## Robert Douglas

BSc PhD (Bath)

## Lecturer

### Contact Details

**Room Number..........: **4.24**Building....................: **Physical Sciences**Phone.......................: **+44 (0)1970 622756**E-Mail........................: ** rsd**Home Page...............: ** Personal

### Biography

Educated at Pates Grammar School, Cheltenham, and a graduate in Mathematics from Bath University. He obtained his PhD in applied mathematical analysis at Bath, specialising in nonlinear analysis of vortices in ideal fluid flow. Dr. Douglas was a Research Fellow at the Universities of Cambridge and Reading, a Visiting Scientist at the U.K. Meteorological Office, and a Lecturer at the University of Surrey before joining the Mathematics Department at Aberystwyth in February 2000.

His interests include bridge (he has won the Mid Wales Open Pairs and was a member of the first Mid Wales team to win the Perry National Championships), watching cricket and rugby, and reading crime novels.

### Research Groups

### Research Interests

**Areas of interest include:**Dr. Douglas is an applied mathematical analyst; he is interested in innovative modelling of physical problems, but his main focus is proving rigorous results using measure theory and convex analysis (and its generalisations). His previous work proved the Cullen-Norbury-Purser conjecture that identified energy minimising solutions of the semigeostrophic equations, a model for weather front formation, with an extra constraint on the equations. For such flows the trajectory map (for fluid particles) must arise from a polar factorisation (that is a vector-valued mapping written as the composition of the gradient of a convex function with a size-preserving mapping); in collaboration with G.R. Burton (University of Bath), Dr. Douglas established sharp conditions for when such a factorisation is guaranteed to exist, and precisely when it is unique. He is an active participant in the Analysis cluster of the Wales Institute for Mathematical and Computational Sciences, organising a workshop in the Mathematical Analysis and Modern Applications series, where optimal mass transfer was the central theme. Dr. Douglas collaborates with mathematicians in Canada, France, Portugal and the U.S.A., and with meteorologists in the U.K. A general theme of his research is to apply physical principles to the rigorous analysis of a model of genuine interest to the practical scientist.

### Current Interests

Dr. Douglas's current research interests concentrate on optimal mass transfer problems and their applications. Some of the problems are motivated by meteorology, such as finding more sophisticated ways of measuring weather forecast error, whilst others are more theoretical. He is also interested in linear analysis of complex fluids. After an overview of optimal mass transfer problems, these topics are described in more detail.

#### Optimal mass transfer

The archetypal optimal mass transfer problem is to find the optimal way to transfer mass from a set

*U*to a set*V*, amongst a set*S*of allowable strategies, where optimality is measured against a non-negative cost function*c = c(x; y).*One interprets*c(x, y)*as being the cost per unit mass of transporting material from*x 2 U to y 2 V*. The set*S*is described by a set of measure-preserving mappings. Depending on the cost function, there may be one, many, or no solutions; since the 1980s considerable progress has been made classifying optimisers for a variety of cost functions. The actual optimal value denes a*Wasserstein distance*, an important concept in the modern theory of pdes. The original problem of Monge was concerned with minimising work done when moving piles of rubble and soil. There is much interest in both theoretical results for new classes of cost functions, and modelling of physical processes via optimal mass transfer, where one chooses suitable cost functions.#### Weather forecast error decomposition

Many weather forecast errors are simply displacements of significant weather in space or time, such as a rainband arriving a few hours later than expected.

*Weather forecast error decomposition*splits the error into two parts: the error due to differences in qualitative features, and the error due to displacement. In its simplest formulation, we find a displaced version of the forecast (in the set of rearrangements of a function) that is a best fit to the actual distribution. The Wasserstein distance between the forecast and its displaced version represents displacement error (and may be thought of as kinetic energy), and the*L*-norm of the distance between the actual distribution and the displaced forecast is the qualitative features error. Total error is calculated by a weighted sum. Well-posedness of this formulation has been shown, and the best-fitting displacements characterised. Recent work is concerned with developing a more sophisticated implementation of this idea: minimise both types of error simultaneously, using a "modified Wasserstein" distance.^{2}#### Weather front formation

The semigeostrophic equations are a model for atmospheric and oceanic flows which can accurately represent such large scale features as fronts, inversion layers, and the drag exerted by mountain ranges. Fronts are described by tracking singularities of a potential (which we think of as representing pressure) as time evolves. Rigorous results have been obtained concerning energy minimisation and duality structure of the variable rotation case, extending the validity of the theory to the whole of the sphere. Current work concerns the existence, properties, and (nonlinear) stability of generalised Lagrangian solutions. This builds on exciting recent advances in the theory of optimal mass transfer and its application to hydrodynamics.

#### Polar inclusion

An integrable vector-valued mapping is said to have a polar factorisation if it can be written as the composition of the gradient of a convex function with a size-preserving mapping; this concept has been used in diverse applications, one of which is enhancing computer images. Previous work has settled the question of when the factorisation is unique, and provided a characterisation of the polar factors. Attention is now focussed on classes of integrable mappings which do not have polar factorisations: progress has been made on the existence and characterisation of a weaker version of polar factorisation, that is

*polar**inclusion*. This concept has parallels with very weak (measure-valued) solutions of ideal fluid flow.#### Relaxation Spectrum Recovery

Visco-elastic fluids exhibit both solid-like and liquid-like behaviour: toothpaste can be modelled in this way. The solid-like behaviour arises because the fluid has memory: roughly speaking, the relaxation spectrum describes for how long and to what extent the fluid remembers the past deformation history. The relaxation spectrum can be recovered from the storage or loss modulus by inverting integral equations; the problem is ill-posed in the sense of Hadamard (i.e. solutions do not depend continuously on the given data). The inverse problem can be reformulated as finding the inverse Fourier transform (in the sense of tempered distributions) of a weighted function. Recent research has established a result for Schwartz distributions, and seeks to extend this result to non-classical distributions. Numerical inversion of this integral equation is of practical interest to rheologists; it is a challenging problem. The aim of this work is to provide a theoretical basis for judging competing numerical schemes.

### Teaching Areas

**Modules Taught**

**MA10510:**Algebra**MA11110:**Mathematical Analysis**MA20110:**Real Analysis**MA21410:**Linear Algebra**MA30210:**Norms and Differential Equations**MA37010:**Lebesgue Integration**MA38010:**Spectral Theory**MAM7020:**Lebesgue Integration**MAM8020:**Spectral Theory**MAM9840:**Major Project**MP12910:**Career Planning and Skills Development**MT11110:**Dadansoddi Mathemategol**MT20110:**Dadansoddiad Real**MT21410:**Algebra Llinol**MT30210:**Normau a Hafaliadau Differol**MTM9840:**Prif Brosiect

### Publications

#### 2016

Nonexistence results for relaxation spectra with compact support. *Inverse Problems* **32** (3) 035006 10.1088/0266-5611/32/3/035006 Cadair

#### 2007

Non-existence of polar factorisations and polar inclusion of a vector-valued mapping. *International Journal of Pure and Applied Mathematics* **41** (3) Cadair Other

#### 2005

Generalized semi-geostrophic theory on a sphere. *Journal of Fluid Mechanics* **531** pp. 123-157. 10.1017/S0022112005003812 Cadair

#### 2003

Uniqueness of the polar factorisation and projection of a vector-valued mapping. *Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire* **20** (3) pp. 405-418. 10.1016/S0294-1449(02)00026-4 Cadair

Large-Amplitude nonlinear stability results for atmospheric circulations. *Quarterly Journal of the Royal Meteorological Society* **129** (591) pp. 1969-1988. 10.1256/qj.01.196 Cadair