Multiplicative Operator Splittings in Nonlinear Di usion: from Spatial Splitting to Multiple Timesteps

Operator splitting is a powerful concept used in many diversed elds of applied mathematics for the design of eeective numerical schemes. Following the success of the additive operator splitting (AOS) in performing an eecient nonlinear diiusion ltering on digital images, we analyze the possibility of using multiplicative operator splittings to process images from diierent perspectives. We start by examining the potential of using fractional step methods to design a multiplicative operator splitting as an alternative to AOS schemes. By means of a Strang splitting, we attempt to use numerical schemes that are known to be more accurate in linear diiusion processes and apply them on images. Initially we implement the Crank-Nicolson and DuFort-Frankel schemes to diiuse noisy signals in one dimension and devise a simple extrapolation that enables the Crank-1 Nicolson to be used with high accuracy on these signals. We then combine the Crank-Nicolson in 1D with various multiplicative operator splittings to process images. Based on these ideas we obtain some interesting results. However, from the practical standpoint, due to the computational expenses associated with these schemes and the questionable beneets in applying them to perform nonlinear diiusion ltering when using long timesteps, we conclude that AOS schemes are simple and eecient compared to these alternatives. We then examine the potential utility of using multiple timestep methods combined with AOS schemes, as means to expedite the diiusion process. These methods were developed for molecular dynamics applications and are used eeciently in biomolecular simulations. The idea is to split the forces exerted on atoms into diierent classes according to their behavior in time, and assign longer timesteps to nonlocal, slowly-varying forces such as the Coulomb and van der Waals interactions , whereas the local forces like bond and angle are treated with smaller timesteps. Multiple timestep integrators can be derived from the Trotter factorization, a decomposition that bears a strong resemblance to a Strang splitting. Both formulations decompose the time propagator into trilateral products to construct multiplicative operator splittings which are second order in time, with the possibility of extending the factorization to higher order expansions. While a Strang splitting is a decomposition across spatial dimensions, where each dimension is subsequently treated with a fractional step, the multiple timestep method is a decomposition across scales. Thus, multiple timestep methods are a realization of the multiplicative operator splitting idea. For certain nonlinear diiusion coeecients with favorable properties, we show that a …