# Symplectic Model Order Reduction with Non-Orthonormal Bases

## TL;DR

This paper introduces a novel symplectic model order reduction method that generates non-orthonormal bases, improving accuracy over traditional orthonormal approaches in structure-preserving MOR for Hamiltonian systems.

## Contribution

It proposes a new method to generate non-orthonormal symplectic bases, overcoming limitations of existing orthonormal-based techniques in structure-preserving MOR.

## Key findings

- Non-orthonormal bases yield lower reduction errors.
- The new method outperforms traditional orthonormal approaches.
- Numerical experiments confirm improved accuracy in elasticity problems.

## Abstract

Parametric high-fidelity simulations are of interest for a wide range of applications. But 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 a 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 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.

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Source: https://tomesphere.com/paper/1902.10523