Numerical quadrature has a huge number of applications in scientific computing, engineering and applied sciences, including the evaluation of special functions, the calculation of Fourier and Laplace transforms, and the implementation of numerical methods for the solution of ODEs and PDEs. Despite being a classical topic in numerical analysis, quadrature remains a highly active research area. In this project we aim to address some fundamental open problems concerning the development, analysis and implementation of efficient quadrature rules for singular and oscillatory integrands on non-smooth domains.
The open problems we wish to address are motivated by our previous and ongoing research into computational acoustic and electromagnetic wave scattering, as modelled for example by the Helmholtz equation and the time-harmonic Maxwell equations. This is a highly active research area, due to the fact that in many applications throughout science and technology including medical imaging, RF and microwave communications and weather/climate prediction, there are important scattering problems for which no satisfactory numerical method currently exists. The main challenges relate to the accurate and efficient treatment of high frequency problems (where the wavelength is small compared to the scatterer) and non-smooth scatterers (where the scatterer has multiple corners, edges and other surface irregularities).
Popular simulation methods, all of which have numerical quadrature at their core, include variational formulations of the underlying PDEs (leading to finite element methods), and boundary and volume integral equation formulations (leading to boundary element method and methods such as the Discrete Dipole Approximation, respectively). Existing quadrature rules available for these methods apply only to situations where the function being integrated (the "integrand''), and the domain over which the integration is carried out, are relatively simple, possessing a certain degree of mathematical "smoothness". However, in each of the applications listed above, one encounters integrands that are singular (blow up to infinity at certain points) and/or highly oscillatory, and integration domains that are highly non-smooth. The former situation arises when the scatterer is large compared to the incident wavelength, and the latter when the scatterer is particularly "rough" or irregular in shape, as is the case e.g. for scattering by trees and vegetation, building facades, the surface of the ocean, certain antenna designs in electrical engineering, and atmospheric particles such as snow/ice crystals and dust aggregates.
This project aims to generalise the theory of numerical quadrature to be able to handle the complicated singular and oscillatory integrals over non-smooth domains that arise in real-world applications, with a particular focus on atmospheric physics, where improved tools for computing scattering of radiation by atmospheric ice crystals would significantly improve current capabilities for remote sensing (and hence weather prediction) and the calculation of radiation balances (and hence climate prediction). The project will deliver new quadrature rules and associated algorithms, and new theoretical results guaranteeing their accuracy and stability. We will develop user-friendly open-source software for quadrature rules and scattering simulations, designed for non-expert practitioners in a range of application areas.
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