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

TL;DR

This work introduces the dynamical variable-separation (DVS) method to efficiently solve parameter-dependent dynamical systems by constructing a time-dependent, low-rank representation $u_N=\sum_{i=1}^{N}\zeta_i(t;\bm{\xi})g_i(\bm{x},t)$. At each enrichment step, two uncoupled evolution equations are solved: a parameter-independent PDE for the space-time basis $g_k$ and a parameter-dependent ODE for the coefficient basis $\zeta_k$, enabling an offline-online decomposition. The affine parameter decomposition and greedy enrichment with an error estimator drive the offline basis construction, while the online stage assembles the solution from precomputed scalars and basis functions, yielding reduced computational complexity. Numerical experiments on linear (reaction-diffusion, heat) and nonlinear (Burgers, Allen-Cahn) systems demonstrate that DVS achieves accurate approximations with favorable online efficiency, often outperforming competing reduced-basis strategies like VS and MTD. The approach provides a practical framework for rapid parametric simulations in design, optimization, and uncertainty quantification.

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.

Figures (14)

• Figure 1: Comparison of the two strategies to choose iteration points.
• Figure 2: Comparison of the relative error.
• Figure 3: Basis fields $g_i(\bm{x},t)$ of the DVS approximation.
• Figure 4: The average relative error corresponding to the number of the separate terms $N$.
• Figure 5: The average relative error corresponding to the different numbers of the separate terms $N$.

Theorems & Definitions (6)

• Remark 3.1
• Definition A.1: Local logarithmic Lipschitz constant
• Lemma A.1: Comparison lemma Wirtz2014aFriess2017DynamicalMR
• Proposition A.1
• proof
• Remark A.1