You may contact a Proposer directly about a specific project or contact the Postgraduate Admissions Secretary with general enquiries.
Title  Mathematical modelling of macromolecular capillary permeability 

Group(s)  Mathematical Medicine and Biology, Scientific Computation 
Proposer(s)  Dr Reuben O'Dea, Dr Matthew Hubbard 
Description  The primary function of blood vessels is to transport molecules to tissues. In diseases such as cancer and diabetes this transport, particularly of large molecules such as albumin, can be an order of magnitude higher than normal. The project is to model transient flow of macromolecules across the vascular wall in physiology and pathology. With additional supervision from Dr Kenton Arkill and Professor David Bates (Medicine), the doctoral student will join a team that includes medical researchers, biophysicists and mathematicians acquiring structural and functional data. Detailed microscale models of vascular wall hydrodynamics and transport properties will be employed; in addition, powerful multiscale homogenisation techniques will be exploited that enable permeability and convection parameters on the nanoscale to be linked through the microscale into translatable information on the tissue scale. Computational simulations will be used to investigate and understand the model behaviour, including, for example, stochastic and multiphysics effects in the complex diffusionconvection nanoscale environment. The project will afford a great opportunity to form an information triangle where modelling outcomes will determine physiological experiments to feedback to the model. Furthermore, the primary results will inform medical researchers on potential molecular therapeutic targets. 
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Title  Discontinuous Galerkin Finite Elements for Moving Boundary Problems 

Group(s)  Scientific Computation 
Proposer(s)  Dr Matthew Hubbard 
Description  Many physical and chemical processes, typified by those related to fluid flow, can be modelled mathematically using partial differential equations. These can usually only be solved in the simplest of situations, but solutions in far more complex cases can be approximated using numerical and computational techniques. Traditional approaches to providing these computational simulations have typically modelled the evolution of the system by approximating the equations on a uniform mesh of points covering a domain with a fixed boundary. However, many situations (consider the spreading of a droplet, for example), naturally suggest a domain which evolves with the flow, while the main focus of interest in others (say the movement of a shock wave up and down an aeroplane wing) is in following the motion of a sharp internal feature. For accuracy and efficiency a computational method should not only approximate the partial differential equations appropriately, but also move the computational mesh in a manner which follows such features. Recent research has developed a finite element approach to the adaptive approximation of timedependent physical problems involving moving boundaries or interfaces. It has been deliberately designed to preserve inherent properties (such as conservation principles and invariances) of the underlying partial differential equations and hence of the system the mathematics is intended to represent. Extremely promising results have been obtained for a wide range of problems in one and two space dimensions, but the applicability of the approach is still limited (as are all moving mesh methods) by the potential for the computational mesh to ``tangle''. The aim of this project will be to develop an alternative approach, derived within the same framework, which takes advantage of the additional flexibility inherent in the discontinuous Galerkin finite element framework. This has the potential to reduce the occurrence of mesh tangling and to greatly improve the robustness of the method when modelling problems involving complex, interacting features and when using different monitor to govern the movement of the mesh. 
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Title  Foundations of adaptive finite element methods for PDEs 

Group(s)  Scientific Computation, Algebra and Analysis 
Proposer(s)  Dr Kris van der Zee 
Description  Foundations of adaptive finite element methods for PDEs Adaptive finite element methods allow the computation of solutions to partial differential equations (PDEs) in the most optimal manner that is possible. In particular, these methods require the least amount of degreesoffreedom to obtain a solution up to a desired accuracy! In recent years a theory has emerged that explains this behaviour. It relies on classical a posteriori error estimation, Banach contraction, and nonlinear approximation theory. Unfortunately, the theory so far applies only to specific model problems. Challenges for students: Depending on the interest of the student, several of these issues (or others) can be addressed. 
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Title  Multiscale models for growing tissues – simulation and analysis. 

Group(s)  Scientific Computation, Mathematical Medicine and Biology 
Proposer(s)  Dr Donald Brown 
Description  A fundamental barrier to advancing the understanding of biological tissue growth lies in its inherently multiscale nature: interactions between processes acting at disparate scales can profoundly influence the emergent dynamics. A unified description of such phenomena requires reconciling insight obtained by theoretical or experimental study at one scale with observations at another. For example, diseases that manifest at the organ scale often arise through the interaction of microscopic events at the cellular scale; moreover, the resulting macroscale changes influence the microscopic dynamics. [1] Effective equations governing an active poroelastic medium. J Collis, DL Brown, ME Hubbard, RD O'Dea Proceedings of the Royal Society A 473 (2198) 
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Title  Partitioneddomain concurrent multiscale modelling 

Group(s)  Scientific Computation, Mathematical Medicine and Biology 
Proposer(s)  Dr Kris van der Zee 
Description  Partitioneddomain concurrent multiscale modelling Multiscale modeling is an active area of research in all scientific disciplines. The main aim is to address problems involving phenomena at disparate length and/or time scales that span several orders of magnitude! An important multiscalemodeling type is known as partitioneddomain concurrent modelling. This type addresses problems that require a finescale model in only a small part of the domain, while a coarse model is employed in the remainder of the domain. By doing this, significant computational savings are obtained compared to a full finescale model. Unfortunately, it is far from trivial to develop a working multiscale model for a particular problem. Challenges for students: Depending on the interest of the student, several of these issues (or others) can be addressed. 
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Title  Phasefield modelling of evolving interfaces 

Group(s)  Scientific Computation, Mathematical Medicine and Biology 
Proposer(s)  Dr Kris van der Zee 
Description  Phasefield modelling of evolving interfaces Evolving interfaces are ubiquitous in nature, think of the melting of the polar ice caps, the separation of oil and water, or the growth of cancerous tumours. Two mathematical descriptions exist to model evolving interfaces: those with sharpinterface descriptions, such as parametric and levelset methods, and those with diffuseinterface descriptions, commonly referred to as phasefield models. Challenges for students: Depending on the interest of the student, one of these issues (or others) can be addressed. 
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Title  Geometric integration of stochastic differential equations 

Group(s)  Scientific Computation, Statistics and Probability 
Proposer(s)  Prof Michael Tretyakov 
Description  For many applications (especially, in molecular dynamics and Bayesian statistics), it is of interest to compute the mean of a given function with respect to the invariant law of the diffusion, i.e. the ergodic limit. To evaluate these mean values in situations of practical interest, one has to integrate large dimensional systems of stochastic differential equations over long time intervals. Computationally, this is a challenging problem. Stochastic geometric integrators play an important role in longtime simulation of dynamical systems with high accuracy and relatively low cost. The project involves construction of new efficient numerical methods for ergodic stochastic differential equations and stochastic numerical analysis of properties of the methods. We require an enthusiastic graduate with a 1st class degree in Mathematics, preferably at MMath/MSc level (in exceptional circumstances a 2:1 class degree, or equivalent, can be considered). We are expecting that the successful applicant has a very good background in Probability and has good computational skills. 
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Title  Numerical methods for stochastic partial differential equations 

Group(s)  Scientific Computation, Statistics and Probability 
Proposer(s)  Prof Michael Tretyakov 
Description  Numerics for stochastic partial differential equations (SPDEs) is one of the central topics in modern numerical analysis. It is motivated both by applications and theoretical study. SPDEs essentially originated from the filtering theory and now they are also widely used in modelling spatially distributed systems from physics, chemistry, biology and finance acting in the presence of fluctuations. The primary objectives of this project include construction, analysis and testing of new numerical methods for SPDEs.

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Title  Bayesian inversion in resin transfer moulding 

Group(s)  Scientific Computation, Statistics and Probability 
Proposer(s)  Prof Michael Tretyakov 
Description  Supervisors: Dr Marco Iglesias^{1}, Dr Mikhail Matveev^{2}, Prof Michael Tretyakov^{1} University of Nottingham, University Park, Nottingham NG7 2RD, UK 1 School of Mathematical Sciences 2 Polymer Composites Group, Faculty of Engineering This project will be based at the University of Nottingham in the School of Mathematical Sciences and the Faculty of Engineering. The use of fibrereinforced composite materials in aerospace and automotive industries and other areas has seen a significant growth over the last two decades. One of the main manufacturing processes for producing advanced composites is resin transfer moulding (RTM). The crucial stage of RTM is injection of resin into the mould cavity to fill empty spaces between fibres; the corresponding process is described by an elliptic PDE with moving boundaries. Imperfections of the preform result in uncertainty of its permeability, which can lead to defects in the final product. Consequently, uncertainty quantification (UQ) of composites’ properties is essential for optimal RTM. One of important UQ problems is quantification of the uncertain permeability. The objectives of this PhD project include (i) to construct, justify and test efficient algorithms for the Bayesian inverse problem within the moving boundary setting and (ii) to apply the algorithms to real data from composite laboratory experiments. Eligibility/Entry Requirements: We require an enthusiastic graduate with a 1st class degree in Mathematics, preferably at MMath/MSc level (in exceptional circumstances a 2:1 class degree, or equivalent, can be considered). We are expecting that the successful applicant has a background in PDEs, Probability and Statistics and has exceptional computational skills. For any enquiries please email: Marco.Iglesias@nottingham.ac.uk or Michael.Tretyakov@nottingham.ac.uk or Mikhail.Matveev@nottingham.ac.uk

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Title  Computational Finance 

Group(s)  Scientific Computation, Statistics and Probability 
Proposer(s)  Prof Michael Tretyakov 
Description  Computational Finance is the key element for successful risk management at investment banks and hedge funds and it is also a growing area on the interface between finance, computational mathematics and applied probability. Pricing and hedging financial derivatives, evaluating risks of default for financial product and firms, satisfying requirements of the Basel Accord, etc.  all require sophisticated modelling and reliable calibration of the models. These aims cannot be achieved without efficient numerical techniques which form the area of computational finance. The project will aim at developing new, efficient computational techniques related to finance. Eligibility/Entry Requirements: We require an enthusiastic graduate with a 1st class degree in Mathematics, preferably at MMath/MSc level (in exceptional circumstances a 2:1 class degree, or equivalent, can be considered). We are expecting that the successful applicant has a good background in Probability and Stochastic Analysis, some knowledge of Finance and has exceptional computational skills. 
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