Prof. V.N. Biktashev: sample descriptions of Ph.D. projects
These are examples of Ph.D. topics I have recently offered but which
have not (yet) started. Each of these descriptions was designed with a
particular candidate in mind. Any actual new project will be designed
with the actual candidate in mind, and may or may not be based on any
of these descriptions.
The project will be concerned with numerical solution of partial
differential equations of 'reaction-diffusion' type, particularly of
the types that are used to describe waves and patterns in mathematical
biology. Specifically, the project will focus on sensitivity of such
solutions to initial conditions and/or perturbations. This sensitivity
will be determined by the method of 'causodynamics', which involves
integration of the adjoint linearized equations backwards in time. A
few selected problems will be considered during the project. The
tentative plan of the research is a follows. At the initial stage,
this will be Turing patterns and propagating pulses in one spatial
dimension and spiral wave solutions in two spatial dimensions. At the
next stage, spatio-temporal chaos (such as generated by
Kuramoto-Sivashinsky equation) in one spatial dimension will be
considered. Further examples will concentrate on more complicated
regimes, relevant for modelling cardiac fibrillation, such as:
competing spiral waves, 'mother rotor' regimes, and two- and
three-dimensional spiral and scroll wave turbulences of various sorts.
The overall aim is to characterize the possibilities to control such
regimes by small perturbations. The project will require from the
candidate some fundamental mathematical knowledge, including linear
algebra, basic dynamical systems theory, asymptotic methods for
ordinary and partial differential equations, and possibly some
elements of functional analysis. It will also require suitable IT
skills, including numerical solution of partial differential
equations. The candidate should be prepared to learn necessary
disciplines and skills during the project, if they do not possess them
already.
Modern mathematical models of cardiac excitation are ''stiff'', in
that they involve processes happening on time scales from fractions of
a millisecond to tens of seconds and longer, which in conjunction with
the spatial complexity of the heart, makes cardiac simulations a
serious computational challenge. On the other hand, the time scale
ratios can be used as small parameters in asymptotic methods. One
possible application for such asymptotics is their use in large-scale
computations of normal and abnormal cardiac excitation patterns, in
hybrid asymptotic-numerical schemes. The proposed Ph.D. project will
make first steps in that direction, based on the recent progress by
the supervisor's group in cardiac asymptotics. The main goal will be a
methodology to combine asymptotic description of excitation waves, in
terms of propagating fronts, with the original partial-differential
equations for spatio-temporal evolution of nonlinear dynamic fields.
This will be done first in one spatial dimension and subsequently
extended to two and three dimensions. Initially we will consider
simple topologies, when the front is a point (in one spatial
dimension), or a manifold without internal borders (in two and three
spatial dimensions). Then we extend it to the case of wavebreaks,
where the wave front has edges within the excitable medium. Finally,
we shall consider extension from the initial modomain (semi-parabolic
PDE systems) to bidomain (with added elliptic equations) models of the
heart.
The project will require knowledge of mathematical biology, asymptotic
and numerical methods for PDEs and software development. Hence,
depending on their background, the candidate may be required to do
relevant modules offered by this and/or other Departments in this
University, to be specified by the supervisor, and pass them at least
at the level acceptable by M.Sc. standard, in his/her first year of
the project.
Spiral waves are a form of self-organization observed in distributed
active media, such as some catalytic chemical reactions or heart
muscle. Spiral waves in heart underlie dangerous arrhythmias. A
peculiar feature of spiral waves is "wave-particle duality": being
waves, they behave like particles when drifting in response to generic
small perturbations. This allows description of their drift in terms
of ordinary differential equations of motion, which are easier to
solve and more amenable to qualitative analysis than the
"reaction-diffusion" nonlinear partial differential equations of the
original models. The success of this asymptotic approach depends on
knowledge of so called response functions, which is critical
eigenfunctions of the adjoint linearized operator. A numerical method
for calculating the response has been developed recently by the
supervisors' group, and its workability has been successfully
demonstrated on a number of concrete models, including models of
cardiac tissue. However, this method involves direct LU decomposition
of a matrix of linearization, which severely limits the achievable
spatial resolution, due to memory demands.
The subject of the present project will be development and
investigation of alternative methods of calculation of response
functions. The methods are likely to be iterative and based on the
Conjugate Gradients idea. Apart from the development of the methods,
the proposed programme will include study of their convergence and
computational efficiency, and test applications to dynamics of spiral
waves in selected models, most likely from cardiac dynamics.
The project will require knowledge of mathematical biology, asymptotic
and numerical methods for PDEs and software development. Hence,
depending on their background, the candidate may be required to do
relevant modules offered by this and/or other Departments in this
University, to be specified by the supervisor, and pass them at least
at the level acceptable by M.Sc. standard, in his/her first year of
the project.
Interaction of predators and prey can lead to a variety of complicated
behaviours, particulary in the spatially extended context. For
instance, theory predicts "pursuit-evasion" waves, where the only
chance for prey to flourish is to flee away from the predators, and
vice versa. The proposed project will look at the problem of
coevolution of prey (bacteria) and predators (phages) as
pursuit-evasion waves in the "trait space" instead of the physical
space. This view has been motivated by in-vitro experiments when
evolution bacteria and phages happens under controlled conditions and
in real time. The project will involve construction and numerical and
analytical studies of mathematical models of such coevolution, with a
view to identify relevant scales of parameters and possible ways to
relate the models with the experimental data.
The project will require knowledge of mathematical biology, dynamical
systems theory and asymptotic and numerical methods for PDEs. Hence,
depending on their background, the candidate may be required to do
relevant modules offered by this and/or other Departments in this
University, to be specified by the supervisor, and pass them at least
at the level acceptable by M.Sc. standard, in his/her first year of
the project.
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V. N. Biktashev at exeter dot ac dot uk
Last modified: Fri Nov 22 10:35:07 GMT 2019