Improved Statistical Methods for Elliptic Stochastic Homogenization Problems : Application of Multi Level- and Multi Index Monte Carlo on Elliptic Stochastic Homogenization Problems

Abstract: In numerical multiscale methods, one relies on a coupling between macroscopic model and a microscopic model. The macroscopic model does not include the microscopic properties that the microscopic model offers and that are vital for the desired solution. Such microscopic properties include parameters like material coefficients and fluxes which may variate microscopically in the material. The effective values of this data can be computed by running local microscale simulations while averaging the microscopic data. One desires the effect of the microscopic coefficients on a macroscopic scale, and this can be done using classical homogenisation theory. One method in the homogenization theory is to use local elliptic cell problems in order to compute the homogenized constants and this results in error where is the wavelength of the microscopic variations and is the size of the simulation domain. However, one could greatly improve the accuracy by a slight modification in the homogenisation elliptic PDE and use a filter in the averaging process to get much better orders of error. The modification relates the elliptic PDE to a parabolic one, that could be solved and integrated in time to get the elliptic PDE's solution.   In this thesis I apply the modified elliptic cell homogenization method with a qth order filter to compute the homogenized diffusion constant in a 2d Poisson equation on a rectangular domain. Two cases were simulated. The diffusion coefficients used in the first case was a deterministic 2d matrix function and in the second case I used stochastic 2d matrix function, which results in a 2d stochastic differential equation (SDE). In the second case two methods were used to determine the expected value of the homogenized constants, firstly the multi-level Monte Carlo (MLMC) and secondly its generalization multi-index Monte Carlo (MIMC). The performance of MLMC and MIMC is then compared when used in the process of the homogenization.   In the homogenization process the finite element notations in 2d were used to estimate a solution of the Poisson equation. The grid spatial steps were varied in a first order differences in MLMC (square mesh) and first order mixed differences in MIMC (which allows for rectangular mesh).