University of Oxford

There is a beautiful problem resulting from arithmetic number theory where a continuous and compactly supported function's 3-fold autoconvolution is constant. In this talk, we optimize the coefficients of a Chebyshev series multiplied by an endpoint singularity to obtain a highly accurate approximation to this constant. Convolving functions with endpoint singularities turns out to be a challenge for standard quadrature routines. However, variable transformations inducing double exponential endpoint decay are used to effectively annihilate the singularities in a way that keeps accuracy high and complexity low.

Flow thought a porous media is usually described by assuming the superficial velocity can be expressed in terms of a constant permeability and a pressure gradient. In poroelastic flows the underlying elastic matrix responds to changes in the fluid pressure. When the elastic deformation is allowed to influence the permeability through the elastic strain, it becomes possible for increased fluid pressure gradient not to result in increased flow, but to decrease the permeability and potentially this may close off or choke the flow. I will talk about a simple model problem for a number of different elastic constitutive models and a number of different permeability-strain models and examine whether there is a general criterion that can be derived to show when, or indeed if, choking can occur for different elasticity-permeability combinations.

We present an algorithm, Parallel-$\ell_0$, for {\em combinatorial compressed sensing} (CCS), where the sensing matrix $A \in \mathbb{R}^{m\times n}$ is the adjacency matrix of an expander graph. The information preserving nature of expander graphs allow the proposed algorithm to provably recover a $k$-sparse vector $x\in\mathbb{R}^n$ from $m = \mathcal{O}(k \log (n/k))$ measurements $y = Ax$ via $\mathcal{O}(\log k)$ simple and parallelizable iterations when the non-zeros in the support of the signal form a dissociated set, meaning that all of the $2^k$ subset sums of the support of $x$ are pairwise different. In addition to the low computational cost, Parallel-$\ell_0$ is observed to be able to recover vectors with $k$ substantially larger than previous CCS algorithms, and even higher than $\ell_1$-regularization when the number of measurements is significantly smaller than the vector length.

This talk concerns the numerical solution of elliptic partial differential equations posed on general smooth surfaces by the Closest Point Method. Based on the closest point representation of the surface, we formulate an embedding equation in a narrow band surrounding the surface, then discretize it using standard finite differences and interpolation schemes. Numerical convergence of the method will be discussed. In order to solve the resulting large sparse linear systems, we propose a specific geometric multigrid method which makes use of the closest point representation of the surface.
 

We discuss the iterative solution of a finite element discretisation of the magma dynamics equations.  These equations share features of the Stokes equations, however, Elman-Silvester-Wathen (ESW) preconditioners for the magma dynamics equations are not optimal. By introducing a new field, the compaction pressure, into the magma dynamics equations, we have developed a new three-field preconditioner which is optimal in terms of problem size and less sensitive to physical parameters compared to the ESW preconditioners.

It is well-known that a matrix $A$ is Hurwitz stable if and only if there exists a positive definite solution to the Lyapunov matrix equation $A X + X A^* = B$, where $B$ is Hermitian negative definite. We present a verified numerical algorithm to rigorously prove the stability of a given matrix $A$ in the presence of rounding errors.  The computational cost of the algorithm is cubic and it is fast since we can cast almost all operations in level 3 BLAS for which interval arithmetic can be implemented very efficiently.  This is a joint work with Andreas Frommer and the results are already published in ETNA in 2013.

“Narrative and Proof”, is an interdisciplinary discussion where one of the UK's leading scientists, Marcus du Sautoy, will argue that mathematical proofs are not just number-based, but also rely on narrative. He will be joined by author Ben Okri, mathematician Roger Penrose, and literature expert Laura Marcus, to consider how narrative shapes the sciences as well as the arts.

The discussion will be chaired by Elleke Boehmer, Professor of World Literature in English, University of Oxford, and will be followed by audience questions and a drinks reception.

The event will take place from 5 to 6:30 pm on Tuesday 20 January 2015 at the Mathematical Institute, Radcliffe Observatory Quarter, Woodstock Road, Oxford. The event is free and open to all, but registration is recommended. 

Please click here to register.

This event is co-hosted by the Mathematical Institute and The Oxford Research Centre in the Humanities (TORCH), and celebrates the launch of TORCH’s Annual Headline Series 2014-15, Humanities and Science.
 

A key question to develop our understanding of turbulence in shear flows is: what is the smallest perturbation to the laminar flow that causes a transition to turbulence, and how does this change with the Reynolds number, R?  Finding this so-called ``minimal seed'' is as yet unachievable in direct numerical simulations of the Navier-Stokes equations. We search for the minimal seed in a low-dimensional model analogue to the full Navier-Stokes in plane sinusoidal flow, developed by Waleffe (1997). A previous such calculation found the minimal seed as the least distance (energy norm) from the origin (laminar flow) to the basin of attraction of another fixed point (turbulent attractor).  However, using a non-linear optimization technique, we found an internal boundary of the basin of attraction of the origin that separates flows which directly relaminarize from flows which undergo transient turbulence. It is this boundary which contains the minimal seed, and we find it to be smaller than the previously calculated minimal seed. We present results over a range of Reynolds numbers up to 2000 and find an R^{-1} scaling law fits reasonably well. We propose a new scaling law which asymptotes to R^{-1} for large R but, using some additional information, matches the minimal seed scaling better at low R.