The goal of this proposal is to advance a research program of developing sublinear-time algorithms for estimating a wide range of natural and important classes of probability distributions. We live in an era of "big data," where the amount of data that can be brought to bear on questions of biology, climate, economics, etc, is vast and expanding rapidly. Much of this raw data frequently consists of example points without corresponding labels. The challenge of how to make sense of this unlabeled data has immediate relevance and has rapidly become a bottleneck in scientific understanding across many disciplines. An important class of big data is most naturally modeled as samples from a probability distribution over a very large domain. The challenge of big data is that the sizes of the domains of the distributions are immense, typically resulting in unacceptably slow algorithms. Scaling up a computational framework to comfortably deal with ever-larger data presents a series of challenges in algorithms. This prompts the basic question: Given samples from some unknown distribution, what can we infer? While this question has been studied for several decades by various different communities of researchers, both the number of samples and running time required for such estimation tasks are not yet well understood, even for some surprisingly simple types of discrete distributions. The proposed research focuses on sublinear-time algorithms, that is, algorithms that run in time that is significantly less than the domain of the underlying distributions. In this project we will develop sublinear-time algorithms for estimating various classes of discrete distributions over very large domains. Specific problems we will address include: (1) Developing sublinear algorithms to estimate probability distributions that satisfy various natural types of "shape restrictions" on the underlying probability density function. (2) Developing sublinear algorithms for estimating complex distributions that result from the aggregation of many independent simple sources of randomness. We believe that highly efficient algorithms for these estimation tasks may play an important role for the next generation of large-scale machine learning applications.

Multiprocessor machines are now predominant, as most laptops, desktops, servers, mobile phones and aircrafts routinely have multiple to many cores. Unfortunately, concurrent programming is error-prone, which now affects everyone given this trend towards more and more concurrency. Let us mention for example a recent concurrency bug found in the PostgreSQL database (see http://archives.postgresql.org/pgsql-hackers/2011-08/msg00330.php). PostgreSQL is one of the most popular database nowadays, and many websites rely on its correct functioning. This bug was particularly difficult to observe (and indeed is not fixed yet) because it only occurred on a multicore machine, and a particular hardware platform, IBM Power. Reproducing such bugs is as hard as observing them; thus testing can hardly discover them. To prove a program free of errors, we would like to devise automated techniques that analyse the code without executing it. Thus, we can relieve programmers from the burden of writing the proofs of their programs. Yet, automatic verification of concurrent programs represents a challenge, whether it aims at proving the full correctness of a program (e.g. a program sorting a list actually sorts the list), or at checking specific properties (e.g. the program is free of data races) short of full correctness. We focus here on the latter: we would like to enhance the scalability of tools checking that a concurrent program does not violate certain safety-critical properties of interest. We would like to show that scalable automatic verification can be achieved by exploiting the rich history of partial orders for modeling concurrency.

One of the greatest scientific challenges of our time is to provide an answer to the dramatic increase in energy demand and costs. The further optimization of devices such as fuel cells, super capacitors and batteries is central to developing cleaner, cheaper, safer, sustainable energy supplies for the 21st century. New battery technologies, for instance, used with intermittent energy sources like solar and wind, could bring new portable energy solutions to the developing world. Electrochemical (EC) reactions, which usually produce or are driven by an electric current, ultimately dictate the behaviour of most energy devices as well as novel devices for memory and logic applications, such as memristors and EC gating devices. Microscopic processes of this kind occur for instance in electrolytic cells, where water can be split into hydrogen and oxygen thanks to an electrical energy supply, or in batteries where an electrical energy is derived from chemical reactions taking place within the cell. In electrochemistry, the gap between theoretical understanding of microscopic phenomena and the macroscopic outcomes of experiments can be wide. New theoretical and computational approaches save time and cost, validate experimental results, identify new pathways for experiments, and predict exciting new effects with huge potential technological advances. In this fellowship I will develop and apply new computational methodologies, which hold the promise of transforming the way we model, analyse and understand crucial EC processes underlying the functioning of EC devices. To illustrate the importance of advancing in this field and the potential impact in the real world of computer simulations we might recall that the most innovative and fuel efficient plane ever, the Boeing 787 Dreamliner, thousands of models of which were sold before it was even built, has been grounded for months because of a problem with its batteries. This engineering blunder and the related huge loss of revenues could have been prevented by the use of better tools for investigating the properties of such sophisticated batteries, testing and optimising their performance, and thus predicting their behaviour under unusual and hazardous conditions. Whilst uch a complex task is still outside the range of present possibilities, computational research is nonetheless progressing steadily. Recently the amazing development of computational power has made possible the modelling of EC problems purely on the basis of microscopic information on the atomic structure and of our knowledge of electronic phenomena. My research follows precisely this approach. The most beneficial result of my research will be developing the ability to model the effect of an applied potential or a current flow through an EC cell. This will enable for the first time direct atomistic simulations of devices such as EC cells for water splitting and hydrogen production, fuel cells, sensors, batteries, memristors and super-capacitors in operating conditions, e.g. under applied potential and current flow. Understanding these phenomena allows for the design of new strategies - going beyond mere trial and error procedures - for improving current energy technology. Mobile phones batteries lasting more than a week, electric or hydrogen fueled cars are not by any means unforseeable and outlandish future outcomes of these improvements. In the shorter term, we can bear in mind that the leading Li-based technology represents a $10 billion industry with 2 billion cells produced per year. A tiny advance in this technology would deliver significant societal benefits.

Optical lithography is a process that utilises light to define a specific pattern within a material. Standard optical lithography is capable of patterning materials in two dimensions and the possible feature size scales with the wavelength of the light. It is research into this process and associated techniques that has been one of the main drivers of the technological revolution, is partly responsible for the reduction of areal density within computer hard drives and the doubling of processor power every 18 months (Moore's Law). As we progress through the 21st century it is likely that 3D architectures on the nanoscale will become important in developing advanced materials for future data processing and storage technologies. Two-photon lithography is a 3D fabrication methodology that has recently been commercialised and is having a huge impact upon science, allowing the fabrication of bespoke 3D geometries on a length-scale of 200nm horizontally and 500nm vertically. Commercial two-photon lithography has made the fabrication of 3D systems on the several-100nm scale accessible to scientists in a variety of fields allowing the realisation of swimming micro-robots for targeted drug delivery, bioscaffolds and a range of photonic and mechanical metamaterials. A significant setback with two-photon lithography is the asymmetry in the lateral and vertical resolution, which limits both the absolute size and the type of geometry that can be realised. In this proposal, we are going to utilise our world-leading expertise in non-linear microscopy to modify a commercial two-photon lithography system and obtain enhanced resolution. We will utilise techniques that have already significantly improved the resolution in fluorescence microscopy in order to achieve a 100nm isotropic resolution. The newly built system will be used by our team to fabricate two types of 3D nanoscale magnetic materials, in geometries and on length-scales that are difficult to achieve using other fabrication methodologies. Our work in this area will pave the way for next generation 3D memory technolgies such as magnetic racetrack memory and help us to understand magnetic charge transport in novel magnetic materials. In addition, we will be working with project partners in the regenerative medicine and photonics communities in order to realise a number of novel 3D nanostructured materials. Firstly, we will work with stem cell researchers in order to fabricate artificial tissues that will be used in stem cell differentiation experiments. Our work here will provide a fascinating insight into the role of nanoscale topography upon stem cell differentiation and may eventually have applications in tissue/organ growth. Secondly, we will work with academics studying photonic crystals - artificial materials that are capable of blocking electromagnetic radiation within a certain range of the spectrum. The majority of 3D photonic crystals that have been made to date are capable of attenuating electromagnetic waves that are outside the visible range of the spectrum, limiting applications in optoelectronics. Our work here will allow the fabrication and measurement of photonic crystals that can be used with visible and infra-red light. This work may pave the way to next generation three-dimensional optical circuits that can be utilised by telecommunication industries. Overall, this project will build an internationally unique instrument and utilise it to fabricate a range of advanced materials. This will put the U.K. at the forefront of 3D lithography technologies and the associated biomedical, magnetic and photonic materials that will be realised using our newly built instrument.

At the 2015 Paris climate conference, 195 countries agreed that global greenhouse gases should peak as soon as possible and that countries should thereafter rapidly reduce their emissions. The process industries must therefore reduce their energy consumption and increase efficiency while maintaining consumer services. Next generation decision-making software at the interface of engineering, computer science, and mathematics is critical for these efficient systems of the future. Already, state-of-the-art computational packages are routine in the process industries; practically every major company uses simulation and optimisation to model production in different modes including: continuous, batch, and semi-continuous production systems. But more efficient industrial systems require simultaneously considering many tightly integrated subsystems which exponentially increase complexity and necessitate many temporal/spatial scales; the resulting decision making problems may not be solvable with current techniques. Increasing efficiency may also jeopardise safety: the process integration required for efficiency implies interchanging heat between processes and may damage safety precautions by transferring disturbances across a plant. During this fellowship, we propose to develop GALINI, new decision-making software constructing and deploying next generation process optimisation tools dealing with combinatorial complexity, disparate temporal/spatial scales, and safety considerations. The GALINI project proposes step-changes in optimisation algorithms that are immediately applicable to efficiency challenges in process systems engineering (PSE): safely operating batch reactors, retrofitting heat-exchanger networks, intermediate blending, and integrating planning and scheduling. We will freely release our software on open-source platform Pyomo and build an international user community. The primary GALINI research aim is to develop optimisation software that pushes the boundary of computational tractability for PSE energy efficiency applications. Effective optimisation software in the process industries answers: How can we best achieve a definite engineering objective? Given constraints such as an existing plant layout or a contractual obligation to produce specific products, the software supports novel engineering by quantitatively comparing the implications of different options and identifying the best decision. GALINI is particularly interested in design: How should we build new facilities or modify existing ones to achieve our design goals with maximum efficiency? The state-of-the-art in decision making for the process industries is represented by commercial modelling software such as AspenTech and gPROMS. Practically every major company in the process industries uses these software tools since the outputs of the simulation or optimisation can be implemented with minimal day-to-day operational disruption and savings can be realised with a payback time as short as 6-12 months. GALINI will develop deterministic global optimisation software for mixed-integer nonlinear programs, a type of optimisation problem highly relevant to energy efficiency and process systems engineering. Energy efficiency instances may exhibit the mathematical property of nonconvexity, i.e. have many locally optimal solutions; global optimisation mathematically guarantees the best process engineering solution. GALINI proposes transformational shifts in algorithms that creatively reimagine the core divide-and-conquer algorithm typically applied to this type of optimisation problem. Our approach is to freely release GALINI to users including those in the process industries, publicise the software, demonstrate its utility, and build a user community that will feed back into software development.