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Dynamically Optimal Projection onto Slow Spectral Manifolds for Linear Systems

Florian Kogelbauer, Ilya Karlin

TL;DR

The paper addresses the problem of projecting arbitrary initial conditions onto a linear slow manifold in systems with transient dynamics arising from non-normal operators, formalized as minimizing $E(x_0, \xi) = \frac{1}{2} \int_{0}^{\infty} \| e^{tL} x_0 - e^{tL} x_{slow}(\xi) \|^2 dt$. It introduces the dynamically optimal projection (DOP) via a temporal variational principle, deriving $\xi^{min}(x_0) = (G^T)^{-1} I(x_0)$ with $G_{ij} = \frac{\langle \hat{x}_i, \hat{x}_j \rangle}{\lambda_i + \lambda_j^*}$ and $I_j(x_0) = \langle (L + \lambda_j^*)^{-1} x_0, \hat{x}_j \rangle$. Key contributions include an explicit projection operator $P_{DOP}$, the fact that it reduces to the orthogonal projection for normal $L$, and demonstrations on a 2D shear system and a linear Grad's 3-component system showing superior capture of transient dynamics. The work provides a spectral, variational tool for reduced-order modeling and data assimilation in linear, non-normal settings, with potential extensions to non-invariant constraints and nonlinear regimes.

Abstract

We derive the dynamically optimal projection onto the linear slow manifold from a temporal variational principle. We demonstrate that the projection captures transient dynamics of the overall dissipative system and leads to a considerably improved fit of reduced trajectories compared to full trajectories. We illustrate these optimal model reduction properties on explicit examples, including the linear three-component Grad's moment system.

Dynamically Optimal Projection onto Slow Spectral Manifolds for Linear Systems

TL;DR

The paper addresses the problem of projecting arbitrary initial conditions onto a linear slow manifold in systems with transient dynamics arising from non-normal operators, formalized as minimizing . It introduces the dynamically optimal projection (DOP) via a temporal variational principle, deriving with and . Key contributions include an explicit projection operator , the fact that it reduces to the orthogonal projection for normal , and demonstrations on a 2D shear system and a linear Grad's 3-component system showing superior capture of transient dynamics. The work provides a spectral, variational tool for reduced-order modeling and data assimilation in linear, non-normal settings, with potential extensions to non-invariant constraints and nonlinear regimes.

Abstract

We derive the dynamically optimal projection onto the linear slow manifold from a temporal variational principle. We demonstrate that the projection captures transient dynamics of the overall dissipative system and leads to a considerably improved fit of reduced trajectories compared to full trajectories. We illustrate these optimal model reduction properties on explicit examples, including the linear three-component Grad's moment system.

Paper Structure

This paper contains 12 sections, 4 theorems, 75 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Spectral Theory and Slow Manifolds
  3. Dynamically Optimal projection from a Variational Principle
  4. Examples
  5. Two-Dimensional Shear System
  6. Three-Component Grad's Moment System
  7. Discussion
  8. Properties of the quadratic form $\mathcal{E}$ \ref{['E']}
  9. Explicit expression of the minimizer
  10. Projection property of DOP \ref{['defPdyn']}
  11. DOP reduces to the canonical orthogonal projection for $L$ normal
  12. Characterization of the Riesz projection through commutativity

Key Result

Proposition 1

The unique minimizer eq:min_problem reads, for $1\le i\le n$.

Figures (4)

  • Figure 1: Illustration of the dynamical projection for system \ref{['dyn2D']} ($\alpha=5$, $\gamma=1$) applied to the initial condition $x_0=(0.4,1.2)$ for which the trajectory to the full system is shown in red. The dynamically optimal projection (solid arrow) moves the initial condition on the slow manifold (x-axis in green) farther away from the global equilibrium compared to the orthogonal projection (dashed arrow).
  • Figure 2: Comparison of the $x$-components of the full solution (black) to the dynamically projected trajectory (red) and the orthogonally projected trajectory (blue) of system \ref{['dyn2D']}.
  • Figure 3: Spectrum of the linear Grad system \ref{['eqGrad']} for $\varepsilon=0.1$ and various wave numbers. The fast real modes (star symbol on the real axis) accumulate at $-\frac{5}{9\varepsilon}$ (vertical dashed line at $\Re{\lambda} = -50/9$), while the slow complex conjugated modes (bell-shaped star symbol) accumulate at $\Re\lambda = -\frac{2}{9\varepsilon}$ (dashed line at $\Re{\lambda} = -20/9$) as $k\to\infty$.
  • Figure 4: Comparison of the pressure components of the full solution (black) to the dynamically projected trajectory (red) and the orthogonally projected trajectory (blue) of system \ref{['eqGrad']} with initial condition orthogonal to the slow manifold at wave number $k=1$. Even though the difference in initial pressure is comparably small, the orthogonally projected pressure trajectory exhibits much larger oscillations compared to the dynamically projected one.

Theorems & Definitions (6)

  • Proposition 1
  • Proposition 2
  • Lemma B.1
  • proof
  • Lemma E.1
  • proof