Abstract
The main goal of this paper is to define and study new methods for the computation of effective coefficients in the homogenization of divergence-form operators with random coefficients. The methods introduced here are proved to have optimal computational complexity and are shown numerically to display small constant prefactors. In the spirit of multiscale methods, the main idea is to rely on a progressive coarsening of the problem, which we implement via a generalization of the Green–Kubo formula. The technique can be applied more generally to compute the effective diffusivity of any additive functional of a Markov process. In this broader context, we also discuss the alternative possibility of using Monte Carlo sampling and show how a simple one-step extrapolation can considerably improve the performance of this alternative method.
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Notes
In order to estimate the slope of the regression line of the set of blue dots in [24, Figure 14], I first drew the straight line going through the extremal blue dots and observed that all the other dots are very close to this line. I then measured the coordinates of the extremal dots to be \((3.59, -\,0.62)\) and \((9.89,-\,3.06)\), respectively, which yields that the dx / dy slope of the line is about \(-\,2.58\). Moreover, the seven dots closest to the leftmost blue point are below this line, while the three dots closest to the rightmost blue point are above this line, and thus essentially any other reasonable choice of pair of points to draw a line and measure the slope from would yield a larger absolute slope.
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Acknowledgements
I would like to thank Josselin Garnier for an inspiring talk which motivated me to revisit this problem, Tony Lelièvre for his helpful feedback and Harmen Stoppels for his precious help with the Julia language. This work has been partially supported by the ANR Grant LSD (ANR-15-CE40-0020-03).
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Communicated by Endre Suli.
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Mourrat, JC. Efficient Methods for the Estimation of Homogenized Coefficients. Found Comput Math 19, 435–483 (2019). https://doi.org/10.1007/s10208-018-9389-9
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DOI: https://doi.org/10.1007/s10208-018-9389-9