Nonparametric, data-based kernel interpolation for particle-tracking simulations and kernel density estimation

•Traditional kernel density estimation uses assumed kernel function form (e.g., Gaussian).•A more intuitive approach uses a kernel that is the shape of the underlying data density.•An iterative, machine learning approach learns the shape of the kernel by using the evolving estimated density.•Our approach has lower error than traditional methods when applied to a range of known densities.•When applied to particle arrival times, the new approach smoothly interpolates and extrapolates the BTC. Traditional interpolation techniques for particle tracking include binning and convolutional formulas that use pre-determined (i.e., closed-form, parameteric) kernels. In many instances, the particles are introduced as point sources in time and space, so the cloud of particles (either in space or time) is a discrete representation of the Green’s function of an underlying PDE. As such, each particle is a sample from the Green’s function; therefore, each particle should be distributed according to the Green’s function. In short, the kernel of a convolutional interpolation of the particle sample “cloud” should be a replica of the cloud itself. This idea gives rise to an iterative method by which the form of the kernel may be discerned in the process of interpolating the Green’s function. When the Green’s function is a density, this method is broadly applicable to interpolating a kernel density estimate based on random data drawn from a single distribution. We formulate and construct the algorithm and demonstrate its ability to perform kernel density estimation of skewed and/or heavy-tailed data including breakthrough curves.