297 research outputs found
Stability of central finite difference schemes for the Heston PDE
This paper deals with stability in the numerical solution of the prominent
Heston partial differential equation from mathematical finance. We study the
well-known central second-order finite difference discretization, which leads
to large semi-discrete systems with non-normal matrices A. By employing the
logarithmic spectral norm we prove practical, rigorous stability bounds. Our
theoretical stability results are illustrated by ample numerical experiments
Pseudospectra in non-Hermitian quantum mechanics
We propose giving the mathematical concept of the pseudospectrum a central
role in quantum mechanics with non-Hermitian operators. We relate
pseudospectral properties to quasi-Hermiticity, similarity to self-adjoint
operators, and basis properties of eigenfunctions. The abstract results are
illustrated by unexpected wild properties of operators familiar from
PT-symmetric quantum mechanics.Comment: version accepted for publication in J. Math. Phys.: criterion
excluding basis property (Proposition 6) added, unbounded time-evolution
discussed, new reference
Singular Modes of the Electromagnetic Field
We show that the mode corresponding to the point of essential spectrum of the
electromagnetic scattering operator is a vector-valued distribution
representing the square root of the three-dimensional Dirac's delta function.
An explicit expression for this singular mode in terms of the Weyl sequence is
provided and analyzed. An essential resonance thus leads to a perfect
localization (confinement) of the electromagnetic field, which in practice,
however, may result in complete absorption.Comment: 14 pages, no figure
Eigenvector statistics in non-Hermitian random matrix ensembles
We study statistical properties of the eigenvectors of non-Hermitian random
matrices, concentrating on Ginibre's complex Gaussian ensemble, in which the
real and imaginary parts of each element of an N x N matrix, J, are independent
random variables. Calculating ensemble averages based on the quantity , where and are left and right eigenvectors of J, we show for large N that
eigenvectors associated with a pair of eigenvalues are highly correlated if the
two eigenvalues lie close in the complex plane. We examine consequences of
these correlations that are likely to be important in physical applications.Comment: 4 pages, no figure
Determination of Inter-Phase Line Tension in Langmuir Films
A Langmuir film is a molecularly thin film on the surface of a fluid; we
study the evolution of a Langmuir film with two co-existing fluid phases driven
by an inter-phase line tension and damped by the viscous drag of the underlying
subfluid. Experimentally, we study an 8CB Langmuir film via digitally-imaged
Brewster Angle Microscopy (BAM) in a four-roll mill setup which applies a
transient strain and images the response. When a compact domain is stretched by
the imposed strain, it first assumes a bola shape with two tear-drop shaped
reservoirs connected by a thin tether which then slowly relaxes to a circular
domain which minimizes the interfacial energy of the system. We process the
digital images of the experiment to extract the domain shapes. We then use one
of these shapes as an initial condition for the numerical solution of a
boundary-integral model of the underlying hydrodynamics and compare the
subsequent images of the experiment to the numerical simulation. The numerical
evolutions first verify that our hydrodynamical model can reproduce the
observed dynamics. They also allow us to deduce the magnitude of the line
tension in the system, often to within 1%. We find line tensions in the range
of 200-600 pN; we hypothesize that this variation is due to differences in the
layer depths of the 8CB fluid phases.Comment: See (http://www.math.hmc.edu/~ajb/bola/) for related movie
Exact Controllability of the Time Discrete Wave Equation: A Multiplier Approach
In this paper we summarize our recent results on the exact boundary controllability of a trapezoidal time discrete wave equation in a bounded domain. It is shown that the projection of the solution in an appropriate space in which the high frequencies have been filtered is exactly controllable with uniformly bounded controls (with respect to the time-step). By classical duality arguments, the problem is reduced to a boundary observability inequality for a time-discrete wave equation. Using multiplier techniques the uniform observability property is proved in a class of filtered initial data. The optimality of the filtering parameter is also analyzed
Spectral Theory of Sparse Non-Hermitian Random Matrices
Sparse non-Hermitian random matrices arise in the study of disordered
physical systems with asymmetric local interactions, and have applications
ranging from neural networks to ecosystem dynamics. The spectral
characteristics of these matrices provide crucial information on system
stability and susceptibility, however, their study is greatly complicated by
the twin challenges of a lack of symmetry and a sparse interaction structure.
In this review we provide a concise and systematic introduction to the main
tools and results in this field. We show how the spectra of sparse
non-Hermitian matrices can be computed via an analogy with infinite dimensional
operators obeying certain recursion relations. With reference to three
illustrative examples --- adjacency matrices of regular oriented graphs,
adjacency matrices of oriented Erd\H{o}s-R\'{e}nyi graphs, and adjacency
matrices of weighted oriented Erd\H{o}s-R\'{e}nyi graphs --- we demonstrate the
use of these methods to obtain both analytic and numerical results for the
spectrum, the spectral distribution, the location of outlier eigenvalues, and
the statistical properties of eigenvectors.Comment: 60 pages, 10 figure
Modelling for Robust Feedback Control of Fluid Flows
This paper addresses the problem of obtaining low-order models of fluid flows for the purpose of designing robust feedback controllers. This is challenging since whilst many flows are governed by a set of nonlinear, partial differential-algebraic equations (the Navier-Stokes equations), the majority of established control theory assumes models of much greater simplicity, in that they are firstly: linear, secondly: described by ordinary differential equations, and thirdly: finite-dimensional. Linearisation, where appropriate, overcomes the first disparity, but attempts to reconcile the remaining two have proved difficult. This paper addresses these two problems as follows. Firstly, a numerical approach is used to project the governing equations onto a divergence-free basis, thus converting a system of differential-algebraic equations into one of ordinary differential equations. This dispenses with the need for analytical velocity-vorticity transformations, and thus simplifies the modelling of boundary sensing and actuation. Secondly, this paper presents a novel and straightforward approach for obtaining suitable low-order models of fluid flows, from which robust feedback controllers can be synthesised that provide~\emph{a~priori} guarantees of robust performance when connected to the (infinite-dimensional) linearised flow system. This approach overcomes many of the problems inherent in approaches that rely upon model-reduction. To illustrate these methods, a perturbation shear stress controller is designed and applied to plane channel flow, assuming arrays of wall mounted shear-stress sensors and transpiration actuators. DNS results demonstrate robust attenuation of the perturbation shear-stresses across a wide range of Reynolds numbers with a single, linear controller
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