High dimensional finite elements for elliptic problems with multiple scales and stochastic data

Multiple scale homogenization problems are reduced to single scale problems in higher dimension. It is shown that sparse tensor product Finite Element Methods (FEM) allow the numerical solution in complexity independent of the dimension and of the length scale. Problems with stochastic input data are reformulated as high dimensional deterministic problems for the statistical moments of the random solution. Sparse tensor product FEM give a deterministic solution algorithm of log-linear complexity for statistical moments.