University of Oxford

Toeplitz matrices enjoy the dual virtues of ubiquity and beauty. We begin this talk by surveying some of the interesting spectral properties of such matrices, emphasizing the distinctions between infinite-dimensional Toeplitz matrices and the large-dimensional limit of the corresponding finite matrices. These basic results utilize the algebraic Toeplitz structure, but in many applications, one is forced to spoil this structure with some perturbations (e.g., by imposing boundary conditions upon a finite difference discretization of an initial-boundary value problem). How do such

perturbations affect the eigenvalues? This talk will address this question for "localized" perturbations, by which we mean perturbations that are restricted to a single entry, or a block of entries whose size remains fixed as the matrix dimension grows. One finds, for a broad class of matrices, that sufficiently small perturbations fail to alter the spectrum, though the spectrum is exponentially sensitive to other perturbations. For larger real single-entry

perturbations, one observes the perturbed eigenvalues trace out curves in the complex plane. We'll show a number of illustrations of this phenomenon for tridiagonal Toeplitz matrices.

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This talk describes collaborative work with Albrecht Boettcher, Marko Lindner, and Viatcheslav Sokolov of TU Chemnitz.

The asymptotic rate of convergence of the trapezium rule is

defined, for smooth functions, by the Euler-Maclaurin expansion.

The extension to other methods, such as Gauss rules, is straightforward;

this talk begins with some special cases, such as Periodic functions, and

functions with various singularities.

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Convergence rates for ODEs (Initial and Boundary value problems)

and for PDEs are available, but not so well known. Extension to singular

problems seems to require methods specific to each situation. Some of

the results are unexpected - to me, anyway.

We develop an algorithm for estimating the local Sobolev regularity index

of a given function by monitoring the decay rate of its Legendre expansion

coefficients. On the basis of these local regularities, we design and

implement an hp--adaptive finite element method based on employing

discontinuous piecewise polynomials, for the approximation of nonlinear

systems of hyperbolic conservation laws. The performance of the proposed

adaptive strategy is demonstrated numerically.

Many questions of interest to both mathematicians and physicists relate

to the behavior of eigenvalues and eigenmodes of the Laplace operator

on a polygon. Algorithmic improvements have revived the old "method

of fundamental solutions" associated with Fox, Henrici and Moler; is it

going to end up competitive with the state-of-the-art method of Descloux,

Tolley and Driscoll? This talk will outline the numerical issues but

give equal attention to applications including "can you hear the shape

of a drum?", localization of eigenmodes, eigenvalue avoidance, and the

design of drums that play chords.

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This is very much work in progress -- with graduate student Timo Betcke.

The seminar will address issues related to numerical simulation

of non-linear behaviour of solid materials to impact loading.

The kinematic and constitutive aspects of the transition from

continuum to discontinuum will be presented as utilised

within an explicit finite element development framework.

Material softening, mesh sensitivity and regularisation of

solutions will be discussed.

The heart can be described as an electrically driven mechanical pump. This

pump couldn't adapt to beat-by-beat changes in circulatory demand if there

was no feedback from the mechanical environment to the electrical control

processes. Cardiac mechano-electric feedback has been studied at various

levels of functional integration, from stretch-activated ion channels,

through mechanically induced changes in cardiac cells and tissue, to

clinically relevant observations in man, where mechanical stimulation of the

heart may either disturb or reinstate cardiac rhythmicity. The seminar will

illustrate the patho-physiological relevance of cardiac mechano-electric

feedback, introduce underlying mechanisms, and show the utility of iterating

between experimental research and mathematical modelling in studying this

phenomenon.

The lattice Boltzmann equation has been used successfully used to simulate

nearly incompressible flows using an isothermal equation of state, but

much less work has been done to determine stable implementations for other

equations of state. The commonly used nine velocity lattice Boltzmann

equation supports three non-hydrodynamic or "ghost'' modes in addition to

the macroscopic density, momentum, and stress modes. The equilibrium value

of one non-hydrodynamic mode is not constrained by the continuum equations

at Navier-Stokes order in the Chapman-Enskog expansion. Instead, we show

that it must be chosen to eliminate a high wavenumber instability. For

general barotropic equations of state the resulting stable equilibria do

not coincide with a truncated expansion in Hermite polynomials, and need

not be positive or even sign-definite as one would expect from arguments

based on entropy extremisation. An alternative approach tries to suppress

the instability by enhancing the damping the non-hydrodynamic modes using

a collision operator with multiple relaxation times instead of the common

single relaxation time BGK collision operator. However, the resulting

scheme fails to converge to the correct incompressible limit if the

non-hydrodynamic relaxation times are fixed in lattice units. Instead we

show that they must scale with the Mach number in the same way as the

stress relaxation time.