Structure-Preserving Model Reduction of Physical Network Systems

This paper considers physical network systems where the energy storage is naturally associated to the nodes of the graph, while the edges of the graph correspond to static couplings. The first sections deal with the linear case, covering examples such as mass-damper and hydraulic systems, which have a structure that is similar to symmetric consensus dynamics. The last section is concerned with a specific class of nonlinear physical network systems; namely detailed-balanced chemical reaction networks governed by mass action kinetics. In both cases, linear and nonlinear, the structure of the dynamics is similar, and is based on a weighted Laplacian matrix, together with an energy function capturing the energy storage at the nodes. We discuss two methods for structure-preserving model reduction. The first one is clustering; aggregating the nodes of the underlying graph to obtain a reduced graph. The second approach is based on neglecting the energy storage at some of the nodes, and subsequently eliminating those nodes (called Kron reduction).</p

A block Newton method for nonlinear eigenvalue problems

We consider matrix eigenvalue problems that are nonlinear in the eigenvalue parameter. One of the most fundamental differences from the linear case is that distinct eigenvalues may have linearly dependent eigenvectors or even share the same eigenvector. This has been a severe hindrance in the development of general numerical schemes for computing several eigenvalues of a nonlinear eigenvalue problem, either simultaneously or subsequently. The purpose of this work is to show that the concept of invariant pairs offers a way of representing eigenvalues and eigenvectors that is insensitive to this phenomenon. To demonstrate the use of this concept in the development of numerical methods, we have developed a novel block Newton method for computing such invariant pairs. Algorithmic aspects of this method are considered and a few academic examples demonstrate its viabilit

Fast Interpolation-based Globality Certificates for Computing Kreiss Constants and the Distance to Uncontrollability

We propose a new approach to computing global minimizers of singular value functions in two real variables. Specifically, we present new algorithms to compute the Kreiss constant of a matrix and the distance to uncontrollability of a linear control system, both to arbitrary accuracy. Previous state-of-the-art methods for these two quantities rely on 2D level-set tests that are based on solving large eigenvalue problems. Consequently, these methods are costly, i.e., $\mathcal{O}(n^6)$ work using dense eigensolvers, and often multiple tests are needed before convergence. Divide-and-conquer techniques have been proposed that reduce the work complexity to $\mathcal{O}(n^4)$ on average and $\mathcal{O}(n^5)$ in the worst case, but these variants are nevertheless still very expensive and can be numerically unreliable. In contrast, our new interpolation-based globality certificates perform level-set tests by building interpolant approximations to certain one-variable continuous functions that are both relatively cheap and numerically robust to evaluate. Our new approach has a $\mathcal{O}(kn^3)$ work complexity and uses $\mathcal{O}(n^2)$ memory, where $k$ is the number of function evaluations necessary to build the interpolants. Not only is this interpolation process mostly "embarrassingly parallel," but also low-fidelity approximations typically suffice for all but the final interpolant, which must be built to high accuracy. Even without taking advantage of the aforementioned parallelism, $k$ is sufficiently small that our new approach is generally orders of magnitude faster than the previous state-of-the-art.Comment: Revision #

A block Newton method for nonlinear eigenvalue problems

We consider matrix eigenvalue problems that are nonlinear in the eigenvalue parameter. One of the most fundamental differences from the linear case is that distinct eigenvalues may have linearly dependent eigenvectors or even share the same eigenvector. This has been a severe hindrance in the development of general numerical schemes for computing several eigenvalues of a nonlinear eigenvalue problem, either simultaneously or subsequently. The purpose of this work is to show that the concept of invariant pairs offers a way of representing eigenvalues and eigenvectors that is insensitive to this phenomenon. To demonstrate the use of this concept in the development of numerical methods, we have developed a novel block Newton method for computing such invariant pairs. Algorithmic aspects of this method are considered and a few academic examples demonstrate its viability. © Springer-Verlag 2009

Étude et modélisation du phénomène de croissance transitoire et de son lien avec la transition Bypass au sein des couches limites tridimensionnelles

The transition from a laminar to a turbulent flow strongly modifies the boundary layer properties. Understanding the mechanisms leading to transition is crucial to reliably predict aerodynamic performances. For boundary layers subjected to high levels of external disturbances, the natural transition due to the amplification of the least stable mode is replaced by an early transition, called Bypass transition. This is the result of non-normal mode interactions that lead to a phenomenon of transient growth of disturbances. These disturbances are known as Klebanoff modes and take the form of streamwise velocity streaks. This thesis aims at understanding this linear mechanism of transient growth and quantifying its influence on the classical modal amplification of disturbances. This is done by computing the so-called optimal perturbations, i.e. the initial disturbances that undergo maximum amplification in the boundary layer. These optimal perturbations are first determined for two-dimensional compressible boundary layers developing over curved surfaces. In particular, we show that Klebanoff modes naturally evolve towards Görtler vortices that occur over concave walls. Three-dimensional boundary layers are then considered. In such configurations, transient growth provides an initial amplitude to crossflow vortices. Finally, applying the tools developed in this thesis to new flow cases such as swept wings provides further understanding of the phenomenon of transient growth for realistic geometries.Le passage du régime laminaire au régime turbulent s’accompagne d’importantes modifications des propriétés physiques de la couche limite. La détermination précise de la transition est donc cruciale dans de nombreux cas pratiques. Lorsque la couche limite se développe dans un environnement extérieur faiblement perturbé, la transition est gouvernée par l’amplification du mode propre le moins stable. Lorsque l’intensité des perturbations extérieures augmente, des interactions multimodales entraînent une amplification transitoire des perturbations. Ce phénomène peut conduire à une transition prématurée, appelée transition Bypass. Les perturbations prennent alors la forme de stries longitudinales de vitesse appelées modes de Klebanoff. L’objectif de cette thèse est d’étudier ce mécanisme linéaire de croissance transitoire et son influence sur l’amplification modale classique des perturbations. Cela passe par la détermination des perturbations les plus amplifiées au sein de la couche limite, appelées perturbations optimales. Ces perturbations optimales sont d’abord calculées pour des couches limites bidimensionnelles et compressibles se développant sur des surfaces courbes. En particulier, on montre que les modes de Klebanoff évoluent vers les tourbillons de Görtler qui se forment sur des parois concaves. Le cas plus général de couches limites tridimensionnelles est ensuite envisagé. Pour de telles configurations, la croissance transitoire fournit une amplitude initiale aux instabilités transversales. Enfin, l’application des outils développés dans cette thèse fournit de nouveaux résultats pour des cas d’écoulements autour de géométries réalistes comme une aile en flèche