Past Industrial and Interdisciplinary Workshops

Elkem is developing a new method for categorising the reactivity between silicon carbide (SiC) powder and silicon monoxide gas (SiO(g)). Experiments have been designed which pass SiO gas through a powdered bed of SiC inside of a heated crucible, resulting in a reaction between the two. The SiO gas is produced via a secondary reaction outside of the SiC bed. Both reactions require specific temperature and pressure constraints to occur. Therefore, we would like to mathematically model the temperature distribution and gas flow within the experimental set-up to provide insight into how we can control the process.

Complexities arise from:

• Endothermic reactions causing heat sinks
• Competing reactions beyond the two we desire
• Dynamically changing properties of the bed, such as permeability

Monumo is interested in computing physical properties of electric motors (torque, efficiency, back EMF) from their designs (shapes, materials, currents). This involves solving Maxwell's equations (non-linear PDEs). They currently compute the magnetic flux, and then use that to compute the other properties of interest. The main challenge they face is that they want to do this for many, many different designs. There seems to be lots of redundancy here, but exploiting it has proved difficult.

As one of the largest retailers in the world, Tesco relies on automated forecasting to help with decision making. A common issue with forecasts is that of the cold start problem; that we must make forecasts for new products that have no history to learn from. Lack of historical data becomes a real problem as it prevents us from knowing how products react to events, and if their sales react to the time of year. We might consider using similar products as a way to produce a starting forecast, but how should we define what ‘similar’ means, and how should we evolve this model as we start getting real live data? We’ll present some examples to hopefully start a fruitful discussion.

Movie Me would like offer dynamical pricing for movie tickets, considering consumer’s demand for the movie, showtime and lead time before the show begins, such that the overall quantity of tickets sold is maximized. We encourage all interested party to join us and especially those interested in data science, optimization and mathematical finance.

Peel Ports operate a number of locks that allow ships to enter and leave the port. The lock gates comprise a single caisson structure which blocks the waterway when closed and retracts into the dockside as the gate opens. Build up of silt ahead of the opening lock gate can prevent it from fully opening or requiring excessive power to move. If the lock is not able to fully open, ships are unable to enter the port, leading to significant operational impacts for the whole port. Peel ports are interested in understanding, and mitigating, this silt build up. 

Alumina is a raw material for aluminium production, and Attila Kovacs made mathematical models for alumina feeding, including heating, melt infiltration, and dissolution. One of his assumptions is that when several alumina particle stick together to form a "raft", these will stay together even if initial frozen cryolite inside this "raft" melts, and even if almost all alumina in the "raft" is dissolved. In reality, the "raft" will break up, either from one of the two mechanisms already mentioned, or from the expansion of gas or water vapor stuck within the "raft". We would therefore like to investigate mathematically when and under which circumstances this splitting up will take place. 

The questions we would like to answer are as follows:

1. Given three distributions pdf1, pdf2 and pdf-so, is it always possible to find a joint-distribution consistent with those 3 one-dimensional distributions?
2. Assuming that we are in a situation where (1) holds, can we find a nonparametric joint-distribution consistent with the 3 given one-dimensional distributions?
3. If (2) leads to an under-determined problem, can we find a joint-distribution that is “as close as possible” to the historical joint distribution?
4. Can we achieve (3) with a neural network?
5. If we observe the marginal and spread distributions for multiple maturities T, can we specify the evolution of pdf(T), possibly using neural differential equations?

MHD equilibrium is an important topic for fusion (and other MHD applications). A tokamak, in principle, is a toroidally symmetric fusion device and so MHD equilibrium can be reduced to solving the time independent MHD equations in axisymmetry. This produces the Grad-Shafranov equation (a two dimensional, nonlinear PDE) which has been solved using various techniques in the fusion community including finite difference, finite elements and spectral methods. A similar PDE exists if there is a plasma column with helical symmetry. Non-axisymmetric plasmas do occur in tokamaks as a result of instabilities and applied fields. However, if there is no symmetry angle there is no PDE to be solved. The current workhorse for finding non-axisymmetric equilibria uses energy minimization to find the equilibrium. New approaches to this problem that can use state of the art techniques are desirable. The speaker has formulated a coupled set of PDEs for the non-axisymmetric MHD equilibrium problem assuming that flux surfaces are nested (i.e. there are no magnetic islands) and has written this in weak form to use finite element method to solve the equations. The questions are around whether there is an optimal way to try to formulate the problem for FEM and to couple the equations, what sort of elements to use, if other solution techniques would be better suited and so on.

The discrete Fourier transform is fundamental in modern communication systems.  It is used to generate and process (i.e. modulate and demodulate) the signals transmitted in 4G, 5G, and wifi systems, and is always implemented by one of the fast Fourier transforms (FFT) algorithms.  It is possible to generalize the FFT to work correctly on input vectors with periodic missing values.   I will consider whether this has applications, such as more general transmitted signal waveforms, or further applications such as spectral density estimation for time series with missing data.  More speculatively, can we generalize to "recursive" missing values, where the non-missing blocks have gaps?   If so, how do we optimally recognize such a pattern in a given time series?

We would like to share two challenges that, if solved, could improve our domestic lives.  

 Firstly, having appliances that are as unobtrusive as possible is a strong desire, unwanted noise can cause a negative impact on relaxation.  A key target for refrigerators is low sound level, a key noise source is the capillary tube.  The capillary tube effects the phase change that is required for the refrigerant to be in the gaseous state in the evaporator for cooling.  Noise is generated during this process due to two phases being present within the flow through the tube.  The challenge is to create a numerical model and analysis of refrigerant flow properties in order to estimate the acoustic behaviour.

 Secondly, we would like to maximise the information that can be gathered from our new range of connected devices.  By analysing the data generated during usage we would like to be able to predict faults and understand user behaviour in more detail.  The challenge regarding fault prediction is the scarcity of the failure data and the impact of false positives.  Due to the number of units in the field, a relatively small fraction of false positives can remove the ROI from such an initiative.  We would like to understand if advanced machine learning methods can be used to reduce this risk.

There are strengths and weaknesses to both mathematical models and machine learning approaches, for instance mathematical models may be difficult to fully specify or become intractable when representing complex natural or built environments whilst machine learning models can be inscrutable (“black box”) and perform poorly when driven outside of the range of data they have been trained on. At the same time measured data from sensors is becoming increasing available.

We have been working to try and bring the best of both worlds together and we would like to discuss our work and the challenges it presents. Such challenges include model simplification or reduction, model performance in previously unobserved extreme conditions, quantification of uncertainty and techniques to parameterise mathematical models from data.

Maths-Whizz is an online, virtual maths tutor for 5-13 year-olds that is designed to behave like a human tutor. Using adaptive assessment and decision-tree algorithms, the virtual tutor guides each student along a personalised learning journey tailored to their needs. As students interact with the tutor, the system captures a range of learning analytics as an automatic by-product. These analytics, collected on a per-lesson and per-question basis, then inform a range of research projects centred on students' learning patterns. This workshop will introduce the mechanics of the Maths-Whizz tutor, as well as its related learning analytics. We will summarise the research behind four InfoMM mini-projects and present open questions we are currently grappling with. Maths-Whizz has supported over a million children and thousands of schools worldwide, from the UK and US to rural Kenya, the DRC and Mexico. In a world of social distancing and widespread school closures, the need for virtual tutoring has never been more paramount to children's learning - and nor has your data analytical expertise!

Modern cellular radio systems such as 4G and 5G use antennas with multiple elements, a technique known as MIMO, and the intention is to increase the capacity of the radio channel.  5G allows even more possibilities, such as massive MIMO, where there can be hundreds of elements in the transmit antenna, and beam-forming (or beam-steering), where the phase of the signals fed to the antenna elements is adjusted to focus the signal energy in the direction of the receivers.  However, this technology poses some difficult optimization problems, and here mathematicians can contribute.   In this talk I will explain the background, and then look at questions such as: what is an appropriate objective function?; what constraints are there?; are any problems of this type convex (or quasi-convex, or difference-of-convex)?; and, can big problems of this type be solved in real time?

We present a simple algorithm that successfully re-constructs a sine wave, sampled vastly below the Nyquist rate, but with sampling time intervals having small random perturbations. We show how the fact that it works is just common sense, but then go on to discuss how the procedure relates to Compressed Sensing. It is not exactly Compressed Sensing as traditionally stated because the sampling transformation is not linear.  Some published results do exist that cover non-linear sampling transformations, but we would like a better understanding as to what extent the relevant CS properties (of reconstruction up to probability) are known in certain relatively simple but non-linear cases that could be relevant to industrial applications.

RF-engineering defines the “perfect” parabolic shape a foldable reflector antenna (e.g. the membrane) should have. In practice it is virtually impossible to design a deployable backing structure that can meet all RF-imposed requirements. Inevitably the shape of the membrane will deviate from its ideal parabolic shape when material properties and pragmatic mechanical design are considered. There is therefore a challenge to model such membranes in order to find the form they take and then use the model as a design tool and perhaps in an optimisation objective function, if tractable. 

The variables we deal with are:
Elasticity of the membrane (anisotropic or orthotropic typ)
Boundary forces (by virtue of the interaction between the membrane and it’s attachment)
Elasticity of the backing structure (e.g. the elasticity properties of the attachment)
Number, location and elasticity of the membrane fixing points

There are also in-orbit environmental effects on such structures for which modelling could also be of value. For example, the structure can undergo thermal shocks and oscillations can occur that are un-dampened by the usual atmospheric interactions at ground level etc. There are many other such points to be considered and allowed for.

Background: The traditional business models for B2B freight and distribution are struggling with underutilised transport capacities resulting in higher costs, excessive environmental damage and unnecessary congestion. The scale of the problem is captured by the European Environmental Agency: only 63% of journeys carry useful load and the average vehicle utilisation is under 60% (by weight or volume). Decarbonisation of vehicles would address only part of the problem. That is why leading sector researchers estimate that freight collaboration (co-shipment) will deliver a step change improvement in vehicle fill and thus remove unproductive journeys delivering over 20% of cost savings and >25% reduction in environmental footprint. However, these benefits can only be achieved at a scale that involves 100’s of players collaborating at a national or pan-regional level. Such scale and level of complexity creates a massive optimisation challenge that current market solutions are unable to handle (modern route planning solutions optimise deliveries only within the “4 walls” of a single business).

Maths challenge: The mentioned above optimisation challenge could be expressed as an extended version of the TSP, but with multiple optimisation objectives (other than distance). Moreover, besides the scale and multi-agent setup (many shippers, carriers and recipients engaged simultaneously) the model would have to operate a number of variables and constraints, which in addition to the obvious ones also include: time (despatch/delivery dates/slots and journey durations), volume (items to be delivered), transport equipment with respective rate-cards from different carriers, et al. With the possible variability of despatch locations (when clients have multi-warehouse setup) this potentially creates a very-large non-convex optimisation problem that would require development of new, much faster algorithms and approaches. Such algorithm should be capable of finding “local” optimums and subsequently improve them within a very short window i.e. in minutes, which would be required to drive and manage effective inter-company collaboration across many parties involved. We tried a few different approaches eg used Gurobi solver, which even with clustering was still too slow and lacked scalability, only to realise that we need to build such an algorithm in-house.

Ask: We started to investigate other approaches like Simulated Annealing or Gravitational Emulation Local Search but this work is preliminary and new and better ideas are of interest. So in support of our Technical Feasibility study we are looking for support in identification of the best approach and design of the actual algorithm that we’ll use in the development of our Proof of Concept.  

This challenge relates to problems (of a mathematical nature) in generating optimal solutions for natural flood management.  Natural flood management involves large numbers of small scale interventions in a much larger context through exploiting natural features in place of, for example, large civil engineering construction works. There is an optimisation problem related to the catchment hydrology and present methods use several unsatisfactory simplifications and assumptions that we would like to improve on.

Background

The RON test is an engine test that is used to measure the research octane number (RON) of a gasoline. It is a parameter that is set in fuels specifications and is an indicator of a fuel to partially explode during burning rather than burn smoothly.

The efficiency of a gasoline engine is limited by the RON value of the fuel that it is using. As the world moves towards lower carbon, predicting the RON of a fuel will become more important.

Typical market gasolines are blended from several hundred hydrocarbon components plus alcohols and ethers. Each component has a RON value and therefore, if the composition is known then the RON can be calculated. Unfortunately, components can have antagonistic or complimentary effects on each other and therefore this needs to be taken into account in the calculation.

Several models have been produced over the years (the RON test has been around for over 60 years) but the accuracy of the models is variable. The existing models are empirically based rather than taking into account the causal links between fuel component properties and RON performance.

Opportunity

BP has developed intellectual property regarding the causal links and we need to know if these can be used to build a functional based model. There is also an opportunity to build a better empirically based model using data on individual fuel components (previous models have grouped similar components to lessen the computing effort)

Optical super-resolution microscopy enables the observations of individual bio-molecules. The arrangement and dynamic behaviour of such molecules is studied to get insights into cellular processes which in turn lead to various application such as treatments for cancer diseases. STFC's Central Laser Facility provides (among other) public access to super-resolution microscope techniques via research grants. The access includes sample preparation, imaging facilities and data analysis support. Data analysis includes single molecule tracking algorithms that produce molecule traces or tracks from time series of molecule observations. While current algorithms are gradually getting away from "connecting the dots" and using probabilistic methods, they often fail to quantify the uncertainties in the results. We have developed a method that samples a probability distribution of tracking solutions using the Metropolis-Hastings algorithm. Such a method can produce likely alternative solutions together with uncertainties in the results. While the method works well for smaller data sets, it is still inefficient for the amount of data that is commonly collected with microscopes. Given the observations of the molecules, tracking solutions are discrete, which gives the proposal distribution of the sampler a peculiar form. In order for the sampler to work efficiently, the proposal density needs to be well designed. We will discuss the properties of tracking solutions and the problems of the proposal function design from the point of view of discrete mathematics, specifically in terms of graphs. Can mathematical theory help to design a efficient proposal function?