Spectral element methods for fractional differential equations, with applications in applied analysis and medical imaging

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Fractional differential equations are of increasing importance in a wide-range of applications, including medical imagining, collective behaviours, finance, image analysis, and elsewhere. These equations are challenging to solve numerically as they involve nonlocal interactions, which if tackled naively lead to dense discretisations that are too computationally difficult to solve, thereby limiting the scope of feasible numerical simulations. This project will develop a state-of-the-art spectral element method for simulating these models based on reducing the equations to highly structured linear systems, using a key observation that singular behaviour can be captured exactly in the development of numerical schemes. This will lead to faster and more accurate simulations facilitating progress in a wide range of applications. We will apply the results to challenging problems arising in applied analysis. Fractional differential equations arise in collective behaviour models such as swarming of animal species, cell movement by chemotaxis, granular media interaction and self-assembly of particles, and give important information about the equilibrium behaviour of such systems. These equations are difficult to solve numerically due to possible blow-up behaviour, where the model develops a singularity in finite time. The proposed scheme will allow for refinement near singularities to concentrate computational power in these difficult regions while keeping control on computational cost, allowing for high performance simulations. We will also tackle real world applications in medical imaging, including ultrasound imagining of the brain. Fractional differential equations have proven powerful tools in designing modern models that capture nonlocal behaviour caused by memory effects in the tissue. The developed spectral element method will facilitate more accurate simulations involving non-trivial geometries, for example ellipsoidal models of the skull, while avoiding inaccuracies in current schemes caused by sharp transitions between the skull and tissue.