Tue, 21 Jan 2014

14:00 - 15:00
L5

Numerical solution of Hamilton—Jacobi—Bellman equations

Iain Smears
(University of Oxford)
Abstract

Hamilton—Jacobi—Bellman (HJB) equations are a class of fully nonlinear second-order partial differential equations (PDE) of elliptic or parabolic type that originate from Stochastic Optimal Control Theory. These PDE are fully nonlinear in the sense that the nonlinear terms include the second partial derivatives of the unknown solution; this strong nonlinearity severely restricts the range of numerical methods that are known to be convergent. These problems have traditionally been solved with low order monotone schemes, often of finite difference type, which feature certain limitations in terms of efficiency and practicability.
In this summary talk of my DPhil studies, we will be interested in the development of hp-version discontinuous Galerkin finite element methods (DGFEM) for the class of HJB equations that satisfy a Cordès condition. First, we will show the novel techniques of analysis used to find a stable and convergent scheme in the elliptic setting, and then we will present recent work on their extension to parabolic problems. The resulting method is very nonstandard, provably of high order, and it even allows for exponential convergence under hp-refinement. We present numerical experiments showing the accuracy, computational efficiency and flexibility of the scheme
Thu, 13 Feb 2014

12:00 - 13:00
L6

Modelling collective motion in biology

Prof. Philip Maini
(University of Oxford)
Abstract

We will present three different recent applications of cell motion in biology: (i) Movement of epithelial sheets and rosette formation, (ii) neural crest cell migrations, (iii) acid-mediated cancer cell invasion. While the talk will focus primarily on the biological application, it will be shown that all of these processes can be represented by reaction-diffusion equations with nonlinear diffusion term.

Tue, 26 Nov 2013

14:30 - 15:00
L5

Small dot, big challenging: on the new benchmark of Top500 and Green500

Shengxin (Jude) Zhu
(University of Oxford)
Abstract

A new benchmark, High Performance Conjugate Gradient (HPCG), finally was introduced recently for the Top500 list and the Green500 list. This will draw more attention to performance of sparse iterative solvers on distributed supercomputers and energy efficiency of hardware and software. At the same time, this will more widely promote the concept that communications are the bottleneck of performance of iterative solvers on distributed supercomputers, here we will go a little deeper, discussing components of communications and discuss which part takes a dominate share. Also discussed are mathematics tricks to detect some metrics of an underlying supercomputer.

Tue, 26 Nov 2013

14:00 - 14:30
L5

Novel numerical techniques for magma dynamics

Sander Rhebergen
(University of Oxford)
Abstract

We discuss the development of finite element techniques and solvers for magma dynamics computations. These are implemented within the FEniCS framework. This approach allows for user-friendly, expressive, high-level code development, but also provides access to powerful, scalable numerical solvers and a large family of finite element discretizations. The ability to easily scale codes to three dimensions with large meshes means that efficiency of the numerical algorithms is vital. We therefore describe our development and analysis of preconditioners designed specifically for finite element discretizations of equations governing magma dynamics. The preconditioners are based on Elman-Silvester-Wathen methods for the Stokes equation, and we extend these to flows with compaction.  This work is joint with Andrew Wathen and Richard Katz from the University of Oxford and Laura Alisic, John Rudge and Garth Wells from the University of Cambridge.

Tue, 19 Nov 2013

14:30 - 15:00
L5

The antitriangular factorisation of saddle point matrices

Jennifer Pestana
(University of Oxford)
Abstract

The antitriangular factorisation of real symmetric indefinite matrices recently proposed by Mastronardi and van Dooren has several pleasing properties. It is backward stable, preserves eigenvalues and reveals the inertia, that is, the number of positive, zero and negative eigenvalues. 

In this talk we show that the antitriangular factorization simplifies for saddle point matrices, and that solving a saddle point system in antitriangular form is equivalent to applying the well-known nullspace method. We obtain eigenvalue bounds for the saddle point matrix and discuss the role of the factorisation in preconditioning. 

Tue, 19 Nov 2013

14:00 - 14:30
L5

Finding integral points on curves via numerical (p-adic) integration: a number theorist's perspective

Jennifer Balakrishnan
(University of Oxford)
Abstract

From cryptography to the proof of Fermat's Last Theorem, elliptic curves (those curves of the form y^2 = x^3 + ax+b) are ubiquitous in modern number theory.  In particular, much activity is focused on developing techniques to discover rational points on these curves. It turns out that finding a rational point on an elliptic curve is very much like finding the proverbial needle in the haystack -- in fact, there is currently no algorithm known to completely determine the group of rational points on an arbitrary elliptic curve.


 I'll introduce the ''real'' picture of elliptic curves and discuss why the ambient real points of these curves seem to tell us little about finding rational points. I'll summarize some of the story of elliptic curves over finite and p-adic fields and tell you about how I study integral points on (hyper)elliptic curves via p-adic integration, which relies on doing a bit of p-adic linear algebra.  Time permitting, I'll also give a short demo of some code we have to carry out these algorithms in the Sage Math Cloud.

Tue, 12 Nov 2013

14:00 - 15:00
L5

Continuous analogues of matrix factorizations

Alex Townsend
(University of Oxford)
Abstract

In this talk we explore continuous analogues of matrix factorizations.  The analogues we develop involve bivariate functions, quasimatrices (a matrix whose columns are 1D functions), and a definition of triangular in the continuous setting.  Also, we describe why direct matrix algorithms must become iterative algorithms with pivoting for functions. New applications arise for function factorizations because of the underlying assumption of continuity. One application is central to Chebfun2. 

Tue, 05 Nov 2013

14:30 - 15:00
L5

Pearcy's 1966 proof and Crouzeix's conjecture

L. Nick Trefethen
(University of Oxford)
Abstract

Crouzeix's conjecture is an exasperating problem of linear algebra that has been open since 2004: the norm of p(A) is bounded by twice the maximum value of p on the field of values of A, where A is a square matrix and p is a polynomial (or more generally an analytic function).  I'll say a few words about the conjecture and
show the beautiful proof of Pearcy in 1966 of a special case, based on a vector-valued barycentric interpolation formula.

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