Manchester University

In many applications we need to find or estimate the $p \ge 1$ largest elements of a matrix, along with their locations. This is required for recommender systems used by Amazon and Netflix, link prediction in graphs, and in finding the most important links in a complex network, for example. 

Our algorithm uses only matrix vector products and is based upon a power method for mixed subordinate norms. We have obtained theoretical results on the convergence of this algorithm via a comparison with rook pivoting for the LU  decomposition. We have also improved the practicality of the algorithm by producing a blocked version iterating on $n \times t$ matrices, as opposed to vectors, where $t$ is a tunable parameter. For $p > 1$ we show how deflation can be used to improve the convergence of the algorithm. 

Finally, numerical experiments on both randomly generated matrices and real-life datasets (the latter for $A^TA$ and $e^A$) show how our algorithms can reliably estimate the largest elements of a matrix whilst obtaining considerable speedups when compared to forming the matrix explicitly: over 1000x in some cases.

We propose the use of a constraint preconditioner for the iterative solution of the linear system arising from the finite element discretization of the coupled Stokes-Darcy system. The Stokes-Darcy system is a set of coupled PDEs that can be used to model a freely flowing fluid over porous media flow. The fully coupled system matrix is large, sparse, non-symmetric, and of saddle point form. We provide for exact versions of the constraint preconditioner spectral and field-of-values bounds that are independent of the underlying mesh width. We present several numerical experiments, using the deal.II finite element library, that illustrate our results in both two and three dimensions. We compare exact and inexact versions of the constraint preconditioner against standard block diagonal and block lower triangular preconditioners to illustrate its favorable properties.

Multiplicative chaos theory originated from the study of turbulence by Kolmogorov in the 1940s and it was mathematically founded by Kahane in the 1980s. Recently the theory has drawn much of attention due to its connection to SLEs and statistical physics.  In this talk I shall present some recent development of multiplicative chaos theory, as well as its applications to Liouville quantum gravity.

Some problems in scientific computing, like the forward simulation of electromagnetic waves in geophysical prospecting, can be
solved via approximation of f(A)b, the action of a large matrix function f(A) onto a vector b. Iterative methods based on rational Krylov
spaces are powerful tools for these computations, and the choice of parameters in these methods is an active area of research.
We provide an overview of different approaches for obtaining optimal parameters, with an emphasis on the exponential and resolvent function, and the square root. We will discuss applications of the rational Arnoldi method for iteratively generating near-optimal absorbing boundary layers for indefinite Helmholtz problems, and for rational least squares vector fitting.

For many tomographic imaging problems there are explicit inversion formulas, and depending on the completeness of the data these are unstable to differing degrees. Increasingly we are solving tomographic problems as though they were any other linear inverse problem using numerical linear algebra. I will illustrate the use of numerical singular value decomposition to explore the (in)stability for various problems. I will also show how standard techniques from numerical linear algebra, such as conjugate gradient least squares, can be employed with systematic regularization compared with the ad hoc use of slowly convergent iterative methods more traditionally used in computed tomography. I will mainly illustrate the talk with examples from three dimensional x-ray tomography but I will also touch on tensor tomography problems.
 

Central extensions of a finite group G correspond to 2-cocycles on G, which give rise to an abelian cohomology group known as the Schur

multiplier of G. Recently, the Schur multiplier was defined in a much more

general setting of a monoidal category. I will explain how to twist algebras by categorical 2-cocycles and will mention the role of

such twists the theory of quantum groups. I will then describe an approach to twisting rational Cherednik algebras by cocycles,

and will discuss possible applications of this new construction to the representation theory of these algebras.

In this talk we will begin by discussing the notion of homogenization as an extension to the continuum assumption and regimes in which it breaks down. We then discuss various approaches to dealing with randomness whilst determining the effective properties of acoustic, thermal and elastic media.  In particular we show how the effective properties depend on the randomness of the microstructure