Refined Subspace Projection for Model Reduction via Interpolation and Tensor Decomposition

Research on nonlinear model order reduction has revealed that as nonlinearity increases, the subspaces capturing dominant information require more complex bases. The complexity is influenced by two main factors: coefficients and approximation criteria. On one hand, it is affected by the characteristics of all coefficients. Therefore, we begin by introducing a generalized Gramian-based method for estimating eigenvalue decay, which demonstrates the factors on the reduced order. On the other hand, existing interpolation conditions based on transfer functions must match multiple features with different bases. This article confirms that kernels, as new criteria, can match all features of a large class of nonlinear systems using the same basis as the linear part. We propose a $\mathit{k}$-dimensional refined space for affine input/output nonlinear systems, in contrast to existing methods that require at least an $\mathit{O(nk)}$-dimensional space to match $\mathit{n}$ transfer functions at $\mathit{k}$ interpolating points, even for $\mathbf{0}$th moment matching. To expand the applicability, we present a rank-$\mathbf{R}$ quadratic approximation to transform general systems into normalized affine input/output nonlinear systems. In terms of computational efficiency, we propose a parallel partial columnwise least-squares method to further reduce the rank. Finally, we provide two numerical examples to illustrate the effectiveness of our method.