In this paper, we investigate the use of multilinear algebra for reducing the order of multidimensional linear time-invariant (MLTI) systems. Our main tools are tensor rational Krylov subspace methods, which enable us to approximate the systems solution within a low-dimensional subspace. We introduce the tensor rational block Arnoldi and tensor rational block Lanczos algorithms. By utilizing these methods, we develop a model reduction approach based on projection techniques. Additionally, we demonstrate how these approaches can be applied to large-scale Lyapunov tensor equations, which are critical for the balanced truncation method, a well-known technique for order reduction. An adaptive method for choosing the interpolation points is also introduced. Finally, some numerical experiments are reported to show the effectiveness of the proposed adaptive approaches.

翻译：本文研究了利用多重线性代数降低多维线性时不变(MLTI)系统阶数的方法。我们的主要工具是张量有理Krylov子空间方法，该方法使我们能够在低维子空间内逼近系统解。我们提出了张量有理块Arnoldi算法和张量有理块Lanczos算法。通过运用这些方法，我们开发了基于投影技术的模型降阶方法。此外，我们展示了如何将这些方法应用于大规模Lyapunov张量方程——这对于平衡截断法（一种著名的降阶技术）至关重要。本文还提出了一种选择插值点的自适应方法。最后，通过数值实验验证了所提自适应方法的有效性。