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. 

Understanding networks - from social media to rabbit warrens

A booklet about networks literacy developed by Mason Porter, Fellow of Somerville College and Professor of Nonlinear and Complex Systems in the University of Oxford's Mathematical Institute, in collaboration with colleagues from the USA, could help people understand all types of networks from social media to rabbit warrens. Mason was part of a team of over 30 network-science researchers, educators, teachers, and students who have written the booklet on networks literacy that schools can adapt to teach students the core concepts about networks.

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 $10$.

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