Parametric model order reduction based on parallel tensor compression

Abstract

In this paper, we for the first time explore the model order reduction (MOR) of parametric systems based on the tensor techniques and a parallel tensor compression algorithm. For the parametric system characterising multidimensional parameter space and nonlinear parametric dependence, we first approximate the system matrices by tensor functions of the parameters, whose first-order coefficients are third-order tensors. In order to effectively reduce the computational cost and the storage burden, we propose a parallel tensor compression algorithm based on Tensor-SVD to deal with the tensors in the tensor functions. Then, we obtain the low-rank approximation in Kruskal form of third-order tensors. After that, by computing the first several expansion coefficients of the state variable with the selected parameter vectors, the projection matrix is constructed to obtain the reduced parametric system. Theoretical analysis shows that the reduced parametric system can match the first several expansion coefficients of the output variable of the original system at the selected parameter vectors. Moreover, the stability of the proposed MOR method is discussed. Finally, the efficiency of the proposed method is illustrated by two numerical examples.

Keywords:

• Parametric system
• model order reduction
• tensor compression
• parallel
• stability

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the Natural Science Foundation of China (NSFC) under grant 11871393 and the International Science and Technology Cooperation Program of Shaanxi Key Research & Development Plan under grant 2019KWZ-08.

Notes on contributors

Zhen Li

Zhen Li is currently pursuing Ph.D at the School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, People's Republic of China. His research interests are model order reduction, tensor theory and parallel computation.

Yao-Lin Jiang

Yao-Lin Jiang is a full professor at the School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, Shaanxi, China. He has published 5 books and about 310 papers in journals. His research interests include theoretical studies of engineering problems, model order reduction, waveform relaxation, numerical solutions of differential equations, circuit simulation, and parallel processing.

Hong-liang Mu

Hong-Liang Mu is currently pursuing master's degree at the School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, People's Republic of China. His research interests are model order reduction and parallel computation.