Title: Spectral element methods for fractional differential equations, with applications in applied analysis and medical imaging
Funder: Engineering and Physical Sciences Research Council

Abstract: 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.

Lay abstract: We propose to develop and analyse a new spectral element method for the numerical solution of fractional differential equations involving fractional Laplacians, which arise naturally in stochastics, medical imagining, kinetic models, and elsewhere. The proposed method will achieve highly structured discretisations of fractional differential equations-involving sparse, low rank, and hierarchical matrices-that will result from precise knowledge of underlying singularity behaviour. This new approach to the long standing problem of solving fractional differential equations will for the first time enable high performance, high accuracy, and large scale simulations of fractional differential equations in 1D, 2D, and 3D. The results of this project will be used to tackle problems in applied analysis and medical imagining. These applications will provide a comprehensive test suite for the proposed scheme, and thereby demonstrate its versatility and scalability. Moreover, the simulations produced as part of this project will have an immediate impact on a wide range of applications, such as ultrasound imaging and therapy of the brain. The project is divided into 4 work packages: WP1 Develop a spectral element method for one-dimensional fractional differential equations. To achieve spectral accuracy we will introduce +a new approach to simultaneously capture multiple algebraic singularities that are inherent in the solutions, focussing in the first instance on the important case of square-root and rational powers of the Laplacian. We will exploit sparsity, low rank, and hierarchical structure in the resulting discretisation to achieve fast complexity. A stretch goal is to tackle irrational powers of the Laplacian. WP2 Develop a spectral element method for multi-dimensional fractional differential equations. We will adapt the one-dimensional scheme to equations involving fractional powers on ellipsoids and ellipsoidal shells (including balls and spherical shells), which have special structure that facilitates the adaptation of the techniques from 1D. This will enable a high performance spectral element method with ellipsoidal elements, which arise naturally in medical imagining applications. A stretch goal is to generalise the approach to non-ellipsoidal geometries such as tetrahedral meshes. WP3 Applications in applied analysis. We will numerically solve equilibrium problems in collective behaviour problems, and solve time-evolution problems such as the fractional Keller-Segel model and two-dimensional quasi-geostrophic equation, which is related to Navier Stokes. The approach developed in WP1-2 will allow for efficient simulations of equilibrium solutions even in the case of multiply supported geometries, and will be able to resolve blow-up behaviour by appropriately decomposing the geometry into a graded mesh. These new numerical simulations will lead to deeper mathematical understanding of the underlying processes. WP4 Applications in medical imaging. We will apply the methodology of WP2 to models of wave propagation in viscoelastic solids which arise in ultrasound through the skull, where the geometries can be effectively modelled using ellipsoidal shells. The ability to design meshes that capture changes in the geometry will facilitating more accurate and efficient simulations, improving the accuracy and reliability of simulations associated with sharp transitions between media, e.g., between the skull and the brain.

Research activity codes: 2.5 RESEARCH DESIGN AND METHODOLOGIES (AETIOLOGY)
Health categories: GENERIC HEALTH RELEVANCE