Tue, 19 Jan 2016

14:30 - 15:00
L5

Sparse information representation through feature selection

Thanasis Tsanas
(University of Oxford)
Abstract
In this talk I am presenting a range of feature selection methods, which are aimed at detecting the most parsimonious subset of characteristics/features/genes. This sparse representation leads always to simpler, more interpretable models, and may lead to improvement in prediction accuracy. I survey some of the state-of-the-art developed algorithms, and discuss a novel approach which is both computationally attractive, and seems to work very effectively across a range of domains, in particular for fat datasets.
Fri, 19 Feb 2016

16:00 - 17:00
L1

North meets South Colloquium

Patrick Farrell + Yufei Zhao
(University of Oxford)
Abstract

Computing distinct solutions of differential equations -- Patrick Farrell

Abstract: TBA

Triangles and equations -- Yufei Zhao

Abstract: I will explain how tools in graph theory can be useful for understanding certain problems in additive combinatorics, in particular the existence of arithmetic progressions in sets of integers. 

Thu, 04 Feb 2016

16:00 - 17:00
L5

Strongly semistable sheaves and the Mordell-Lang conjecture over function fields

Damian Rössler
(University of Oxford)
Abstract

We shall describe a new proof of the Mordell-Lang conjecture in positive characteristic, in the situation where the variety under scrutiny is a smooth subvariety of an abelian variety. 
Our proof is based on the theory of semistable sheaves in positive characteristic, in particular on  Langer's theorem that the Harder-Narasimhan filtration of sheaves becomes strongly semistable after a finite number of iterations of Frobenius pull-backs. Our proof produces a numerical upper-bound for the degree of the finite morphism from an isotrivial variety appearing in the statement of the Mordell-Lang conjecture. This upper-bound is given in terms of the Frobenius-stabilised slopes of the cotangent bundle of the variety.

Thu, 18 Feb 2016

16:00 - 17:00
L5

(Joint Number Theory and Logic) On a modular Fermat equation

Jonathan Pila
(University of Oxford)
Abstract

I will describe some diophantine problems and results motivated by the analogy between powers of the modular curve and powers of the multiplicative group in the context of the Zilber-Pink conjecture.

Tue, 24 Nov 2015

14:30 - 15:00
L5

Geometric integrators in optimal control theory

Sina Ober-Blobaum
(University of Oxford)
Abstract
Geometric integrators are structure-peserving integrators with the goal to capture the dynamical system's behavior in a most realistic way. Using structure-preserving methods for the simulation of mechanical systems, specific properties of the underlying system are handed down to the numerical solution, for example, the energy of a conservative system shows no numerical drift or momentum maps induced by symmetries are preserved exactly. One particular class of geometric integrators is the class of variational integrators. They are derived from a discrete variational principle based on a discrete action function that approximates the continuous one. The resulting schemes are symplectic-momentum conserving and exhibit good energy behaviour. 
 
For the numerical solution of optimal control problems, direct methods are based on a discretization of the underlying differential equations which serve as equality constraints for the resulting finite dimensional nonlinear optimization problem. For the case of mechanical systems, we use variational integrators for the discretization of optimal control problems. By analyzing the adjoint systems of the optimal control problem and its discretized counterpart, we prove that for these particular integrators optimization and discretization commute due to the symplecticity of the discretization scheme. This property guarantees that the convergence rates are preserved for the adjoint system which is also referred to as the Covector Mapping Principle. 
Tue, 17 Nov 2015

14:30 - 15:00
L5

A GPU Implementation of the Filtered Lanczos Procedure

Jared Aurentz
(University of Oxford)
Abstract

This talk describes a graphics processing unit (GPU) implementation of the Filtered Lanczos Procedure for the solution of large, sparse, symmetric eigenvalue problems. The Filtered Lanczos Procedure uses a carefully chosen polynomial spectral transformation to accelerate the convergence of the Lanczos method when computing eigenvalues within a desired interval. This method has proven particularly effective when matrix-vector products can be performed efficiently in parallel. We illustrate, via example, that the Filtered Lanczos Procedure implemented on a GPU can greatly accelerate eigenvalue computations for certain classes of symmetric matrices common in electronic structure calculations and graph theory. Comparisons against previously published CPU results suggest a typical speedup of at least a factor of $10$.

Tue, 17 Nov 2015

14:00 - 14:30
L5

A fast hierarchical direct solver for singular integral equations defined on disjoint boundaries and application to fractal screens

Mikael Slevinsky
(University of Oxford)
Abstract
Olver and I recently developed a fast and stable algorithm for the solution of singular integral equations. It is a new systematic approach for converting singular integral equations into almost-banded and block-banded systems of equations. The structures of these systems lend themselves to fast direct solution via the adaptive QR factorization. However, as the number of disjoint boundaries increases, the computational effectiveness deteriorates and specialized linear algebra is required.

Our starting point for specialized linear algebra is an alternative algorithm based on a recursive block LU factorization recently developed by Aminfar, Ambikasaran, and Darve. This algorithm specifically exploits the hierarchically off-diagonal low-rank structure arising from coercive singular integral operators of elliptic partial differential operators. The hierarchical solver involves a pre-computation phase independent of the right-hand side. Once this pre-computation factorizes the operator, the solution of many right-hand sides takes a fraction of the original time. Our fast direct solver allows for the exploration of reduced-basis problems, where the boundary density for any incident plane wave can be represented by a periodic Fourier series whose coefficients are in turn expanded in weighted Chebyshev or ultraspherical bases.
 
A fractal antenna uses a self-similar design to achieve excellent broadband performance. Similarly, a fractal screen uses a fractal such as a Cantor set to screen electromagnetic radiation. Hewett, Langdon, and Chandler-Wilde have shown recently that the density on the nth convergent to a fractal screen converges to a non-zero element in the suitable Sobolev space, resulting in a physically observable and persistent scattered field as n tends to infinity. We use our hierarchical solver to show numerical results for prefractal screens.
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