Past Numerical Analysis Group Internal Seminar

19 January 2016
14:30
Thanasis Tsanas
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.
  • Numerical Analysis Group Internal Seminar
24 November 2015
14:30
Sina Ober-Blobaum
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. 
  • Numerical Analysis Group Internal Seminar
24 November 2015
14:00
Peter McCullagh
Abstract
The $\alpha$-permanent of a square matrix is a determinant-style sum, with $\alpha=-1$ corresponding to the determinant, $\alpha=1$ to the ordinary permanent, and $\alpha=0$ to the Hamiltonian sum over cyclic permutations.  Exact computation of permanents is notoriously difficult; numerical computation using the best algorithm for $\alpha=1$ is feasible for matrices of order about 25--30; numerical computation for general $\alpha$ is feasible only for $n < 12$.  I will describe briefly how the $\alpha$-permanent arises in statistical work as the probability density function of the Boson point process, and I will discuss the level of numerical accuracy needed for statistical applications.  My hope is that, for sufficiently large matrices, it may be possible to develop a non-stochastic polynomial-time approximation of adequate accuracy.
  • Numerical Analysis Group Internal Seminar
17 November 2015
14:30
Jared Aurentz
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 $10$.

  • Numerical Analysis Group Internal Seminar
17 November 2015
14:00
Mikael Slevinsky
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.
  • Numerical Analysis Group Internal Seminar
10 November 2015
14:00
Philippe Toint
Abstract

The talk will describe a new "Brute Force Optimizer" whose objective is to provide a very versatile derivative-free Matlab package for bound-constrained optimization, with the distinctive feature that it can be trained to improve its own performance on classes of problems specified by the user (rather than on a single-but-wide problem class chosen by the algorithm developer).  In addition, BFO can be used to optimize the performance of other algorithms and provides facilities for mixed-integer and multilevel problems, including constrained equilibrium calculations.

  • Numerical Analysis Group Internal Seminar
3 November 2015
14:30
Niall Bootland
Abstract

Modelling two-phase, incompressible flow with level set or volume-of-fluid formulations results in a variable coefficient Navier-Stokes system that is challenging to solve computationally. In this talk I will present work from a recent InFoMM CDT mini-project which looked to adapt current preconditioners for one-phase Navier-Stokes flows. In particular we consider systems arising from the application of finite element methodology and preconditioners which are based on approximate block factorisations. A crucial ingredient is a good approximation of the Schur complement arising in the factorisation which can be computed efficiently.

  • Numerical Analysis Group Internal Seminar

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