Let$X$be a compact Hausdorff space. A kernel function on$X$×$X$, enjoying additional properties, naturally defines a semi-inner product structure on certain subspaces of all finite signed Borel measures on$X$. This paper discusses the question of completeness of such spaces.
We propose a generalized multiscale finite element method (GMsFEM) based on clustering algorithm to study the elliptic PDEs with random coefficients in the multi-query setting. Our method consists of offline and online stages. In the offline stage, we construct a small number of reduced basis functions within each coarse grid block, which can then be used to approximate the multiscale finite element basis functions. In addition, we coarsen the corresponding random space through a clustering algorithm. In the online stage, we can obtain the multiscale finite element basis very efficiently on a coarse grid by using the pre-computed multiscale basis. The new GMsFEM can be applied to multiscale SPDE starting with a relatively coarse grid, without requiring the coarsest grid to resolve the smallest-scale of the solution. The new method offers considerable savings in solving multiscale SPDEs. Numerical results are
Let X, Y be realcompact spaces or completely regular spaces consisting of X, Y -points. Let X, Y be a linear bijective map from X, Y (resp. X, Y ) onto X, Y (resp. X, Y ). We show that if X, Y preserves nonvanishing functions, that is,