# Structure Preserving Model Reduction of Parametric Hamiltonian Systems

## TL;DR

This paper introduces a structure-preserving greedy model reduction method for parametric Hamiltonian systems that ensures stability and efficiency in long-term simulations by maintaining symplectic structure.

## Contribution

It presents a novel greedy basis selection algorithm that preserves symplectic structure and demonstrates exponential convergence for parametric Hamiltonian systems.

## Key findings

- Ensures stability of reduced models over long-time integration.
- Achieves exponential convergence of the greedy algorithm.
- Preserves symplectic structure when combined with empirical interpolation.

## Abstract

While reduced-order models (ROMs) have been popular for efficiently solving large systems of differential equations, the stability of reduced models over long-time integration is of present challenges. We present a greedy approach for ROM generation of parametric Hamiltonian systems that captures the symplectic structure of Hamiltonian systems to ensure stability of the reduced model. Through the greedy selection of basis vectors, two new vectors are added at each iteration to the linear vector space to increase the accuracy of the reduced basis. We use the error in the Hamiltonian due to model reduction as an error indicator to search the parameter space and identify the next best basis vectors. Under natural assumptions on the set of all solutions of the Hamiltonian system under variation of the parameters, we show that the greedy algorithm converges with exponential rate. Moreover, we demonstrate that combining the greedy basis with the discrete empirical interpolation method also preserves the symplectic structure. This enables the reduction of the computational cost for nonlinear Hamiltonian systems. The efficiency, accuracy, and stability of this model reduction technique is illustrated through simulations of the parametric wave equation and the parametric Schrodinger equation.