# Variance Reduction: Noise Reduction with No Approximation

**Principal Investigator**
Nicolas Hadjiconstantinou

**Project Website**
http://web.mit.edu/ngh-group/research3-variance.html

Although powerful, Monte Carlo simulations of the Boltzmann transport equation suffer from statistical noise (uncertainty), which can make simulations prohibitively expensive in the low-signal limit. The statistical uncertainty of such approaches can be drastically reduced using the variance-reduction method of control variates, which in the present context can be applied by simulating only the deviation from equilibrium. In other words, by noting that small signals (e.g. low Mach number, or low temperature differences) result from small deviations from equilibrium, simulating the deviation from equilibrium can be used as a general solution to these limitations, without introducing an approximation.

Simulating only the deviation from equilibrium is achieved by decomposing the non-equilibrium distribution into an equilibrium and a (small) non-equilibrium component, and developing stochastic particle dynamics for simulating only the non-equilibrium component. Variance reduction is achieved because the Monte Carlo procedure is used to evaluate only a small part of the solution (the equilibrium part and its moments are known analytically and thus do not contribute to the noise).

This formulation can capture arbitrarily small devations from equilibrium, since as the signal becomes smaller the uncertainty becomes smaller and thus the cost of achieving the same signal-to-noise ratio. In contrast, the cost of regular Monte Carlo grows quadratically for the same signal-to-noise ratio as the signal decreases. We show a comparison between a variance-reduced and a regular Monte Carlo simulation of phonon transport in porous Silicon.