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Stochastic homogenization: Theory and numerics

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In this chapter, we pursue two related goals. First, we derive a theoretical stochastic homogenization result for the stochastic forward problem introduced in the first chapter. The key ingredient to obtain this result is the use of the Feynman-Kac formula for the complete electrode model. The proof is constructive in the sense that it yields a strategy to achieve our second goal, the numerical approximation of the effective conductivity. In contrast to periodic homogenization, which is well understood, numerical homogenization of random media still poses major practical challenges. In order to cope with these challenges, we propose a new numerical method inspired by a highly efficient stochastic method from the physics literature that was invented by Torquato, Kim and Cule [159]. From a mathematical point of view, the novelty of the proposed method lies in the fact that it is based on the aforementioned stochastic homogenization result, that is, on simulation of the underlying diffusion process rather than on the usual discretization of the so-called auxiliary problem.