  A central question in numerical homogenization of partial differential
equations with multiscale coefficients is the accurate computation of effective
quantities, such as the homogenized coefficients. Computing homogenized
coefficients requires solving local corrector problems followed by upscaling
relevant local data. The most naive way of computing homogenized coefficients
is by solving a local elliptic problem, which is known to suffer from the
so-called resonance error dominating all other errors inherent in multiscale
computations. A far more efficient modelling strategy, based on adding an
exponential correction term to the standard local elliptic problem, has
recently been proved to result in exponentially decaying error bounds with
respect to the size of the local geometry. The questions in relation with the
accuracy and computational efficiency of this approach has been previously
addressed in the context of periodic homogenization. The present article
concerns the extension of mathematical and numerical study of this modified
elliptic corrector problem to stochastic homogenization problems. In
particular, we assume a stationary, ergodic micro-structure and i) establish
the well-posedness of the corrector equation, ii) analyse the bias (or the
systematic error) originating from additional exponential correction term in
the model. Numerical results corroborating our theoretical findings are
presented.
