A Computationally Optimal Randomized Proper Orthogonal Decomposition Technique

TL;DR

This paper introduces a randomized proper orthogonal decomposition method that significantly reduces computational costs for model reduction in large-scale systems, outperforming existing techniques in efficiency and effectiveness.

Contribution

The paper presents a novel RPOD* technique that is computationally optimal, requiring fewer snapshots and outperforming traditional BPOD methods in efficiency and accuracy.

Findings

RPOD* is orders of magnitude cheaper than BPOD.

RPOD* achieves better performance than BPOD output projection.

RPOD* requires minimal snapshots for optimal results.

Abstract

In this paper, we consider the model reduction problem of large-scale systems, such as systems obtained through the discretization of partial differential equations. We propose a computationally optimal randomized proper orthogonal decomposition (RPOD*) technique to obtain the reduced order model by perturbing the primal and adjoint system using Gaussian white noise. We show that the computations required by the RPOD* algorithm is orders of magnitude cheaper when compared to the balanced proper orthogonal decomposition (BPOD) algorithm and BPOD output projection algorithm while the performance of the RPOD* algorithm is much better than BPOD output projection algorithm. It is optimal in the sense that a minimal number of snapshots is needed. We also relate the RPOD* algorithm to random projection algorithms. The method is tested on two advection-diffusion equations.