Speaker
Description
Large-scale Hamiltonian dynamics are governed by
$$
\dot x = J\nabla H(x),\quad
J=\begin{bmatrix}0&I\\-I&0\end{bmatrix},
$$
and arise from spatial discretizations of conservative PDEs as well as in molecular and multibody models. In multi-query, control, and real-time settings, projection-based model order reduction (MOR) is essential, but generic reduced spaces may destroy the symplectic geometry, leading to energy gain or loss and poor long-time behavior. Symplectic MOR therefore seeks a symplectic trial basis $V\in\mathbb{R}^{2n\times 2r}$ with $V^TJV=J_{2r}$, so that the reduced system remains Hamiltonian.
A key offline bottleneck of symplectic MOR is symplectic basis generation from snapshot data, commonly via the complex SVD (cSVD) or the SVD-like decomposition, which can dominate the cost when $n$ and/or the number of snapshots are large.
Building on randomized numerical linear algebra, we present basis construction methods that use random sketching to accelerate the matrix factorizations underlying these basis generation techniques, while preserving symplecticity by construction. In particular, the randomized cSVD (rcSVD) and a randomized SVD-like decomposition yield substantial speedups while retaining accuracy close to their deterministic counterparts in numerical tests [1].
Beyond efficiency, we highlight a priori error analysis for rcSVD-based symplectic MOR. The resulting bounds relate the projection error to sketch size (oversampling) and power-iteration depth, thereby quantifying the runtime--accuracy trade-off, motivating practical parameter choices [2].
References:
[1] R. Herkert et. al. Randomized symplectic model order reduction for Hamiltonian systems. In LSSC 2023, pages 99--107. Springer, Cham, 2024.
[2] R. Herkert et. al. Error analysis of randomized symplectic model order reduction for Hamiltonian systems. LAA, 729:67--99, 2026.