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.

用于粒子跟踪的传统插值技术包括使用预定（即，封闭形式，参数）内核的合并和卷积公式。在许多情况下，粒子是作为时间和空间的点源引入的，因此粒子的云（无论是空间还是时间）都是底层PDE格林函数的离散表示。这样，每个粒子都是格林函数的样本；因此，每个粒子应根据格林函数进行分布。简而言之，粒子样本“云”的卷积插值的内核应该是云本身的副本。这个想法提出了一种迭代方法，通过该方法可以在对格林函数进行插值的过程中辨别内核的形式。当格林函数是密度时 该方法广泛适用于基于从单个分布中得出的随机数据对内核密度估计进行插值。我们制定并构造了该算法，并演示了其对包括突破曲线在内的偏斜和/或重尾数据进行核密度估计的能力。