A New Reduced Basis Method for Parabolic Equations Based on Single-Eigenvalue Acceleration

TL;DR

This paper introduces the Single Eigenvalue Acceleration Method (SEAM), a novel reduced basis approach for parabolic equations that leverages principal eigenfunctions to improve efficiency under specific discretization assumptions.

Contribution

The paper presents a new RB method, SEAM, which uses principal eigenfunctions of the solution matrix for efficient basis construction in parabolic equations.

Findings

Numerical experiments demonstrate the efficiency of SEAM.

The method effectively approximates solutions with reduced computational cost.

Abstract

In this paper, we develop a new reduced basis (RB) method, named as Single Eigenvalue Acceleration Method (SEAM), for second-order parabolic equations with homogeneous Dirichlet boundary conditions. The high-fidelity numerical method adopts the backward Euler scheme and conforming finite elements for the temporal and spatial discretization, respectively. Under the assumption that the time step size is sufficiently small and time steps are not very large, we show that the singular value distribution of the high-fidelity solution matrix $U$ is close to that of a rank one matrix. We select the eigenfunction associated with the principal eigenvalue of the matrix $U^\top U$ as the basis of the Proper Orthogonal Decomposition (POD) method to obtain SEAM and a parallel SEAM. Numerical experiments confirm the efficiency of the new method.