University of Edinburgh

I will explain how, given a crepant morphism with one-dimensional fibres between 3-folds, it is possible to use noncommutative deformations to run the MMP in a satisfyingly algorithmic fashion.  As part of this, a flop is viewed homologically as the solution to a universal property, and so is constructed not by changing GIT, but instead by changing the algebra. Carrying this extra information of the new algebra allows us to continue to flop, and thus continue the MMP, without having to calculate everything from scratch. Proving things in this manner does in fact have other consequences too, and I will explain some them, both theoretical and computational.

The accurate and efficient solution of linear systems $Ax=b$ is very important in many engineering and technological applications, and systems of this form also arise as subproblems within other algorithms. In particular, this is true for interior point methods (IPM), where the Newton system must be solved to find the search direction at each iteration. Solving this system is a computational bottleneck of an IPM, and in this talk I will explain how preconditioning and deflation techniques can be used, to lessen this computational burden.  This work is joint with Jacek Gondzio.

The Maxwell equation is an intermediate linear model for

Einstein's equation lying between the scalar wave equation and the

linearised Einstein equation. This talk will present the 5 key

estimates necessary to prove a uniform bound on an energy and a

Morawetz integrated local energy decay estimate for the nonradiating

part.

The major obstacles, relative to the scalar wave equation are: that a

scalar equation must be found for at least one of the components,

since there is no known decay estimate directly at the tensor level;

that the scalar equation has a complex potential; and that there are

stationary solutions and, in the nonzero $a$ Kerr case, it is more

difficult to project away from these stationary solutions.

If time permits, some discussion of a geometric proof using the hidden

symmetries will be given.

This is joint work with L. Andersson and is arXiv:1310.2664.

Molecular modelling has become a valuable tool and is increasingly part of the standard methodology of chemistry, physics, engineering and biology. The power of molecular modelling lies in its versatility: as potential energy functions improve, a broader range of more complex phenomena become accessible to simulation, but much of the underlying methodology can be re-used. For example, the Verlet method is still the most popular molecular dynamics scheme for constant energy molecular dynamics simulations despite it being one of the first to be proposed for the purpose.

One of the most important challenges in molecular modelling remains the computation of averages with respect to the canonical Gibbs (constant temperature) distribution, for which the Verlet method is not appropriate. Whereas constant energy molecular dynamics prescribes a set of equations (Newton's equations), there are many alternatives for canonical sampling with quite different properties. The challenge is therefore to identify formulations and numerical methods that are robust and maximally efficient in the computational setting.

One of the simplest and most effective methods for sampling is based on Langevin dynamics which mimics coupling to a heat bath by the incorporation of random forces and an associated dissipative term. Schemes for Langevin dynamics simulation may be developed based on the familiar principle of splitting. I will show that the invariant measure ('long term') approximation may be strongly affected by a simple re-ordering of the terms of the splitting. I will describe a transition in weak numerical order of accuracy that occurs (in one case) in the t->infty limit.

I will also entertain some more radical suggestions for canonical sampling, including stochastic isokinetic methods that enable the use of greatly enlarged timesteps for expensive but slowly-varying force field components.

The Einstein equation from general relativity is a

quasilinear hyperbolic, geometric PDE (when viewed in an appropriate

coordinate system) for a manifold. A particularly interesting set of

known, exact solutions describe black holes. The wave and Maxwell

equations on these manifolds are models for perturbations of the known

solutions and have attracted a significant amount of attention in the

last decade. Key estimates are conservation of energy and Morawetz (or

integrated local energy) estimates. These can be proved using both

Fourier analytic methods and more geometric methods. The main focus of

the talk will be on decay estimates for solutions of the Maxwell

equation outside a slowly rotating Kerr black hole.

The standard Taylor series approach to the higher-order approximation of vector SDEs requires simulation of iterated stochastic integrals, which is difficult. The talk will describe an approach using methods from optimal transport theory which avoid this difficulty in the case of non-degenerate diffusions, for which one can attain arbitrarily high order pathwise approximation in the Vaserstein 2-metric, using easily generated random variables.

I will discuss properties of stochastic differential equations and numerical algorithms for sampling Gibbs (i.e smooth) measures. Methods such as Langevin dynamics are reliable and well-studied performers for molecular sampling.   I will show that, when the objective of simulation is sampling of the configurational distribution, it is possible to obtain a superconvergence result (an unexpected increase in order of accuracy) for the invariant distribution.   I will also describe an application of thermostats to the Hamiltonian vortex method in which the energetic interactions with a bath of weak vortices are treated as thermal fluctuations

This talk will consist of two parts. In the first part we will present a motivating application from oil reservoir simulation, namely finding the location and trajectory of an oil producing well which maximises oil production. We will show how such a problem can be tackled through the use of radial basis function (RBF) approximation (also known as Kriging or Gaussian process regression) and a branch and bound global optimization algorithm.

In the second part of the talk we will show how one can improve the branch and bound algorithm through the use of Lipschitz continuity of the RBF approximation. This leads to an entirely new global optimization algorithm for twice differentiable functions with Lipschitz continuous Hessian. The algorithm makes use of recent cubic regularisation techniques from local optimization to obtain the necessary bounds within the branch and bound algorithm.

We show that the steepest-descent and Newton's methods for unconstrained nonconvex optimization

under standard assumptions may both require a number of iterations and function evaluations

arbitrarily close to the steepest-descent's global worst-case complexity bound. This implies that

the latter upper bound is essentially tight for steepest descent and that Newton's method may be as

slow as the steepest-descent method in the worst case. Then the cubic regularization of Newton's

method (Griewank (1981), Nesterov & Polyak (2006)) is considered and extended to large-scale

problems, while preserving the same order of its improved worst-case complexity (by comparison to

that of steepest-descent); this improved worst-case bound is also shown to be tight. We further

show that the cubic regularization approach is, in fact, optimal from a worst-case complexity point

of view amongst a wide class of second-order methods. The worst-case problem-evaluation complexity

of constrained optimization will also be discussed. This is joint work with Nick Gould (Rutherford

Appleton Laboratory, UK) and Philippe Toint (University of Namur, Belgium).

We present progress on an Interior Point based multi-step solution approach for stochastic programming problems. Our approach works with a series of scenario trees that can be seen as successively more accurate discretizations of an underlying probability distribution and employs IPM warmstarts to "lift" approximate solutions from one tree to the next larger tree.