Adaptive Regularisation

Many physical phenomena can be modelled using differential equations. However, in general, mathematicians are not able to solve these analytically. For example, we know a given fluid can be modelled well using a Navier-Stokes equation, but we cannot solve the equation exactly, so we cannot predict what the fluid does over time. Hence to gain some knowledge on how the fluid is behaving we often turn to numerical approximations. Therefore we must design a scheme which can be run on a computer to simulate what our fluid does. Having access to "good" numerical approximations is very important; in particular, it is important to be able to quantify how accurate the numerical approximation is. This quantification allows us to determine whether to trust the simulation we generate.

A posteriori error analysis is used to assess the accuracy of a given numerical approximation. It allows us to know when and where the simulation misbehaves and gives us the option to correct it by "adapting" the numerical scheme. This is called an adaptive procedure. Adaptive procedures allow us to make the simulation more efficient, in terms of computational time, allowing for more complex simulations to be carried out faster.

One of the research aims of this project is to propose an alternative methodology to tackle the cases when a posteriori analysis fails. For example, when a jet's speed exceeds the sound barrier, shock waves form. Mathematically these are discontinuities in the underlying medium. This phenomena is exceptionally difficult to simulate and the subject of much research. In particular, the a posteriori analysis, our assessment of the simulation, does not provide any useful information.

Another aim of this research is to lay the groundwork towards an application in the area of "data assimilation". Data assimilation is a technique useful when observations are available at specific points in time. Perhaps you are studying the evolution of a hurricane and have access to air pressure from certain weather monitoring stations at certain times. The mathematical model which is derived can then be updated based on these observations at the times they are observed. Data assimilation is a systematic way to provide such updates, and it allows for accurate prediction of how the hurricane evolves based on what has happened. But how are these incorporated into the numerical simulation? Current methodologies enforce that the mathematical model agrees with the observations on average.

The numerical schemes developed in this project will develop the foundations for the design of simulations where the observations can be incorporated into the mathematical model in a "pointwise" sense, rather than on average. This is extremely important and will aid, among other applications, the development of more accurate weather prediction software.

Numerical analysis is arguably the backbone of applicable mathematics. It underpins many other areas of research that occur ubiquitously throughout the STEM areas. The proposal, if successful, will yield a state of the art approach to a variety of challenging problems and would thus have extremely high impact throughout many of these areas benefiting academics and practitioners alike. Conservation laws arising in modelling inviscid flow are widely used in the engineering community; Hamilton-Jacobi equations arise from optimal control problems which are used in, for example, data assimilation, a popular technique aimed at incorporating observations into mathematical models used by Meteorologists; Non-variational problems typically arise as linearisations of fully nonlinear problems used to model a huge range of physical problems coming from topics such as (geophysical) fluid dynamics, data assimilation, differential geometry, image processing and game theory.

Given the large range of areas in which this research could be applied, the potential applications are vast. A particular application which would be focussed upon is that to weather and climate through data assimilation both from the angle of optimal control problems as well as from fully nonlinear partial differential equations. This will be realised through the submission of a PhD proposal for a project to the EPSRC funded CDT in Mathematics of Planet Earth jointly based between Imperial and Reading. Potential industrial beneficiaries to this specific problem include the Met Office, the European Centre for Medium-Range Weather Forecasts, the National Centre for Atmospheric Science and the National Physical Laboratory.