Finding the optimal reparametrization in shape analysis of curves or surfaces is a computationally demanding task. The problem can be phrased as an optimisation problem on the infinite dimensional group of orientation preserving diffeomorphisms $\mathrm{Diff}^+(\Omega)$, where $\Omega$ is the domain where the curves or surfaces are defined.
We consider the composition of a finite number of elementary diffeomprphisms
\begin{equation}
\label{elem_diff}
\varphi_{\ell}:=\mathrm{id}+\sum_{j=1}^M \lambda_j^{\ell} f_j,\qquad \ell=1,\dots , L,\qquad \varphi\approx \varphi_L\circ \cdots \circ\varphi_1,
\end{equation}
where $\{f_i\}_{i=1}^{\infty}$, in $T_{\mathrm{id}}\mathrm{Diff}^+(\Omega)$ is an orthonormal basis, and we optimise simultaneously on all the parameters $\{\lambda_j^{\ell}\}$ for $j=1,\dots ,M$ and $\ell=1,\dots, L$. The obtained algorithm is similar to a deep neural network and its implementation can be carried out using PyTorch. Properties and analysis of the method will be discussed as well as numerical results.[-]

Classical model reduction techniques project the governing equations onto linear subspaces of the high-dimensional state-space. However, for problems with slowly decaying Kolmogorov-n-widths such as certain transport-dominated problems, classical linear-subspace reduced order models (ROMs) of low dimension might yield inaccurate results. Thus, the reduced space needs to be extended to more general nonlinear manifolds. Moreover, as we are dealing with Hamiltonian systems, it is crucial that the underlying symplectic structure is preserved in the reduced model.
To the best of our knowledge, existing literatures addresses either model reduction on manifolds or symplectic model reduction for Hamiltonian systems, but not their combination. In this talk, we bridge the two aforementioned approaches by providing a novel projection technique called symplectic manifold Galerkin, which projects the Hamiltonian system onto a nonlinear symplectic trial manifold such that the reduced model is again a Hamiltonian system. We derive analytical results such as stability, energy-preservation and a rigorous a-posteriori error bound. Moreover, we construct a weakly symplectic convolutional autoencoder in order to computationally approximate the nonlinear symplectic trial manifold. We numerically demonstrate the ability of the method to outperform structure-preserving linear subspace ROMs results for a linear wave equation for which a slow decay of the Kolmogorov-n-width can be observed.[-]

Port-Hamiltonian (PH) systems' theory is a powerful framework for modeling, simulation and control of interconnected multiphysical systems. A consistent numerical treatment of PH systems requires the preservation of the underlying Dirac structure, which represents interconnections and energy conversions. In this talk it will be shown how symplectic integration can be used to define discrete-time PH systems and to implement nonlinear controls in slowly sampled systems with high accuracy.[-]

The multiplier approach is applied to a class of port-Hamiltonian systems with boundary dissipation to establish exponential decay. The exponential stability of port-Hamiltonian systems has been studied and sufficient conditions obtained. Here the decay rate $Me^{-\alpha t}$ is established with $M$ and $\alpha$ are in terms of system parameters. This approach is illustrated by several examples, in particular, boundary stabilization of a piezoelectric beam with magnetic effects.[-]