## Abstracts of Annual Meeting, 14th March 2014

## Ahmed Elsheikh

### Calibration of petroleum reservoirs using compressed sensing techniques

**Abstract:**

In this talk, I will present a casual introduction to the field of compressed sensing. In many applications, signals can be approximated using a linear combination of global basis functions. Given a large set of basis functions, sparse reconstruction is concerned with picking a subset of these basis functions to represent the signal. In recent years, it was shown that a signal with a sparse representation could be exactly recovered from a small set of measurements. These ideas have reaching applications since most computational models rely on poorly known parameterfields that need to be recovered from few measurements. For example, the permeability field in a petroleum reservoir is only known at few spatial locations. Inferring the distributed permeability field enables better predictions of the flow in the reservoir that can assist the petroleum engineers with operational design and production optimization. I will show how to apply some compressed sensing ideas for inferring and calibrating subsurface petroleum reservoirs.

## Ilias Diakonikolas

### A Complexity-Theoretic View on Unsupervised Learning

**Abstract:**

Valiant's Probably Approximately Correct (PAC) learning model brought a computational complexity perspective to the study of machine learning. The PAC framework deals with *supervised* learning problems, where data points are labeled by some target function and the goal is to infer a high-accuracy hypothesis that is close to the target function. The PAC learning model and its variants provide a useful and productive setting for studying how the complexity of learning different types of Boolean functions scales with the complexity of the functions being learned.

A large portion of contemporary machine learning deals with *unsupervised* learning problems. In problems of this sort data points are unlabeled, so there is no "target function" to be learned; instead the goal is to infer some structure from a sample of unlabeled data points.

This talk will focus on the problem of learning an unknown probability distribution given access to independent samples drawn from it. Analogous to the PAC model for learning Boolean functions, the broad goal here is to understand how the complexity of learning different types of distributions scales with the complexity of the distributions being learned. We survey recent results in this area and identify questions for future work.

## Ben Leimkuhler

### Computing averages using stochastic numerical methods

**Abstract:**

Many applications rely on the use of models which are fundamentally stochastic in nature, as when some physical variable is strongly affected by perturbations due to unresolved degrees of freedom. Examples include a molecular system in contact with a reservoir consisting of many neighboring atoms or a truncated discrete model for a fluid dynamics system. A popular model that incorporates uncertainty is Langevin dynamics, whereby a conservative model (e.g. Newton's equations for an N-body system) is supplemented with dissipative and random perturbations in such a way that a prescribed invariant distribution is obtained. In the stochastic setting, the purpose of computational modelling is typically to calculate the average of an observable function, say the energy, the pressure, or an indicator of average size or flexibility. As the models involved are stochastic differential equations, we must utilize a numerical method to perform the computation of the average, and it then becomes a question of understanding the properties of various numerical schemes, such as the convergence order, which must be defined carefully to be of relevance for the applications of interest.

In this talk, I will discuss the weak (in the sense of averages) convergence of numerical methods for SDEs (for both Langevin dynamics and stochastic gradient systems), with particular attention to the order of accuracy in the t->infinity limit. I will show that it is possible to describe the weak error by a perturbation series in the stepsize. This work has turned up new numerical methods, including a simple replacement for the popular Euler-Maruyama method for stochastic gradient systems that has an unexpected weak order increase in the long time limit. I will also demonstrate that the indicated schemes are practical and efficient for convincing applications problems, including systems arising in the molecular modelling setting.

## Joan Simon

### The singular marriage between quantum mechanics and gravity

**Abstract:**

Losing information about any system generates entropy. I will review how this mechanism operates in quantum mechanics, its relevance for the foundations of statistical mechanics and the physics of black holes and the nature of spacetime.

## Christina Cobbold

### Modelling with-in host infection: the interplay of ecology and genetics

**Abstract:**

Mathematical models of infectious diseases have a long history with early models offering important insight into epidemics and guidance for designing effective vaccination strategies.

Increasing availability of genetic data offers new opportunities to understand the fundamental mechanisms of how an infection plays out within a single host. In this talk I will present a mathematical model for African sleeping sickness, a potentially fatal disease caused by the parasite trypanosome. Typansomes are among the many parasites that exhibit genetic variation as a mechanism to sustain chronic infections within their hosts. Other examples where genetic variation is important include influenza and malaria infection. In the case of the trypansome parasite it obtains protection from the hosts immune response by switching between genetically distinct parasite variants. Typical trypanosome infections consist of oscillations in the level of infection present in the hosts, where each peak is composed of a group of genetically distinct parasite variants. In this talk I will present a mathematical model used to investigate within-host dynamics of African trypanosomes and examine the role and limitations of parasite genetic variation.

In particular I will discuss how trypansome infection may play out differently in different types of hosts.