Genetic algorithms and particular integral equations arising from hot stellar winds and solar flare

This thesis is concerned with the application of genetic algorithm techniques to the inversion of integral equations arising from a stellar wind and solar flare hard x-ray problem. The opening chapter sets the context of this work giving a brief background and introduction to integral equations in astronomy, search methods, genetic algorithms, solar flares and hot stellar winds. In the following chapter 2 the development of a new genetic algorithm, the so called genetic farming algorithm, is described. The aim was to develop a more suitable genetic algorithm for the type of problems considered in this study. As this was a new approach, some time is spent introducing genetic algorithm operators and numerical issues, which in turn are used to explain the development the genetic farming algorithm. The main difference between the developed genetic farming algorithm and other genetic algorithms was the selection operator. Classically, selection operators probabilistically choose candidate strings from the current population related to fitness, with better solutions given a higher chance of being selected. The genetic farming selection operator ranked the candidate solutions from best to worst by fitness. The best was then copied into the next population (elitism), while the worst was combined with the best to produce two new solutions, only one of which was copied into the new population, while the other was discarded. The rest of the population was then paired 2nd with 3rd, 4th with 5th etc. in each case, to produce two new candidate solutions. The genetic farming algorithm removes altogether the difficult balance between probability of crossover and scaling of the fitness operator. The genetic farming algorithm, by requiring less recoding between applications, was designed to be more generally applicable. To illustrate some of its features, comparison testing is given between this and another classical genetic algorithm. This testing was designed to highlight particular features of the genetic farming algorithm, such as the superior effectiveness of the mutation and cross over operators. The improved generality of the genetic farming method is demonstrated in chapters following these comparison tests by the successful implementation across a broad range of problems. Some necessary numerical issues are also introduced in chapter 2. These were required for the application of optimisation techniques, such as genetic algorithms, to integral equations. It is shown how the necessary discretisation of the integral equations results in the problem being ill-posed and how this may be dealt with by the introduction of regularisation techniques. The first practical application of the genetic farming method is shown in chapter 3 on the recovery of hot stellar wind velocity profiles from simulated recombination emission lines. It is shown how for hot stellar winds the wind velocity profile could be recovered by considering the wind opacity and total flux of recombination lines formed within the wind. This resulted in an integral equation for which the genetic algorithm was tasked with recovering the velocity profile. The inversion of this formulation is made difficult by an occultation factor which arises from the accounting for the light falling behind the star. Previous work had failed to utilise the whole formulation with the occultation factor; thus the inversion including it is a new result. These results show that the full formulation of this inversion problem, and not the approximations used by previous authors, is essential in providing reasonable results. Chapter 4 concerns practical issues for the velocity recovery using the emission line method: specifically the actual range of wavelengths required for the method to be effective are derived. It is discovered that the required wavelengths were limited to a wavelength range related to the efficiency of the wind. The other main task, given in chapter 5, for the genetic algorithm was to recover electron flux data from solar flare hard x-ray data. It is explained how different formulations of the integral equation describing the hard x-ray emission from solar flares have previously been used: speciflcally different electron-ion interaction bremsstrahlung cross sections. It has until now not been made clear what impact this may have on the recoveries. The genetic farming algorithm is used and shows that at low noise levels the choice of cross section characterising these equations was not very important. However, at high levels of noise the less accurate approximation of the cross section actually recovered better results than the true cross section. Both of these are new results. These examples demonstrate that the genetic farming technique is a useful and generally applicable one for the inversion of integral equations.