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Explore a lecture on variance reduction techniques in random homogenization, focusing on special quasirandom structures. Delve into the joint work of William Minvielle, C. Le Bris, and F. Legoll from École des Ponts and INRIA. Examine the homogenization of a random, linear elliptic second order partial differential equation on a bounded domain in Rd, with a random diffusion coefficient matrix field. Learn about the convergence to a homogenized problem and the deterministic constant matrix A* derived from the corrector function. Understand the practical approximation of the corrector problem on a bounded domain QN and the resulting random, apparent homogenized matrix A*N(ω). Discover a variance reduction approach to obtain approximations of A* with smaller variance, reducing statistical error. Investigate conditions for selecting finite supercell environments to solve the corrector equation, including exact fraction in a bi-composite. This 47-minute lecture, part of the Hausdorff Trimester Program on Optimal Transportation and its Applications, was presented at the Hausdorff Center for Mathematics on January 29, 2015.
