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.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.

Hierarchical model reduction of nonlinear partial differential equations based on the adaptive empirical projection method and reduced basis techniques

“…As the dependency of the range of the nonlinear operator on the parameter and the spatial variables is in general non-smooth, we expect that we need many collateral basis functions and interpolating functionals to obtain an accurate approximation. To speed up the (online) computations often localized approximations are considered for instance by constructing (offline) a partition of the parameter space [23,25,64] or the time domain [22] and computing local collateral bases associated with each element of the partition. At the online stage, the correct basis is chosen following a certain criterion.…”

“…However, in all partitioning methods based on the EIM [22,23,25,48,64] the number of interpolating points equals the number of (local) collateral basis functions, which may lead to an insufficient resolution of the (nonsmooth) collateral basis functions and thus a considerably less accurate approximation. Therefore, we propose to perform an adaptive partitioning of the spatial domain driven by a suitable error indicator until a certain tolerance is reached and define the global interpolant as a sum of the local interpolants.…”

PEBL-ROM: Projection-error based local reduced-order models

“…Note that other types of error measures already have been used in partitioning procedures, i.e., an RB error estimator in the hp-RB approach [11] or the empirical interpolation error in the implicit partitioning approach for function approximation [10].…”

“…These considerations have motivated the recent development of novel local model reduction approaches in which smaller local subspaces are defined and the reduced-order models marches from one subspace to another one within each single simulation [1][2][3]. Local subspaces can be defined in time [1,2], parameter space [4][5][6], solution features [7] or state-space [3,6,[8][9][10].…”