A Dynamical Variable-separation Method for Parameter-dependent Dynamical Systems

TL;DR

This paper introduces a dynamical variable-separation method that efficiently solves parameter-dependent dynamical systems by low-rank approximation, decoupling equations, and offline-online computation stages, demonstrating improved efficiency.

Contribution

The paper develops a novel dynamical variable-separation approach with a greedy algorithm for low-rank approximation, enabling efficient decoupling and solution of parameter-dependent systems.

Findings

Reduces computational complexity compared to existing methods.

Effective for both linear and nonlinear systems.

Demonstrates high efficiency in numerical experiments.

Abstract

This paper proposes a dynamical Variable-separation method for solving parameter-dependent dynamical systems. To achieve this, we establish a dynamical low-rank approximation for the solutions of these dynamical systems by successively enriching each term in the reduced basis functions via a greedy algorithm. This enables us to reformulate the problem to two decoupled evolution equations at each enrichment step, of which one is a parameter-independent partial differential equation and the other one is a parameter-dependent ordinary differential equation. These equations are directly derived from the original dynamical system and the previously separate representation terms. Moreover, the computational process of the proposed method can be split into an offline stage, in which the reduced basis functions are constructed, and an online stage, in which the efficient low-rank representation of the solution is employed. The proposed dynamical Variable-separation method is capable of offering reduced computational complexity and enhanced efficiency compared to many existing low-rank separation techniques. Finally, we present various numerical results for linear/nonlinear parameter-dependent dynamical systems to demonstrate the effectiveness of the proposed method.