Spectral element methods for fractional differential equations, with applications in applied analysis and medical imaging
Lead Research Organisation:
Imperial College London
Department Name: Mathematics
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.
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.
Planned Impact
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.
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.
Organisations
Publications
Gutleb T
(2022)
Computation of Power Law Equilibrium Measures on Balls of Arbitrary Dimension
in Constructive Approximation
Han A
(2022)
The impact of price transparency and competition on hospital costs: a research on all-payer claims databases.
in BMC health services research
Magaletti F
(2022)
A positivity-preserving scheme for fluctuating hydrodynamics
in Journal of Computational Physics
Miscouridou M
(2022)
Classical and Learned MR to Pseudo-CT Mappings for Accurate Transcranial Ultrasound Simulation.
in IEEE transactions on ultrasonics, ferroelectrics, and frequency control
Papadopoulos I
(2023)
Preconditioners for Computing Multiple Solutions in Three-Dimensional Fluid Topology Optimization
in SIAM Journal on Scientific Computing
Papadopoulos I
(2022)
Numerical analysis of a topology optimization problem for Stokes flow
in Journal of Computational and Applied Mathematics
Papadopoulos I
(2022)
Numerical Analysis of a Discontinuous Galerkin Method for the Borrvall--Petersson Topology Optimization Problem
in SIAM Journal on Numerical Analysis
Papadopoulos, I.P.A
(2022)
A sparse spectral method for fractional differential equations in one-spacial dimension
in arXiv
Papadopoulos, I.P.A
(2022)
Numerical analysis of the SIMP model for the topology optimization of minimizing compliance in linear elasticity
in arXiv
Stanziola A
(2022)
Transcranial ultrasound simulation with uncertainty estimation
in arXiv
Stanziola A
(2023)
j-Wave: An open-source differentiable wave simulator
in SoftwareX
Stanziola A
(2021)
A Helmholtz equation solver using unsupervised learning: Application to transcranial ultrasound
in Journal of Computational Physics
Stanziola, A
(2022)
A Learned Born Series for Highly-Scattering Media
in arXiv
Title | ApproxFun.jl |
Description | ApproxFun is a package for approximating functions. It is in a similar vein to the Matlab package Chebfun and the Mathematica package RHPackage. |
Type Of Technology | Software |
Year Produced | 2021 |
Open Source License? | Yes |
Impact | ApproxFun.jl is a highly used package with almost 400 Github stars, and 37 contributors. |
URL | https://zenodo.org/record/6327592 |
Title | ClassicalOrthogonalPolynomials.jl v0.5.1 |
Description | A Julia package for classical orthogonal polynomials and expansions |
Type Of Technology | Software |
Year Produced | 2021 |
Open Source License? | Yes |
Impact | Forms the basis for the grant by giving convenient framework for manipulating orthogonal polynomials. |
URL | https://github.com/JuliaApproximation/ClassicalOrthogonalPolynomials.jl |
Title | JuliaApproximation/ContinuumArrays.jl: v0.10.0 |
Description | ContinuumArrays v0.10.0 Diff since v0.9.6 Merged pull requests: Support transforms with quasi matrices (#122) (@dlfivefifty) |
Type Of Technology | Software |
Year Produced | 2021 |
Impact | Lays the basis for working with continuum arrays as used in ClassicalOrthogonalPolynomials.jl, needed for the grant. |
URL | https://zenodo.org/record/5151530 |
Title | RadialPiecewisePolynomials.jl |
Description | Implements a spectral element method for disks and annuli. To be combined with spectral methods for a fractional time derivative in a plug-and-play manner. |
Type Of Technology | Software |
Year Produced | 2023 |
Open Source License? | Yes |
Impact | The first sparse spectral element method for weak formulations on disks and annuli. |
URL | https://github.com/ioannisPApapadopoulos/RadialPiecewisePolynomials.jl |
Title | SemiclassicalOrthogonalPolynomials.jl v0.3.3 |
Description | Computes Semiclassical Jacobi Polynomials and related operators. |
Type Of Technology | Software |
Year Produced | 2022 |
Open Source License? | Yes |
Impact | Used in constructing multivariate orthogonal polynomials, including on annuli needed for the project. |
URL | https://github.com/JuliaApproximation/SemiclassicalOrthogonalPolynomials.jl |
Title | ioannisPApapadopoulos/SumSpaces.jl: v0.0.1 |
Description | No description provided. |
Type Of Technology | Software |
Year Produced | 2022 |
Open Source License? | Yes |
Impact | Used in constructing frames in 1D which allows for solving fractional PDEs required in the project. |
URL | https://zenodo.org/record/7185131 |
Title | j-Wave: Differentiable acoustic simulations in JAX |
Description | j-Wave is a library of simulators for acoustic applications. Is heavily inspired by k-Wave (a big portion of j-Wave is a port of k-Wave in JAX), and its intended to be used as a collection of modular blocks that can be easily included into any machine learning pipeline. Following the philosophy of JAX, j-Wave is developed with the following principles in mind: to be differentiable, to be fast via jit compilation, easy to run on GPUs, easy to customize. |
Type Of Technology | Software |
Year Produced | 2022 |
Open Source License? | Yes |
Impact | j-Wave was used in a recent modelling intercomparison, and to study uncertainty in transcranial ultrasound simulation. |
URL | https://github.com/ucl-bug/jwave |
Title | jaxdf - JAX-based Discretization Framework |
Description | jaxdf is a JAX-based package defining a coding framework for writing differentiable numerical simulators with arbitrary discretizations. The intended use is to build numerical models of physical systems, such as wave propagation, or the numerical solution of partial differential equations, that are easy to customize to the user's research needs. Such models are pure functions that can be included into arbitrary differentiable programs written in JAX: for example, they can be used as layers of neural networks, or to build a physics loss function. |
Type Of Technology | Software |
Year Produced | 2022 |
Open Source License? | Yes |
Impact | jaxdf is being used by researchers interested in differentiable models, and forms the basis for the j-Wave acoustic simulation software. |
URL | https://github.com/ucl-bug/jaxdf |
Description | Minisymposium at CSE23 |
Form Of Engagement Activity | Participation in an activity, workshop or similar |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Professional Practitioners |
Results and Impact | Organisation of a minisymposium involving 8 talks at CSE23 titled "Applications and implementations of fast spectral methods". The audience was around 50 people and the talks were received very well. |
Year(s) Of Engagement Activity | 2023 |
URL | https://www.siam.org/conferences/cm/conference/cse23 |
Description | Numerics & Acoustics Workshop, Imperial, 2022 |
Form Of Engagement Activity | Participation in an activity, workshop or similar |
Part Of Official Scheme? | No |
Geographic Reach | National |
Primary Audience | Professional Practitioners |
Results and Impact | A 3-day workshop with around 25 people on numerics and acoustics. There was an open-ended format that encouraged thorough conversations. |
Year(s) Of Engagement Activity | 2022 |
Description | Presented at CSE23 |
Form Of Engagement Activity | A talk or presentation |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Professional Practitioners |
Results and Impact | Presented at CSE23 on "sparse spectral methods for fractional PDEs". |
Year(s) Of Engagement Activity | 2023 |
URL | https://www.siam.org/conferences/cm/conference/cse23 |
Description | Presented at Young Researcher's minisymposium at GAMM 2022 |
Form Of Engagement Activity | A talk or presentation |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Professional Practitioners |
Results and Impact | Presentation at GAMM 2022. |
Year(s) Of Engagement Activity | 2022 |
URL | https://jahrestagung.gamm-ev.de/annual-meeting-2022/annual-meeting/ |
Description | Presented at the University of Leicester CSE Mathematics Seminar |
Form Of Engagement Activity | A talk or presentation |
Part Of Official Scheme? | No |
Geographic Reach | Regional |
Primary Audience | Professional Practitioners |
Results and Impact | Presented the work on sparse spectral methods for fractional PDEs at the University of Leicester CSE Mathematics Seminar |
Year(s) Of Engagement Activity | 2022 |