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# Novel Preconditioned Iterative Methods for Radial Basis Linear Systems

This project will develop effective solvers for linear systems in these RBF methods. In particular, we will focus on certain iterative methods (Krylov subspace methods) that start with an initial guess of the solution that is improved at each step.

One

26 January 2018

3 years

## Eligibility

Qualifications: BSc (Hons) 2:1 in Mathematics or in a related discipline.

Funding:   The studentship covers UK/EU tuition fees and comes with an annual tax-free stipend at the standard UK rate. International students who can fund the difference between UK/EU and International fee rates are also encouraged to apply.﻿

## Project Details

This project will develop effective solvers for linear systems in these RBF methods. In particular, we will focus on certain iterative methods (Krylov subspace methods) that start with an initial guess of the solution that is improved at each step. For these ill-conditioned RBF problems, finding matrices known as preconditioners that accelerate the solution process are essential. Thus, at the core of this project will be the proposal, and analysis, of new solvers for RBF-based PDE solvers. The preconditioners will be tested on real-world applications.
Radial basis functions (RBFs) have several advantages when used to numerically approximate the solution of partial differential equations (PDEs) in applications. However, solving linear systems that arise within RBF-based solvers is challenging. This project will develop and mathematically analyse new preconditioned iterative solvers of these linear systems, and will test their performance on interesting applications.
Mathematical modelling is increasingly used to investigate and understand phenomena and forecast future events, particularly when experimentation is prohibitive or costly. However, real-world problems are often posed on complicated domains and involve scattered data, e.g. in geophysical and biological applications, or are inherently high-dimensional, e.g. in quantum physics, finance and systems biology.
Complex geometry, scattered data and high dimensionality can be difficult for some numerical methods for PDEs to deal with. However, these problem features are handled relatively easily by radial basis function approaches. RBF methods represent the solution of PDEs as a combination of radial basis functions that can be placed anywhere in the computational domain. The suitability of RBFs for complex problems is evidenced by their use in applications, including fluid flow, geophysics, plasma physics, finance and biology.
Despite their advantages for dealing with complex, real-world problems, RBF methods can be difficult to implement. This is because obtaining the combination of radial basis functions that describes the PDE solution requires the solution of one or more challenging (i.e. ill-conditioned) systems of linear equations.
This project will develop effective solvers for linear systems in these RBF methods. In particular, we will focus on certain iterative methods (Krylov subspace methods) that start with an initial guess of the solution that is improved at each step. For these ill-conditioned RBF problems, finding matrices known as preconditioners that accelerate the solution process are essential. Thus, at the core of this project will be the proposal, and analysis, of new solvers for RBF-based PDE solvers. The preconditioners will be tested on real-world applications.
Applicants should have or expect to obtain a good (I or II(i)) honours degree in mathematics or in a related discipline. This project would suit students with an interest in linear algebra and/or numerical analysis. Experience of numerical mathematics and/or programming would be beneficial, but is not essential.
The successful applicant will be part of a vibrant postgraduate community, and will have access to a range of training opportunities, including the Scottish Mathematical Sciences Training Centre (https://smstc.ac.uk/).