The phenomenon of poor algorithmic scalability is a critical problem in large-scale machine learning and data science. This has led to a resurgence in the use of first-order (Hessian-free) algorithms from classical optimisation. One major drawback is that first-order methods tend to converge extremely slowly. However, there exist techniques for efficiently accelerating them.
    
The topic of this talk is the Dual Regularisation Nonlinear Acceleration algorithm (DRNA) (Geleta, 2017) for nonconvex optimisation. Numerical studies using the CUTEst optimisation problem set show the method to accelerate several nonconvex optimisation algorithms, including quasi-Newton BFGS and steepest descent methods. DRNA compares favourably with a number of existing accelerators in these studies.
    
DRNA extends to the nonconvex setting a recent acceleration algorithm due to Scieur et al. (Advances in Neural Information Processing Systems 29, 2016). We have proven theorems relating DRNA to the Kylov subspace method GMRES, as well as to Anderson's acceleration method and family of multi-secant quasi-Newton methods.
 

In this talk we describe a new approach that enables the use of elliptic PDEs with white noise forcing to sample Matérn fields within the multilevel Monte Carlo (MLMC) framework.

When MLMC is used to quantify the uncertainty in the solution of PDEs with random coefficients, two key ingredients are needed: 1) a sampling technique for the coefficients that satisfies the MLMC telescopic sum and 2) a numerical solver for the forward PDE problem.

When the dimensionality of the uncertainty in the problem is infinite (i.e. coefficients are random fields), the sampling techniques commonly used in the literature are Karhunen–Loève expansions or circulant embeddings. In the specific case in which the coefficients are Gaussian fields of Mat ́ern covariance structure another sampling technique available relies on the solution of a linear elliptic PDE with white noise forcing.

When the finite element method (FEM) is used for the forward problem, the latter option can become advantageous as elliptic PDEs can be quickly and efficiently solved with the FEM, the sampling can be performed in parallel and the same FEM software can be used without the need for external packages. However, it is unclear how to enforce a good stochastic coupling of white noise between MLMC levels so as to respect the MLMC telescopic sum. In this talk we show how this coupling can be enforced in theory and in practice.

Shortly after Mason & Skinner introduced the so-called ambitwistor strings, Berkovits came up with a pure-spinor analogue of the theory, which was later shown to provide the supersymmetric version of the Cachazo-He-Yuan amplitudes. In the heterotic version, however, both models give somewhat unsatisfactory descriptions of the supergravity sector.

In this talk, I will show how the original pure-spinor version of the heterotic ambitwistor string can be modified in a consistent manner that renders the supergravity sector treatable. In addition to the massless states, the spectrum of the new model --- which we call sectorized heterotic string --- contains a single massive level. In the limit in which a dimensionful parameter is taken to infinity, these massive states become the unexpected massless states (e.g. a 3-form potential) first encountered by Mason & Skinner."

Topologists have the Steenrod squares, a collection of additive homomorphisms on the Z/2 cohomology of a space M. They can be defined axiomatically and are often be regarded as algebraic operations on cohomology groups (for many purposes). However, Betz and Cohen showed that they could be viewed geometrically. 

Symplectic geometers have quantum cohomology, which on a symplectic manifold M is a deformation of singular cohomology using holomorphic spheres.

The geometric definition of the Steenrod square extends to quantum cohomology. This talk will describe the Steenrod square and quantum cohomology in terms of the intersection product, and then give a description of this quantum Steenrod square by putting these both together. We will describe some properties of the quantum squares, such as the quantum Cartan formula, and perform calculations in certain cases.