A Krylov enhanced proper orthogonal decomposition method for efficient nonlinear model reduction

A model order reduction method is proposed for approximating nonlinear partial differential equations (PDEs). The method attempts to combine the desirable attributes of Krylov reduction and proper orthogonal decomposition (POD) and is entitled Krylov enhanced POD (KPOD). The method approximates nonlinear input/output (I/O) behavior using a sequence of state dependent Krylov subspaces which are obtained via simulation of a nonlinear finite element discretized system. POD is then implemented to extract, from the sequence of subspaces, a projection basis that describes the state dependent variance of the Krylov subspaces. Due to Krylov’s moment matching property, the variance describes how the I/O of a sequence of Krylov reduced models would vary as the original model’s state varies. Galerkin projection is performed using the extracted basis to yield a low dimensional I/O approximation of the original nonlinear model. Reduced order models of an electro-thermal switch generated using the conventional POD and KPOD are presented and used to compare the I/O approximation capabilities of each method. Our results indicate that KPOD reduced models are capable of accurately modeling I/O using a sequence of Krylov subspaces generated from a single nonlinear discretized model simulation while POD can require multiple nonlinear discretized model simulations to generate a reduced model with similar I/O accuracy. The proposed method can be a potential asset for practical applications in the area of I/O behavior prediction and design optimization of nonlinear systems.