# Structure-Preserving Model-Reduction of Dissipative Hamiltonian Systems

## TL;DR

This paper introduces a structure-preserving model reduction technique for dissipative Hamiltonian systems that maintains energy dissipation and stability by coupling the original system with a heat bath, enabling accurate and stable simulations.

## Contribution

It presents a novel reduction method that preserves symplectic structure and dissipation in Hamiltonian systems, improving stability and accuracy over traditional methods.

## Key findings

- Method preserves energy dissipation and stability.
- Numerical simulations demonstrate accuracy and stability.
- Applicable to dissipative wave equations and electric circuits.

## Abstract

Reduced basis methods are popular for approximately solving large and complex systems of differential equations. However, conventional reduced basis methods do not generally preserve conservation laws and symmetries of the full order model. Here, we present an approach for reduced model construction, that preserves the symplectic symmetry of dissipative Hamiltonian systems. The method constructs a closed reduced Hamiltonian system by coupling the full model with a canonical heat bath. This allows the reduced system to be integrated with a symplectic integrator, resulting in a correct dissipation of energy, preservation of the total energy and, ultimately, in the stability of the solution. Accuracy and stability of the method are illustrated through the numerical simulation of the dissipative wave equation and a port-Hamiltonian model of an electric circuit.