  Quasi-equilibrium approximation is a widely used closure approximation
approach for model reduction with applications in complex fluids, materials
science, etc. It is based on the maximum entropy principle and leads to
thermodynamically consistent coarse-grain models. However, its high
computational cost is a known barrier for fast and accurate applications.
Despite its good mathematical properties, there are very few works on the fast
and efficient implementations of quasi-equilibrium approximations. In this
paper, we give efficient implementations of quasi-equilibrium approximations
for antipodally symmetric problems on unit circle and unit sphere using
polynomial and piecewise polynomial approximations. Comparing to the existing
methods using linear or cubic interpolations, our approach achieves high
accuracy (double precision) with much less storage cost. The methods proposed
in this paper can be directly extended to handle other moment closure
approximation problems.
