Spatially correlated noise and variance minimization in stochastic simulations

Brownian simulation methods have become a popular approach in computational rheology with the introduction of the CONNFFESSIT algorithm and the method of Brownian configuration fields in the 1990s. Jourdain et al. [B. Jourdain, C.L. Bris, T. Lelievre, On a variance reduction technique for micro-macro simulations of polymeric fluids, J. Non-Newton. Fluid Mech. 122 (2004) 91-106] pointed out that both methods can be viewed as variants that differ in the spatial correlation of the noise, which can be viewed as a computational parameter for statistical error minimization. We formulate an optimization problem of variance minimization with respect to the choice of noise correlation. Our analysis takes place in an infinite-dimensional function space. We solve the optimization problem analytically for the shear flow of a Hookean dumbbell model at steady state. Interestingly, we find that spatially uncorrelated noise, i.e., CONNFFESSIT minimizes the statistical error, although the precise meaning of this statement can only be interpreted as a limit of finite-dimensional approximations.