Implicit partitioning methods for unknown parameter sets

Abstract

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 equations 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 (Haasdonk et al., Math. Comput.Model. Dyn. Syst. 17(4), 423---442, 2011) and affine decompositions (Eftang and Stamm, Int. J. Numer. Methods Eng. 90(4), 412---428, 2012) 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) (Barrault et al., C. R. Math. Acad. Sci. Paris 339(9), 667---672, 2004 and Tonn, 2012), 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. The main idea is to perform several EIMs such that different parts of the family of possible input functions are covered. For a new input in the online stage, the coefficients of the resulting affine decomposition are used for the assignment to the appropriate EIM. An application includes PPDEs with stochastic influences (Chen et al., Math. Model. Numer. Anal. 48(4), 943---953, 2014, Haasdonk et al., SIAM/ASA J. Uncertain. Quantif. 1, 79---105, 2013 and Wieland, 2013). For some probability space, the parameter domain is now associated with the set of possible random outcomes. Hence, the elements of the parameter set are not parameters in classical sense and there often is no explicit description of the class of outcomes.