Nonlinear Model Reduction via Discrete Empirical Interpolation

TLDR

Standard POD‑Galerkin reduces dimensionality but retains the original nonlinear evaluation cost, whereas the empirical interpolation method (EIM) lowers this cost to be proportional to the reduced variables. The paper proposes a discrete empirical interpolation method (DEIM) that dramatically reduces the computational complexity of POD‑based reduced‑order models for nonlinear PDEs and ODEs, and presents a simplified finite‑dimensional EIM with an error bound. DEIM is applicable to ODEs from finite‑difference discretizations of time‑dependent PDEs or parametrically dependent steady‑state problems, and can be extended to arbitrary nonlinear ODE systems with minor modifications. Applying DEIM to the FitzHugh–Nagumo equations reduced the state space from 1024 to about five variables with negligible error over long‑time integration, and similar reductions and accuracy were achieved in higher spatial dimensions.

Abstract

A dimension reduction method called discrete empirical interpolation is proposed and shown to dramatically reduce the computational complexity of the popular proper orthogonal decomposition (POD) method for constructing reduced-order models for time dependent and/or parametrized nonlinear partial differential equations (PDEs). In the presence of a general nonlinearity, the standard POD-Galerkin technique reduces dimension in the sense that far fewer variables are present, but the complexity of evaluating the nonlinear term remains that of the original problem. The original empirical interpolation method (EIM) is a modification of POD that reduces the complexity of evaluating the nonlinear term of the reduced model to a cost proportional to the number of reduced variables obtained by POD. We propose a discrete empirical interpolation method (DEIM), a variant that is suitable for reducing the dimension of systems of ordinary differential equations (ODEs) of a certain type. As presented here, it is applicable to ODEs arising from finite difference discretization of time dependent PDEs and/or parametrically dependent steady state problems. However, the approach extends to arbitrary systems of nonlinear ODEs with minor modification. Our contribution is a greatly simplified description of the EIM in a finite-dimensional setting that possesses an error bound on the quality of approximation. An application of DEIM to a finite difference discretization of the one-dimensional FitzHugh–Nagumo equations is shown to reduce the dimension from 1024 to order 5 variables with negligible error over a long-time integration that fully captures nonlinear limit cycle behavior. We also demonstrate applicability in higher spatial dimensions with similar state space dimension reduction and accuracy results.