Dr Mark Lukas

Room: S&C/3.028

Research

Numerical analysis, ill-posed and inverse problems, optimisation.

Projects

1. Inverse Problems

For many mathematical models arising in science and technology, it is possible to meaningfully formulate both a direct and an inverse problem. The direct problem is to find the effect g from a cause f, given that the model K is completely specified. For example, the direct problem in heat conduction is to find the temperature distribution in a body at some later time given an initial temperature distribution, boundary condition and any heat input (making up the cause) and the heat equation with known parameters (thermal conductivity, specific heat, density and the domain) as the fully specified model. This problem has been studied for a long time (over 150 years) and there are well-known analytic and numerical methods available to solve it. In an inverse problem, the effect g is known (in practice, usually imprecisely as measured data) and it is desired to find either the cause f or some aspect of the model K. For example, in the heat conduction model above, we might have measurements of the current temperature distribution and we want to find the unknown initial temperature distribution. Or assuming we know the initial and some intermediate temperatures too, we may want to find some unknown parameter in the model, for example the thermal conductivity as a function of position or even the shape of the boundary of the domain. Such problems have been studied in depth for a comparatively short time (about 40 years). Over this period, many inverse problems have become crucial for modern technology in areas such as remote sensing and medical imaging. Although inverse problems can be of many different kinds, they usually have a few common features. First it is often difficult to show (or not even known) if there is a solution; for example, in the problem above does the exact temperature distribution identify the unknown thermal conductivity, and if so is it unique? In addition, inverse problems are usually ill-posed, meaning that even if there is a unique solution, it is sensitive to errors (measurement errors or even roundoff errors) in the data (like for an ill-conditioned system of linear equations). This means that special regularisation techniques are needed to stabilise these problems and obtain good approximate solutions. The following are some inverse problems that could be studied as honours projects:

(i) Tomography

This is the subject of reconstructing the internal structure of a three dimensional body using certain measurements made around the body. In X-ray computer assisted tomography (CAT scans), a picture of (the density distribution of) a plane cross-section of the body is found from measurements of the attenuation of X-rays emitted in a beam from multiple sources to detectors on the other side. Mathematically, the problem is to invert the Radon transform, which associates with any line in the plane the integral of the distribution along that line

(ii) Transmissivity Estimation

Steady state flow in a groundwater aquifer can be modelled as an elliptic partial differential equation. The problem is to estimate the transmissivity (reflecting the porosity) from measurements of the groundwater level and the recharge/discharge of water.

(iii) Integral Equations of the First Kind

These equations arise in many areas including hydrology, spectroscopy, degraded image restoration, stereology (estimation of the size distributions of embedded spheres from the measured size distribution of circles in random plane sections) and remote sensing (for example, estimating the temperature profile of the atmosphere from satellite measurements or prospecting for ore bodies using magnetic field data obtained from plane flights across the land).

2. Surface Estimation from Scattered Data

Spline functions in one variable can be extended to splines in two or more variables in several ways. One approach results in what are known as thin plate splines. These have proved to be useful in finding interpolating and smoothed surfaces from scattered data. However more work is required to find efficient methods for constructing these splines for large data sets.

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