduced to a fast summation of irregularly spaced smooth kernel functions. For a scattered smooth particle system of M smooth kernels in R, where the Fourier coefficients have a decay of the type 1/ω, we present an O(M + n log n+ N) time, Fourier based algorithm to compute N approximate, irregular samples of a level set surface and its derivatives within a relative L2 error norm ǫ, where n is O(M ǫ). Specifically, a truncated Gaussian of the form e 2 has the above decay, and n grows as √ b. In the case when the N output points are samples on a uniform grid, the back transform can be done exactly using a Fast Fourier transform algorithm, giving us an algorithm with O(M + n log n + N logN) time complexity, where n is now approximately half its previously estimated value.