Implicit Partitioning Methods for Unknown Parameter Domains

The key condition for the application of the Reduced Basis Method (RBM) to Parametrized Partial Differential Equations (PPDEs) is the availability of affine decompositions of the systems in parameter and space. The efficiency of the RBM depends on both the number of reduced basis functions and the number of affine terms. A possible way to reduce the costs is a partitioning of the parameter domain. One creates separate RB spaces [6] and affine decompositions [4] on each subdomain. Since the solutions are supposed to be smooth in parameter, the variation of the solutions on a subdomain becomes small and only few basis functions and affine terms are needed. Based upon the Empirical Interpolation Method (EIM), we generalize the existing partitioning concepts to arbitrary input functions with possibly unknown, high-dimensional, or even without direct parameter dependencies. No a-priori information about the input is necessary. We create affine decomposition and partitions without the knowledge of either an explicit description of the parameter domain or of the form of the partitions. An application includes PPDEs with stochastic influences [7,12]. For a probability space (Ω,F ,P), the parameter domain is now associated with Ω. Each element ω ∈ Ω represents a stochastic event. Hence, ω is not a parameter in classical sense and there usually is no explicit description of Ω.