Computational Methods

Differential equations on metric graphs: continuum and numerical methods

Supervisor(s): Geoff Vasil

Networks appear throughout the real world. Classic examples include the humble spiderweb, the not-so-humble neuronal network of the human brain, the Manhattan road network, electricity grids, river systems, social networks, vascular networks... These networks facilitate the spread of mechanical waves, electricity, sound, traffic, information, water, blood... The idea of this thesis is to develop a framework for understanding differential equations (the models of how these quantities spread) on metric graphs (networks) when the networks are high-density. Under a microscope, the capillary system of the human lung may be recognisable as any other branching tree or plant in your garden. However, as a whole, the 300 billion capillaries “fill in” the lung into which they are embedded and which they service. A feature of a such a network is that it can route and reroute oxygen and nutrients at various scales–from distributing locally between small groups of capillaries, to distributing globally on the scale of the entire embedding space.

This large-scale capability enables networks to be robust to damage or defects, for example. Modelling this capability is crucial for understanding how these systems work. For a simple but foundational differential equation, which models waves, heat, and electrons, we determine how the density and structure of the network, as well as the shape and size of the embedding space, affect these large-scale dynamics. We find a new differential equation and framework, which we hope will be instructive in understanding large-scale behaviour of many extremely large networks which have remained elusive so far.

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Numerical framework for solving PDE-constrained optimization problems from multiscale particle dynamics

Supervisor(s): Ben Goddard, John Pearson

The modelling of natural phenomena with mathematical equations has been widely applied in all spheres of the natural and social sciences. In particular, partial differential equations (PDEs) have been utilized to model systems from biology, physics, chemistry, and social science. Since computer algorithms have been developed to solve a range of complex partial differential equations, many industrial applications in engineering, medical sciences, and chemical systems have been able to predict quantities of interest via numerical solutions of PDE models.

The PDEs of interest in this thesis primarily describe a system of interacting ‘particles’. Such a formulation is able to describe several systems including swarming and flocking of animals, opinions of individuals in a population, yeast fermentation during brewing of beer, growth of cancer cells, and interactions of electrons. These interacting PDEs are generally non-linear and non-local and present several numerical challenges.

The concept of optimization comes from the pragmatic standpoint that there are almost never infinite resources at our disposal to accomplish a task. To accomplish a desired result, limited resources must be managed effectively.  The design of fast, robust, and effective computer algorithms to solve optimization problems has been a focus in the field of numerical analysis.

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Preconditioned iterative methods for optimal control problems with time-dependent PDEs as constraints

Supervisor(s): John Pearson

Optimal control problems often arise in industrial and real-life applications. In particular, this class of problems often concerns finding the “work” that should act on a given physical system in order to obtain a desired outcome, with a cost that is minimal. A simple example of an optimal control problem is given by the control of the temperature in a room. For instance, suppose during winter one wishes to keep the temperature of a room around 200C. In order to obtain this, one may turn on the heating for the whole day. However, this choice may impact one’s finances, therefore one wishes to find a solution to this problem in a more efficient way.

This can be done by solving an optimal control problem, in which the model drives the temperature of the room towards a desired one while reducing the energy provided to the room for the heating. Other examples are given by the control of the flow of a fluid in a pipe, the separation of some chemicals, or the modification of the atmospheric conditions of some system. In this thesis, we delevop fast and robust solvers for the numerical solution of optimal control problems. Specifically, this work focuses on the development of suitable preconditioners for the linear systems arising from a range of optimal control problems. For a large part of this thesis, we focus our study on time- dependent problems, and we also consider time-independent systems.

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