A Posteriori Error Estimation for DEIM Reduced Nonlinear Dynamical Systems

In this work an efficient approach for a-posteriori error estimation for POD-DEIM reduced nonlinear dynamical systems is introduced. The considered nonlinear systems may also include time and parameter-affine linear terms as well as parametrically dependent inputs and outputs. The reduction process involves a Galerkin projection of the full system and approximation of the system’s nonlinearity by the DEIM method [Chaturantabut & Sorensen (2010)]. The proposed a-posteriori error estimator can be efficiently decomposed in an offline/online fashion and is obtained by a one dimensional auxiliary ODE during reduced simulations. Key elements for efficient online computation are partial similarity transformations and Matrix-DEIM approximations of the nonlinearity Jacobians. The theoretical results are illustrated by application to an unsteady Burgers equation and a cell apoptosis model.