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

Abstract

In this paper we extend the hierarchical model reduction framework based on reduced basis techniques recently introduced in [46] for the application to nonlinear partial differential equations. The major new ingredient to accomplish this goal is the introduction of the adaptive empirical projection method, which is an adaptive integration algorithm based on the (generalized) empirical interpolation method [4,40]. Different from other partitioning concepts for the empirical interpolation method we perform an adaptive decomposition of the spatial domain. We project both the variational formulation and the range of the nonlinear operator onto reduced spaces. Those reduced spaces combine the full dimensional (finite element) space in an identified dominant spatial direction and a reduction space or collateral basis space spanned by modal orthonormal basis functions in the transverse direction. Both the reduction and the collateral basis space are constructed in a highly nonlinear fashion by introducing a parametrized problem in the transverse direction and associated parametrized operator evaluations, and by applying reduced basis methods to select the bases from the corresponding snapshots. Rigorous a priori and a posteriori error estimators, which do not require additional regularity of the nonlinear operator are proven for the adaptive empirical projection method and then used to derive a rigorous a posteriori error estimator for the resulting hierarchical model reduction approach. Numerical experiments for an elliptic nonlinear diffusion equation demonstrate a fast convergence of the proposed dimensionally reduced approximation to the solution of the full-dimensional problem. Runtime experiments verify a close to linear scaling of the reduction method in the number of degrees of freedom used for the computations in the dominant direction.

Adaptive basis construction and improved error estimation for parametric nonlinear dynamical systems

“…From our experience in the numerical tests, setting different tolerances for the RB and EIM approximation, with tol EI < tol RB proves to give the best approximation. A similar observation is also noted in Reference 50; however, no simultaneous enrichment is considered there. Still, it is not entirely clear how small the (D)EIM approximation tolerance has to be when compared to the RB tolerance.…”

“…Previous works dealing with the simultaneous enrichment of RB and (D)EIM bases have noted the issue of (D)EIM plateaus. In fact, two different notions of plateauing have been observed and presented in References 5,16,48‐50, and 51. In References 5,50, and 51, the authors note that when the number of basis elements of the EIM approximation is fixed at some small value, an increase in the number of RB vectors does not result in an improvement in the overall error.…”

A-posteriori reduced basis error-estimates for a semi-discrete in space quasilinear parabolic PDE

“…In practice, however, in order to obtain an accurate reduced-order model and corresponding small errorestimates, it will be necessary to construct RB-and EIM-bases in such a way that the size of residuals and EIM-errors is balanced appropriately. We do not address this issue here and refer for instance to [19,60].…”

(Parametrized) First Order Transport Equations: Realization of Optimally Stable Petrov--Galerkin Methods

“…Alternatively, to obtain a computational more feasible offline stage one might let the strong greedy run on a small test set with relatively high tolerance and use a hierarchical a posteriori error estimator on the large(r) training set, which was proposed in a slightly different context in [24]. Another idea might be to keep a second test training set during the greedy algorithm.…”