University of Oxford

Wave scattering problems arise in numerous applications in acoustics, electromagnetics and linear elasticity. In the boundary element method (BEM) one reformulates the scattering problem as an integral equation on the scatterer boundary, e.g. using Green’s identities, and then seeks an approximate solution of the boundary integral equation (BIE) from some finite-dimensional approximation space. The conventional choice is a space of piecewise polynomials; however, in the “high frequency” regime when the wavelength is small compared to the size of the scatterer, it is computationally expensive to resolve the highly oscillatory wave solution in this way. The hybrid numerical-asymptotic (HNA) approach aims to reduce the computational cost by enriching the BEM approximation space with oscillatory functions, carefully chosen to capture the high frequency asymptotic solution behaviour. To date, the HNA methodology has been implemented almost exclusively in a Galerkin variational framework. This has many attractive features, not least the possibility of proving rigorous convergence results, but has the disadvantage of requiring numerical evaluation of high dimensional oscillatory integrals. In this talk I will present the results of some investigations carried out with my MSc student Emile Parolin into collocation-based implementations, which involve lower-dimensional integrals, but appear harder to analyse in terms of convergence and stability.

The relationship between the so-called ancient (backwards) solutions to the Navier-Stokes equations in the space or in a half space and the global well-posedness of initial boundary value problems for these equations will be explained. If time permits I will sketch details of an equivalence theorem and a proof of smoothness properties of mild bounded ancient solutions in the half space, which is a joint work with Gregory Seregin

We study a simple one-dimensional coupled heat wave system, obtaining a sharp estimate for the rate of energy decay of classical solutions. Our approach is based on the asymptotic theory of $C_0$-semigroups and in particular on a result due to Borichev and Tomilov (2010), which reduces the problem of estimating the rate of energy decay to finding a growth bound for the resolvent of the semigroup generator. This technique not only leads to an optimal result, it is also simpler than the methods used by other authors in similar situations and moreover extends to problems on higher-dimensional domains. Joint work with C.J.K. Batty (Oxford) and L. Paunonen (Tampere).

A period is a certain type of number obtained by integrating algebraic differential forms over algebraic domains. Examples include pi, algebraic numbers, values of the Riemann zeta function at integers, and other classical constants.
Difficult transcendence conjectures due to Grothendieck suggest that there should be a Galois theory of periods.
I will explain these notions in very introductory terms and show how to set up such a Galois theory in certain situations.
I will then discuss some applications, in particular to Kim's method for bounding $S$-integral solutions to the equation $u+v=1$, and possibly to high-energy physics.

In this talk I will give an overview of the algorithms used by Chebfun to numerically compute polynomial and trigonometric minimax approximations of continuous functions. I'll also present Chebfun's capabilities to compute best approximations on compact subsets of an interval and how these methods can be used to design digital filters.

In this talk we will explore the convergence of Krylov methods when used to solve $Lu = f$ where $L$ is an unbounded linear operator.  We will show that for certain problems, methods like Conjugate Gradients and GMRES still converge even though the spectrum of $L$ is unbounded. A theoretical justification for this behavior is given in terms of polynomial approximation on unbounded domains.    

We review a class of systems of non-linear PDEs, derived from the Cahn--Hilliard and Ohta--Kawasaki functionals, that describe the energy evolution of diblock copolymers. These are long chain molecules that can self assemble into repeating patterns as they cool. We are particularly interested in finite element numerical methods that approximate these PDEs in the two-phase (in which we model the polymer only) and three-phase (in which we imagine the polymer surrounded by, and interacting with, a void) cases.

We present a brief derivation of the underlying models, review a class of numerical methods to approximate them, and showcase some early results from our codes.

The talk will present ongoing work on medical image reconstruction from x-ray scanners. A suitable method for reconstruction of these undersampled systems is compressed sensing. The presentation will show respective reconstruction methods and their analysis. Furthermore, work in progress about extensions of the standard approach will be shown.

When the Computing Laboratory discarded its hardcopy journals around 2008, I kept the first ten years or so of each of six classic numerical analysis journals, starting from volume 1, number 1.  This will not be a seminar in the usual sense but a mutual exploration.  Come prepared to look through a few of these old volumes yourself and perhaps to tell the group of something interesting you find.  Bring a pen and paper.  All are welcome.

Mathematics of Computation, from 1943
SIAM Journal, from 1953
Numerische Mathematik, from 1959
BIT, from 1961
SIAM Journal on Numerical Analysis, from 1964
Journal of the IMA, from 1965