Our research in this area has three broad themes:
Approximation theory
Approximation theory is concerned with how functions can best be approximated with simpler functions and with quantitatively characterizing the errors introduced thereby. The objective is to make the approximation as close as possible to the actual function so that the approximant can be used as a proxy for the study of the original function. Moreover, other functions related to the given one, e.g. the derivative and the integral, can then be calculated with more ease via its approximant.
Approximation theory is often associated with the study of orthogonal polynomials, since classic theory and practice of function approximation employ orthogonal polynomials, e.g. Chebyshev polynomials. With orthogonal polynomials, smooth functions can be approximated typically with an accuracy close to that of the underlying computer’s floating point arithmetic. Our current research of approximation theory revolves about polynomial-based approximation of functions and operators and the numerical computing with these approximants, instead of discrete numbers, adopting the idea of ‘computing with functions’. Our research in approximation theory offers fundamental tools to numerical methods for differential or integral equations.
Geometric integration
Conservation laws describe some of the most fundamental properties of a given PDE. They apply universally, that is, each conservation law is satisfied by all solutions of the PDE. Conservation laws are topological rather than geometric, but the fundamental strategy of geometric integration still applies: we aim to find discretisations that put the error where it has little qualitative effect.
In her celebrated 1918 paper, Emmy Noether proved that the Lie group symmetry of a Lagrangian guarantees conservation laws, as many as the dimension of the Lie group. Her proof is constructive, and the laws are referred to as Noether’s laws. The work of Noether has gained public attention recently with the publication of an article in the New York Times where the result is “consider[ed] …as important as Einstein’s theory of relativity; it undergirds much of today’s vanguard research in physics, including the hunt for the almighty Higgs boson.”
Noether’s laws
The main idea of current research at Kent is to embed, a priori, any symmetries of a given variational model into the numerical approximation. This is achieved by constructing an invariant approximate Lagrangian, then to prove a relevant discrete version of Noether’s theorem, and then to show the discrete laws and Euler Lagrange equations converge to the smooth laws in some useful sense. This was known to be achievable for simple finite difference models.
Prof Elizabeth Mansfield, with her post-doctoral research fellow Tristan Pryer (now at the University of Reading), completed this program for the finite element method, for the standard physical symmetries which act linearly on the base space: translations in time and space, rotations, and scalings. There is interest in achieving a similar result for the discontinuous Galerkin method, and a variety of interpolation methods.
When considering both Lie group symmetry and continuum limits, it is important to have a secure mechanism by which discrete invariants may converge to differential invariants. To this end, with Gloria Marı Beffa in Madison, Wisconsin, we constructed a multi-dimensional multi-space which has the jet bundle embedded into a space coordinated by the coefficients of multivariable Lagrange interpolants. Applying the modern, Lie group based moving frame technology to multi-space, we can simultaneously consider the differential invariants used to build the invariant Lagrangian and the invariant approximations. The development of a Noether’s theorem for multi-space is underway. There is also interest in constructing an analogue of multi-space based on B-splines, rather than Lagrange interpolation.
Numerical preservation of conservation laws
Our research uses computer algebra to discover finite difference methods that preserve multiple conservation laws. Early results show that this can yield highly accurate methods fairly systematically. However, as this is a new research area, there are many open questions!
Graph by Ana Rojo-Echeburua showing a Euler Elastica and two discrete approximations.
Numerical solutions of differential equations
Numerical methods for differential equations and integral equations are methods used to find numerical approximations to the solutions to (ordinary or partial) differential equations (ODEs or PDEs) and integral equations (IEs), where ODEs include boundary value problems (BVPs) and initial value problems (IVPs).
Our research in numerical differential equations focuses on pseudo-spectral and spectral methods. The main idea of these methods is to write the solution of the equation as a sum of certain “basis functions” (for example, as a Fourier series or a series of orthogonal polynomials) and then to solve for the unknown coefficients in the sum so that the solution satisfies the equation as well as possible. Spectral methods can be used to solve ODEs, PDEs, IEs, and eigenvalue problems involving all kinds of commonly-seen operators including differentiation, integration (definite or indefinite), convolution, etc. When applying spectral methods to time-dependent PDEs, the solution is written as a sum of basis functions but with time-dependent coefficients; substituting this in the PDE yields a system of ODEs in the coefficients which can be solved using suitable numerical method for ODEs. Eigenvalue problems for ODEs are similarly converted to matrix eigenvalue problems and hence can be solved by employing discrete eigen-solvers.