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Model order reduction via Lie groups

Yannik P. Wotte, Patrick Buchfink, Silke Glas, Federico Califano, Stefano Stramigioli

Abstract

Lie groups and their actions are ubiquitous in the description of physical systems, and we explore implications in the setting of model order reduction (MOR). We present a novel framework of MOR via Lie groups, called MORLie, in which high-dimensional dynamical systems on manifolds are approximated by low-dimensional dynamical systems on Lie groups. In comparison to other Lie group methods we are able to attack non-equivariant dynamics, which are frequent in practical applications, and we provide new non-intrusive MOR methods based on the presented geometric formulation. We also highlight numerically that MORLie has a lower error bound than the Kolmogorov $N$-width, which limits linear-subspace methods. The method is applied to various examples: 1. MOR of a simplified deforming body modeled by a noisy point cloud data following a sheering motion, where MORLie outperforms a naive POD approach in terms of accuracy and dimensionality reduction. 2. Reconstructing liver motion during respiration with data from edge detection in ultrasound scans, where MORLie reaches performance approaching the state of the art, while reducing the training time from hours on a computing cluster to minutes on a mobile workstation. 3. An analytic example showing that the method of freezing is analytically recovered as a special case, showing the generality of the geometric framework.

Model order reduction via Lie groups

Abstract

Lie groups and their actions are ubiquitous in the description of physical systems, and we explore implications in the setting of model order reduction (MOR). We present a novel framework of MOR via Lie groups, called MORLie, in which high-dimensional dynamical systems on manifolds are approximated by low-dimensional dynamical systems on Lie groups. In comparison to other Lie group methods we are able to attack non-equivariant dynamics, which are frequent in practical applications, and we provide new non-intrusive MOR methods based on the presented geometric formulation. We also highlight numerically that MORLie has a lower error bound than the Kolmogorov -width, which limits linear-subspace methods. The method is applied to various examples: 1. MOR of a simplified deforming body modeled by a noisy point cloud data following a sheering motion, where MORLie outperforms a naive POD approach in terms of accuracy and dimensionality reduction. 2. Reconstructing liver motion during respiration with data from edge detection in ultrasound scans, where MORLie reaches performance approaching the state of the art, while reducing the training time from hours on a computing cluster to minutes on a mobile workstation. 3. An analytic example showing that the method of freezing is analytically recovered as a special case, showing the generality of the geometric framework.

Paper Structure

This paper contains 33 sections, 16 theorems, 110 equations, 8 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related literature
  3. Notation
  4. Background
  5. Lie groups and their actions
  6. The Kolmogorov N-width
  7. Problem formulation
  8. MOR via Lie groups
  9. ROM dynamics
  10. The group Kolmogorov N-width
  11. Optimization in MORLie
  12. Search spaces for $(G,\Phi)$
  13. Search space for reduced vector fields
  14. Intrusive and non-intrusive MORLie
  15. Optimization Strategy
  16. ...and 18 more sections

Key Result

Theorem 2.4

If $\Phi$ is transitive, then $\mathcal{O}(x) = \mathcal{M}$, and it is identical for all $x\in\mathcal{M}$. If $\Phi$ is free, then $\mathcal{O}(x)$ is isomorphic to $G$ as a manifold, for all $x \in \mathcal{M}$. If $\Phi$ is free and proper, then $\mathcal{M}/G := \{\mathcal{O}(x) \;|\; x \in \ma

Figures (8)

  • Figure 1: Summary of MORLie: Given a full order model (FOM) based on a vector field $X\in \Gamma(T\mathcal{M})$, a reduced order model (ROM) on a lower-dimensional Lie group $G$ is expressed using an action $\Phi:G\times\mathcal{M}\rightarrow\mathcal{M}$ and a reduction map $\rho:\mathcal{M}\rightarrow\mathfrak{g}$. The goal is to choose $(G,\Phi,\rho)$ such that $\bar{x}(t) = \Phi(g(t),x_0)$ approximately follows the FOM dynamics $\dot{\bar{x}} \approx X(\bar{x})$.
  • Figure 2: Example \ref{['ssec:radial_oscillator']}, MOR of the radial oscillator, showing trajectories of the full order model (FOM) and the reduced order model (ROM).
  • Figure 3: Example \ref{['ssec:rigid-pointcloud']}, rigidly evolving pointlcloud with noise (blue circles) and reconstructed solution (red stars).
  • Figure 4: Example \ref{['ssec:rigid-pointcloud']}, reconstruction errors and singular values for rigidly evolving pointcloud.
  • Figure 5: Example \ref{['ssec:sheering-pointclouds']}, two sheering pointclouds with noise (blue circles) and reconstructed solution (red stars).
  • ...and 3 more figures

Theorems & Definitions (43)

  • Definition 2.1: Group action
  • Definition 2.2: Properties of an action
  • Definition 2.3: Orbit of an action
  • Theorem 2.4: Properties of the orbit, Holm2009
  • Definition 2.5: Infinitesimal generator
  • Definition 2.6: Distribution
  • Definition 2.7: Distribution induced by $G,\Phi$
  • Theorem 2.8: Properties of the induced distribution
  • Proof 2.8.1
  • Theorem 4.1: MorLie reduction, reconstruction and induced dynamics
  • ...and 33 more