Improving the numerical stability of fast matrix multiplication

Fast algorithms for matrix multiplication, namely those that perform asymptotically fewer scalar operations than the classical algorithm, have been considered primarily of theoretical interest. Apart from Strassen's original algorithm, few fast algorithms have been efficiently implemented or used in practical applications. However, there exist many practical alternatives to Strassen's algorithm with varying performance and numerical properties. Fast algorithms are known to be numerically stable, but because their error bounds are slightly weaker than the classical algorithm, they are not used even in cases where they provide a performance benefit. We argue in this paper that the numerical sacrifice of fast algorithms, particularly for the typical use cases of practical algorithms, is not prohibitive, and we explore ways to improve the accuracy both theoretically and empirically. The numerical accuracy of fast matrix multiplication depends on properties of the algorithm and of the input matrices, and we consider both contributions independently. We generalize and tighten previous error analyses of fast algorithms and compare their properties. We discuss algorithmic techniques for improving the error guarantees from two perspectives: manipulating the algorithms, and reducing input anomalies by various forms of diagonal scaling. Finally, we benchmark performance and demonstrate our improved numerical accuracy

A methodology for passenger-centred rail network optimisation

Optimising the allocation of limited resources, be they existing assets or investment, is an ongoing challenge for rail network managers. Recently, methodologies have been developed for optimising the timetable from the passenger perspective. However, there is a gap for a decision support tool which optimises rail networks for maximum passenger satisfaction, captures the experience of individual passengers and can be adapted to different networks and challenges. Towards building such a tool, this thesis develops a novel methodology referred to as the Sheffield University Passenger Rail Experience Maximiser (SUPREME) framework. First, a network assessment metric is developed which captures the multi-stage nature of individual passenger journeys as well as the effect of crowding upon passenger satisfaction. Second, an agent-based simulation is developed to capture individual passenger journeys in enough detail for the network assessment metric to be calculated. Third, for the optimisation algorithm within SUPREME, the Bayesian Optimisation method is selected following an experimental investigation which indicates that it is well suited for ‘expensive-to-compute’ objective functions, such as the one found in SUPREME. Finally, in case studies that include optimising the value engineering strategy of the proposed UK High Speed Two network when saving £5 billion initial investment costs, the SUPREME framework is found to improve network performance by the order of 10%. This thesis shows that the SUPREME framework can find ‘good’ resource allocations for a ‘reasonable’ computational cost, and is sufficiently adaptable for application to many rail network challenges. This indicates that a decision support tool developed on the SUPREME framework could be widely applied by network managers to improve passenger experience and increase ticket revenue. Novel contributions made by this thesis are: the SUPREME methodology, an international comparison between the Journey Time Metric and Disutility Metric, and the application of the Bayesian Optimisation method for maximising the performance of a rail network