Convolution Kernels and Stability of Threshold Dynamics Methods

Threshold dynamics and its extensions have proven useful in computing interfacial motions in applications as diverse as materials science and machine learning. Certain desirable properties of the algorithm, such as unconditional monotonicity in two-phase flows and gradient stability more generally, hinge on positivity properties of the convolution kernel and its Fourier transform. Recent developments in the analysis of this class of algorithms indicate that sometimes, as in the case of certain anisotropic curvature flows arising in materials science, these properties of the convolution kernel cannot be expected. Other applications, such as machine learning, would benefit from as great a level of flexibility in choosing the convolution kernel as possible. In this paper, we establish certain desirable properties of threshold dynamics, such as gamma convergence of its associated energy, for a substantially wider class of kernels than has been hitherto possible. We also present variants of the algorithm that extend some of these properties to even wider classes of convolution kernels.