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

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.

Societal impact: Fractional differential equation are of increasing importance in a wide-range of areas with direct relevance to society. The project will directly produce simulations for medical imagining of the brain, which will lead to higher accuracy simulations and more efficient image reconstruction. We will also investigate simulations of collective behaviour models which will lead to deeper understanding of behaviour of systems with complicated interactions such as swarming of animals or cell movement. Going beyond the applications pursued in this proposal, the results may also lead to more efficient methods for image processing, where fractional differential equations have proven effective for image denoising in a way that sharp edges are preserved.

Economic impact: The proposed tools are relevant to a wide range of industries, including healthcare, image processing, and elsewhere. Fractional differential equations arise naturally in stochastic models with heavy tails which may facilitate more sophisticated financial models that better capture so-called black swan events. The results of the project will be released in open-source software to facilitate direct impact in industry, and consulting with industry will be pursued to increase uptake by potential users.

Academic impact: The project tackles state-of-the-art models in fractional differential equations which are attracting significant interest in pure, applied, and computational mathematics. It will lead to deeper understanding of difficult models, and provide a new computational tool to effectively model such equations. It will help fuel collaboration between numerical analysts, applied analysts, and practitioners in medical imagining, and help to establish and deepen links between researchers at Imperial and UCL.

Human resources: The appointed postdoctoral researchers will be trained in a wide range of mathematical areas such as numerical analysis, spectral methods, and fractional calculus, that will provide them with unique skills that are useful inside academia and in industry. They will further have the opportunity to develop an independent research program building on this proposal, increasing their employability. They will have access to a wide range of seminars in both pure and applied mathematics at Imperial, such as the Imperial-UCL Numerics Seminar, the Applied PDE Seminar, the Pure Analysis Seminar, and the Department Colloquium, that will expose them to a range of research areas.