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.maml-macromolecular

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 diffusion-convection 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.

Relevant Publications
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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 time-dependent 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.

Relevant Publications
  • M.J.Baines, M.E.Hubbard, P.K.Jimack, Velocity-based moving mesh methods for nonlinear partial differential equations, Commun Comput Phys, 10(3):509-576, 2011.
  • M.J.Baines, M.E.Hubbard, P.K.Jimack, A moving mesh finite element algorithm for the adaptive solution of time-dependent partial differential equations with moving boundaries, Appl Numer Math, 54:450-469, 2005.
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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
(Or- Why do adaptive methods work so well?) 

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 degrees-of-freedom 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:
* How can the theory be extended to, for example, nonsymmetric problems, nonlinear problems, or time-dependent problems?
* What about nonstandard discretisation techniques such as, discontinuous Galerkin, isogeometric analysis, or virtual element methods? 

Depending on the interest of the student, several of these issues (or others) can be addressed.
Also, the student is encouraged to suggest a second supervisor, possibly from another group! 

Relevant Publications
  • J.M. Cascon, C. Kreuzer, R.H. Nochetto, and K.G. Siebert, Quasi-optimal convergence rate for an adaptive finite element method, SIAM J. Numer. Anal. 26 (2008), pp. 2524-2550
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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.

This project builds on recent work [1] in which a new description of the coupled growth, flow, transport and deformation of a porous material is derived. The model links explicitly the microscale dynamics with emergent tissue-level behaviour, and therefore provides significant computational challenges.

The PhD studentship aims to develop new and efficient approaches with which to simulate and understand such models, exploiting the (multiscale) finite element method. Moreover, there is flexibility to engage in novel model development, or data integration (for example by comparison with results from experimental collaborators), depending on student background and interests.

Results from this work will be of wide relevance to problems in, for example, tissue engineering, geophysics and industry.

[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)

Relevant Publications
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Title Partitioned-domain concurrent multiscale modelling
Group(s) Scientific Computation, Mathematical Medicine and Biology
Proposer(s) Dr Kris van der Zee
Description

Partitioned-domain concurrent multiscale modelling
(Or- How does one get cheap, but accurate, models?)

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 multiscale-modeling type is known as partitioned-domain concurrent modelling. This type addresses problems that require a fine-scale 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 fine-scale model. Unfortunately, it is far from trivial to develop a working multiscale model for a particular problem.

Challenges for students:
* How can one couple, e.g., discrete (particle) systems with continuum (PDE) models?
* Or a fine-scale PDE with a coarse-scale PDE?
* How can one decide on the size and location of the fine-scale domain?
* Is it possible to proof the numerically observed efficiency of concurrent multiscale models? 
* Can the multiscale methodology be applied to biological growth phenomena (e.g., tumours) where one couples cell-based (agent-based) models with continuum PDE models?

Depending on the interest of the student, several of these issues (or others) can be addressed.
Also, the student is encouraged to suggest a second supervisor, possibly from another group!

Relevant Publications
  • J.T. Oden, S. Prudhomme, A. Romkes, and P.T. Bauman, Multiscale modeling of physical phenomena: Adaptive control of models, SIAM J. Sci. Comput. 28 (2006), pp. 2359-2389
Other information
Title Phase-field modelling of evolving interfaces
Group(s) Scientific Computation, Mathematical Medicine and Biology
Proposer(s) Dr Kris van der Zee
Description

Phase-field modelling of evolving interfaces
(Or – How does one effectively model and simulate interfacial phenomena?)

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 sharp-interface descriptions, such as parametric and level-set methods, and those with diffuse-interface descriptions, commonly referred to as phase-field models.

Challenges for students:
* Can one develop a phase-field model for a particular interfacial phenomenon?
* What are the foundational laws underpinning phase-field models?
* What is the connection between sharp-interface models and phase-field models?
* Can one design stable time-stepping schemes for phase-field models?
* Or efficient adaptive spatial discretisation methods?

Depending on the interest of the student, one of these issues (or others) can be addressed.
Also, the student is encouraged to suggest a second supervisor, possibly from another group! 

Relevant Publications
  • H. GOMEZ, K.G. VAN DER ZEE, Computational Phase-Field Modeling, in Encyclopedia of Computational Mechanics, Second Edition, E. Stein, R. de Borst and T.J.R. Hughes, eds., to appear
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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 long-time 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.

Relevant Publications
  • Milstein, G.N. and Tretyakov, M.V. (2004) Stochastic Numerics for Mathematical Physics. Series: Scientific Computation, Springer
Other information

Web-page http://www.maths.nott.ac.uk/personal/pmzmt

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.

 

Relevant Publications
Other information

Web-page http://www.maths.nott.ac.uk/personal/pmzmt

Title Bayesian inversion in resin transfer moulding
Group(s) Scientific Computation, Statistics and Probability
Proposer(s) Prof Michael Tretyakov
Description

Supervisors: Dr Marco Iglesias1, Dr Mikhail Matveev2, Prof Michael Tretyakov1

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 fibre-reinforced 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

 

Relevant Publications
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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.

Relevant Publications
Other information

Web-page http://www.maths.nott.ac.uk/personal/pmzmt