This paper proposes a dynamical variable-separation method for solving parameter-dependent dynamical systems. To achieve this, we establish a dynamical low-rank approximationfor the solutions of these dynamical systems by successively enriching each term in the reduced basisfunctions via a greedy algorithm. This enables us to reformulate the problem to two decoupledevolution equations at each enrichment step, one of which is a parameter-independent PDE and theother a parameter-dependent ODE. These equations are directly derived from the original dynamicalsystem and the previously separate representation terms. Moreover, the computational processof the proposed method can be split into an offline stage, in which the reduced basis functionsare constructed, and an online stage, in which the efficient low-rank representation of the solutionis employed. The proposed dynamical variable-separation method is capable of offering reducedcomputational complexity and enhanced efficiency compared to many existing low-rank separationtechniques. Finally, we present various numerical results for linear/nonlinear parameter-dependentdynamical systems to demonstrate the effectiveness of the proposed method.

Publication:

SIAM J.SCI COMPUT.

http://dx.doi.org/10.1137/24M168427X

Author:

LIANG CHEN

School of Mathematics, Hunan University, Changsha 410082, People's Republic of China

chl@hnu.edu.cn

YARU CHEN

School of Mathematics, Hunan University, Changsha 410082, People's Republic of China

yrchen@hnu.edu.cn

QIUQI LI

School of Mathematics, Hunan University, Changsha 410082, People's Republic of China

qli28@hnu.edu.cn

TAO ZHOU

LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190,People's Republic of China

tzhou@lsec.cc.ac.cn