Linear model reduction using spectral proper orthogonal decomposition

TL;DR

This paper introduces a novel spectral method for linear model reduction that efficiently captures entire trajectories using SPOD modes, outperforming traditional spatial basis methods in accuracy and speed.

Contribution

The paper develops the spectral solution operator projection (SSOP) method for linear systems, leveraging SPOD modes to improve trajectory representation and prediction accuracy.

Findings

SSOP significantly reduces modeling error compared to POD-Galerkin and balanced truncation.

The method is computationally efficient, with CPU times comparable to or less than benchmarks.

A data-free space-time variant of SSOP also outperforms balanced truncation in accuracy.

Abstract

Most model reduction methods reduce the state dimension and then temporally evolve a set of coefficients that encode the state in the reduced representation. In this paper, we instead employ an efficient representation of the entire trajectory of the state over some time interval of interest and then solve for the static coefficients that encode the trajectory on the interval. We use spectral proper orthogonal decomposition (SPOD) modes, which are provably optimal for representing long trajectories and substantially outperform any representation of the trajectory in a purely spatial basis (e.g., POD). We develop a method to solve for the SPOD coefficients that encode the trajectories for forced linear dynamical systems given the forcing and initial condition, thereby obtaining the accurate prediction of the dynamics afforded by the SPOD representation of the trajectory. The method, which we refer to as spectral solution operator projection (SSOP), is derived by projecting the general time-domain solution for a linear time-invariant system onto the SPOD modes. We demonstrate the new method using two examples: a linearized Ginzburg-Landau equation and an advection-diffusion problem. In both cases, the error of the proposed method is orders of magnitude lower than that of POD-Galerkin projection and balanced truncation. The method is also fast, with CPU time comparable to or lower than both benchmarks in our examples. Finally, we describe a data-free space-time method that is a derivative of the proposed method and show that it is also more accurate than balanced truncation in most cases.