Structured interpolation for multivariate transfer functions of quadratic-bilinear systems

TL;DR

This paper proposes definitions for structured variants of the symmetric subsystem and generalized transfer functions of quadratic-bilinear systems and provides conditions for structure-preserving interpolation by projection and considers the construction of multivariate interpolants in frequency domain for structured quadratics-bilInear systems.

Abstract

High-dimensional/high-fidelity nonlinear dynamical systems appear naturally when the goal is to accurately model real-world phenomena. Many physical properties are thereby encoded in the internal differential structure of these resulting large-scale nonlinear systems. The high-dimensionality of the dynamics causes computational bottlenecks, especially when these large-scale systems need to be simulated for a variety of situations such as different forcing terms. This motivates model reduction where the goal is to replace the full-order dynamics with accurate reduced-order surrogates. Interpolation-based model reduction has been proven to be an effective tool for the construction of cheap-to-evaluate surrogate models that preserve the internal structure in the case of weak nonlinearities. In this paper, we consider the construction of multivariate interpolants in frequency domain for structured quadratic-bilinear systems. We propose definitions for structured variants of the symmetric subsystem and generalized transfer functions of quadratic-bilinear systems and provide conditions for structure-preserving interpolation by projection. The theoretical results are illustrated using two numerical examples including the simulation of molecular dynamics in crystal structures.

Figures (7)

• Figure 1: Schematic illustration of the Toda lattice with $\ell$ particles Wer21. Atoms in a one-dimensional crystal structure are represented as point masses and connected by exponential springs modeling the forces between the particles.
• Figure 2: Time simulation of the time-delay example: The best reduced-order models from each generating approach are shown. All reduced-order models can recover the system behavior for the given input signal, but the interpolation-based reduced-order models perform around four orders of magnitude better in terms of accuracy than the model generated by POD(avg).
• Figure 3: Sigma plots showing $\lVert G(\mathfrak{i} \omega)) \rVert_{2}$ of the first symmetric subsystem transfer function of the time-delay example: The best reduced-order models from each generating approach are shown. All reduced-order models can recover the system behavior for the given input signal, but the interpolation-based reduced-order models perform around six orders of magnitude better in terms of accuracy than the model generated by POD(avg).
• Figure 4: Second symmetric subsystem transfer function relative approximation errors $\mathop{\mathrm{relerr}}\nolimits(\omega_{1}, \omega_{2})$ of the time-delay example: The best reduced-order models from each generating approach are shown. The errors of both interpolation-based reduced-order models are at least four orders of magnitude better than those of the POD(avg) model.
• Figure 5: Time simulation of the Toda lattice example: The best reduced-order models from each generating approach are shown. Only the interpolation-based reduced-order models recover the system behavior over the full time interval, while POD(avg) becomes unstable after about $60$ s.

Theorems & Definitions (10)

• Definition 1
• Definition 2
• Proposition 1: Wer21
• Proposition 2: Wer21
• Theorem 1
• proof
• Proposition 3: Wer21
• Remark 1
• Corollary 1
• proof