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Low-rank approximation and model reduction.
Get PDFThe basic idea of model reduction is to represent a complex linear dynamical system by a much simpler one. This may refer to many different techniques, but in this dissertation
we focus on projection-based model reduction of linear systems. The projection is based on the dominant eigen-spaces of energy functions for ingoing and outgoing signals of the system.
These energy functions are called Gramians of the system and can be obtained as the solutions of Stein equations. When the system matrices are large and sparse, it is not
obvious how to compute efficiently these solutions or their dominant eigen-spaces. In fact, direct methods ignore sparsity in the Stein equations and are not very attractive
for parallelization. Their use is then limited if the state dimension N of the system is large.
The complexity of these methods is roughly O(N3) floating point operations and they require about O(N2) words of memory.
This thesis provides some new ideas of recursive projection-based model reduction for time-varying systems as well as time-invariant systems. We present three algorithms for the recursive computation of the projection. These algorithms combine ideas of two classical methods — namely Balanced Truncation and Krylov subspaces — to produce a
low-rank approximation of the Gramians or the input/output map of the system. We show the practical relevance of our results with real world benchmark examples. We also present some new ideas for second order systems. Such systems have a special structure which one wants to preserve in the reduced order model. We show how to adapt our projection based method to such systems
A Schur--Parlett Algorithm for Computing Matrix Functions
No full textAn algorithm for computing matrix functions is presented. It employs a Schur
decomposition with reordering and blocking followed by the block form of a
recurrence of Parlett, with functions of the nontrivial diagonal blocks
evaluated via a Taylor series. A parameter is used to balance the conflicting
requirements of producing small diagonal blocks and keeping the separations of
the blocks large. The algorithm is intended primarily for functions having a
Taylor series with an infinite radius of convergence, but it can be adapted for
certain other functions, such as the logarithm. Novel features introduced here
include a convergence test that avoids premature termination of the Taylor
series evaluation and an algorithm for reordering and blocking the Schur form.
Numerical experiments show that the algorithm is competitive with existing
special-purpose algorithms for the matrix exponential, logarithm, and cosine.
Nevertheless, the algorithm can be numerically unstable with the default choice
of its blocking parameter (or in certain cases for all choices), and we explain
why determining the optimal parameter appears to be a very difficult problem. A
MATLAB implementation is available that is much more reliable than the function
\texttt{funm} in MATLAB~6.5 (R13)
Three-dimensional airway reopening: the steady propagation of a semi-indefinite bubble into a buckled elastic tube
Get PDFWe consider the steady propagation of an air finger into a buckled elastic tube initially filled with viscous fluid. This study is motivated by the physiological problem of pulmonary airway reopening. The system is modelled using geometrically nonlinear Kirchhoff–Love shell theory coupled to the free-surface Stokes equations. The resulting three-dimensional fluid–structure-interaction problem is solved numerically by a fully coupled finite element method.
The system is governed by three dimensionless parameters: (i) the capillary number, Ca=[mu]U/[sigma]*, represents the ratio of viscous to surface-tension forces, where [mu] is the fluid viscosity, U is the finger's propagation speed and [sigma]* is the surface tension at the air–liquid interface; (ii) [sigma]=[sigma]*/(RK) represents the ratio of surface tension to elastic forces, where R is the undeformed radius of the tube and K its bending modulus; and (iii) A[infty infinity]=A*[infty infinity]/(4R2), characterizes the initial degree of tube collapse, where A*[infty infinity] is the cross-sectional area of the tube far ahead of the bubble.
The generic behaviour of the system is found to be very similar to that observed in previous two-dimensional models (Gaver et al. 1996; Heil 2000). In particular, we find a two-branch behaviour in the relationship between dimensionless propagation speed, Ca, and dimensionless bubble pressure, p*b/([sigma]*/R). At low Ca, a decrease in p*b is required to increase the propagation speed. We present a simple model that explains this behaviour and why it occurs in both two and three dimensions. At high Ca, p*b increases monotonically with propagation speed and p*b/([sigma]*/R) [is proportional to] Ca for sufficiently large values of [sigma] and Ca. In a frame of reference moving with the finger velocity, an open vortex develops ahead of the bubble tip at low Ca, but as Ca increases, the flow topology changes and the vortex disappears.
An increase in dimensional surface tension, [sigma]*, causes an increase in the bubble pressure required to drive the air finger at a given speed; p*b also increases with A*[infty infinity] and higher bubble pressures are required to open less strongly buckled tubes. This unexpected finding could have important physiological ramifications. If [sigma]* is sufficiently small, steady airway reopening can occur when the bubble pressure is lower than the external (pleural) pressure, in which case the airway remains buckled (non-axisymmetric) after the passage of the air finger. Furthermore, we find that the maximum wall shear stresses exerted on the airways during reopening may be large enough to damage the lung tissue
High-frequency self-excited oscillations in a collapsible-channel flow
Get PDFHigh-Reynolds-number asymptotics and numerical simulations are used to describe two-dimensional, unsteady, pressure-driven flow in a finite-length channel, one wall of which contains a section of membrane under longitudinal tension. Asymptotic predictions of stability boundaries for small-amplitude, high-frequency, self-excited oscillations are derived in the limit of large membrane tension. The oscillations are closely related to normal modes of the system, which have a frequency set by a balance between membrane tension and the inertia of the fluid in the entire channel. Oscillations can grow by extracting kinetic energy from the mean Poiseuille flow faster than it is lost to viscous dissipation. Direct numerical simulations, based on a fully coupled finite-element discretization of the equations of large-displacement elasticity and the Navier–Stokes equations, support the predicted stability boundaries, and are used to explore larger-amplitude oscillations at lower tensions. These are characterized by vigorous axial sloshing motions superimposed on the mean flow, with transient secondary instabilities being generated both upstream and downstream of the collapsible segment
Free Lie algebras and Adams operations
No full textLet be a group and a field. For any finite-dimensional -module and any positive integer , let denote the th homogeneous component of the free Lie -algebra generated by (a basis of) . Then can be considered as a -module, called the th Lie power of . The paper is concerned with identifying this module up to isomorphism. A simple formula is obtained which expresses in terms of certain linear functions on the Green ring. When is not divisible by the characteristic of these linear functions are Adams operations. Some results are also obtained which clarify the relationship between Adams operations defined by means of exterior powers and symmetric powers and operations introduced by Benson. Some of these results are put into a more general setting in an appendix by Stephen Donkin
Nonlinear magnetic diffusion and magnetic helicity transport in galactic dynamos
Get PDFWe study a simple model for the solar dynamo in the framework of the Parker migratory dynamo, with a nonlinear dynamo saturation mechanism based on magnetic helicity conservation arguments. We find a parameter range in which the model demonstrates a cyclic behaviour with properties similar to that of Parker dynamo with the simplest form of algebraic alpha -quenching. We compare the nonlinear current helicity evolution in this model with data for the current helicity evolution obtained during 10 years of observations at the Huairou Solar Station of China. On one hand, our simulated data demonstrate behaviour comparable with the observed phenomenology, provided that a suitable set of governing dynamo parameters is chosen. On the other hand, the observational data are shown to be rich enough to reject some other sets of governing parameters. We conclude that, in spite of the very preliminary state of the observations and the crude nature of the model, the idea of using observational data to constrain our ideas concerning magnetic field generation in the framework of the solar dynamo appears promising
When gap solitons become embedded solitons: a generic unfolding
Get PDFA two-parameter unfolding is considered of single-pulsed homoclinic orbits to an equilibrium with two real and two zero eigenvalues in fourth-order reversible dynamical systems. One parameter controls the linearisation, with a transition occurring between a saddle-centre and a hyperbolic equilibrium. In the saddle-centre region, the homoclinic orbit is of codimension-one, which is controlled by the second generic parameter, whereas when the equilibrium is hyperbolic the homoclinic orbit is structurally stable. A geometric approach reveals the homoclinic orbits to the saddle to be generically destroyed either by developing an algebraically decaying tail or through a fold, depending on the sign of the perturbation of the second parameter. Special cases of different actions of Z2-symmetry are considered, as is the case of the system being Hamiltonian. Application of these results is considered to the transition between embedded solitons (corresponding to the codimension-one-homoclinic orbits) and gap solitons (the structurally stable ones) in nonlinear wave systems. The theory is shown to match numerical experiments on two models arising in nonlinear optics and on a form of fifth-order Korteweg de Vries equation
The accumulation of boundary doubling for modified tent maps
Get PDFWe describe the transition to chaos via boundary doubling for a
particularly simple class of map. In this two parameter family of maps the accumulation of boundary doubling occurs on a curve in parameter space. We characterize this curve and use relate the form of this curve to a novel set of difference equations with proportional delay
Model reduction of second order systems
Get PDFIn this paper we propose new model reduction methods which preserve the polynomial form of a
given second order system. We also give numerical results to illustrate that even when imposing
such restrictions, one still obtains approximation errors which are comparable to those obtained
via balanced truncation. The advantage of our approach is that preserving the system structure
may better reflect the physical properties of the system we want to approximate