Renormalization group and perfect operators for stochastic differential equations

We develop renormalization group methods for solving partial and stochastic differential equations on coarse meshes. Renormalization group transformations are used to calculate the precise effect of small scale dynamics on the dynamics at the mesh size. The fixed point of these transformations yields a perfect operator: an exact representation of physical observables on the mesh scale with minimal lattice artifacts. We apply the formalism to simple nonlinear models of critical dynamics, and show how the method leads to an improvement in the computational performance of Monte Carlo methods.