Nonlinear model reduction via discrete empirical interpolation

Abstract

This thesis proposes a model reduction technique for nonlinear dynamical systems based upon combining Proper Orthogonal Decomposition (POD) and a new method, called the Discrete Empirical Interpolation Method (DEIM). The popular method of Galerkin projection with POD basis reduces dimension in the sense that far fewer variables are present, but the complexity of evaluating the nonlinear term generally remains that of the original problem. DEIM, a discrete variant of the approach from [11], is introduced and shown to effectively overcome this complexity issue. State space error estimates for POD-DEIM reduced systems are also derived. These [Special characters omitted.] error estimates reflect the POD approximation property through the decay of certain singular values and explain how the DEIM approximation error involving the nonlinear term comes into play. An application to the simulation of nonlinear miscible flow in a 2-D porous medium shows that the dynamics of a complex full-order system of dimension 15000 can be captured accurately by the POD-DEIM reduced system of dimension 40 with a factor of [Special characters omitted.] (1000) reduction in computational time.