Error Analysis of Randomized Symplectic Model Order Reduction for Hamiltonian systems

TL;DR

This paper analyzes the error bounds of randomized symplectic basis generation methods, specifically rcSVD, for Hamiltonian systems, showing that with proper hyperparameters, their projection error is comparable to traditional cSVD methods.

Contribution

It introduces two error bounds for rcSVD in symplectic MOR, demonstrating that with optimal hyperparameters, the error is within a constant factor of cSVD, enhancing efficiency.

Findings

Error bounds for rcSVD depend on hyperparameters

Proper hyperparameter choice makes rcSVD projection error comparable to cSVD

Numerical experiments confirm efficiency and theoretical bounds

Abstract

Solving high-dimensional dynamical systems in multi-query or real-time applications requires efficient surrogate modelling techniques, as e.g., achieved via model order reduction (MOR). If these systems are Hamiltonian systems their physical structure should be preserved during the reduction, which can be ensured by applying symplectic basis generation techniques such as the complex SVD (cSVD). Recently, randomized symplectic methods such as the randomized complex singular value decomposition (rcSVD) have been developed for a more efficient computation of symplectic bases that preserve the Hamiltonian structure during MOR. In the current paper, we present two error bounds for the rcSVD basis depending on the choice of hyperparameters and show that with a proper choice of hyperparameters, the projection error of rcSVD is at most a constant factor worse than the projection error of cSVD. We provide numerical experiments that demonstrate the efficiency of randomized symplectic basis generation and compare the bounds numerically.