Symplectic Model Order Reduction with Non-Orthonormal Bases

Patrick Buchfink,Bernard Haasdonk,Ashish Bhatt

doi:10.3390/mca24020043

Abstract

Parametric high-fidelity simulations are of interest for a wide range of applications. However, the restriction of computational resources renders such models to be inapplicable in a real-time context or in multi-query scenarios. Model order reduction (MOR) is used to tackle this issue. Recently, MOR is extended to preserve specific structures of the model throughout the reduction, e.g., structure-preserving MOR for Hamiltonian systems. This is referred to as symplectic MOR. It is based on the classical projection-based MOR and uses a symplectic reduced order basis (ROB). Such an ROB can be derived in a data-driven manner with the Proper Symplectic Decomposition (PSD) in the form of a minimization problem. Due to the strong nonlinearity of the minimization problem, it is unclear how to efficiently find a global optimum. In our paper, we show that current solution procedures almost exclusively yield suboptimal solutions by restricting to orthonormal ROBs. As a new methodological contribution, we propose a new method which eliminates this restriction by generating non-orthonormal ROBs. In the numerical experiments, we examine the different techniques for a classical linear elasticity problem and observe that the non-orthonormal technique proposed in this paper shows superior results with respect to the error introduced by the reduction.

Highlights

Simulations enable researchers of all fields to run virtual experiments that are too expensive or impossible to be carried out in the real world
We prove that Complex SVD is the optimal solution of the Proper Symplectic Decomposition (PSD) on the set of orthonormal, symplectic bases
We show that PSD Complex SVD is the solution of the PSD in the subset of symplectic, orthonormal reduced order basis (ROB)

Summary

Introduction

Simulations enable researchers of all fields to run virtual experiments that are too expensive or impossible to be carried out in the real world. High-fidelity models are indispensable to represent the simulated process accurately. These high-fidelity simulations typically come with the burden of a large computational cost such that an application in real time or an evaluation for many different parameters is impossible respecting the given restrictions of computational resources at hand. One class of high-fidelity models are systems of ordinary differential equations (ODEs) with a high order, i.e., a high dimension in the unknown variable. Such models typically arise from fine discretizations of time-dependent partial differential equations (PDEs). Examples are linear elastic models [2] or gyro systems [3]

Objectives

Results

Conclusion