Non-linear model-order reduction based on tensor decomposition and matrix product

In this study, based on tensor decomposition and matrix product, the authors investigate two model-order reduction (MOR) methods for the quadratic-bilinear (QB) systems which are equivalently transformed from the non-linear input-output systems. Since the quadratic term coefficient of the QB system can be considered as the matricisation of a tensor, they propose two computationally efficient ways to obtain the reduced system by using tensor calculus. First, the Tucker decomposition of tensors is used to deal with the quadratic term coefficient of the QB system. The transformational matrix is constructed by applying the analysis of matrix product. Then, they get the reduced QB system which can match the first several expansion coefficients of the original output. Besides, they propose another MOR method based on the CANDECOMP/PARAFAC decomposition. These two methods can avoid large computational complexity in the process of computing the reduced system. Moreover, the error estimation and stability of the authors' MOR methods are discussed. The efficiency of their MOR methods is illustrated by two numerical examples and they show their competitiveness when compared to the proper orthogonal decomposition method.