In the spirit of renormalization group, we propose an alternative discretization procedure, namely, coarse graining the system instead of uniform sampling. This procedure was tested in 1D on several models such as diffusion equation, model A, and Swift-Hohenberg equation. We find that coarse graining enables us to faithfully reproduce results obtained from a fine lattice by using a lattice with grid size as much as 8 times larger. Simulation is further speeded up since one can now choose a bigger time step size. Higher dimension should yield better result.
Further work is needed to study corresponding coarse graining of the evolution equation itself.
Dynamics of Forward Rate Curve. Term structure of interest rates is very important for pricing derivatives. Various models have been devised to study the dynamics of spot rate movement, which is then used to build the whole term structure. In contrast, we model the forward rate curve itself. Specifically, forward rate curve is decomposed into two parts, one is a two factor component dependent on spot rate and the forward rate with longest time to maturity, the remainder is treated like a string with line tension following Bouchaud et al .