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Insights from the parallel implementation of efficient algorithms for the fractional calculus
Banks, Nicola E.
Banks, Nicola E.
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2015-07
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Abstract
This thesis concerns the development of parallel algorithms to solve fractional
differential equations using a numerical approach. The methodology
adopted is to adapt existing numerical schemes and to develop prototype
parallel programs using the MatLab Parallel Computing Toolbox (MPCT).
The approach is to build on existing insights from parallel implementation of
ordinary differential equations methods and to test a range of potential candidates for parallel implementation in the fractional case. As a consequence
of the work, new insights on the use of MPCT for prototyping are presented,
alongside conclusions and algorithms for the effective implementation of parallel
methods for the fractional calculus.
The principal parallel approaches considered in the work include:
- A Runge-Kutta Method for Ordinary Differential Equations including
the application of an adapted Richardson Extrapolation Scheme
- An implementation of the Diethelm-Chern Algorithm for Fractional
Differential Equations
- A parallel version of the well-established Fractional Adams Method for
Fractional Differential Equations
- The adaptation for parallel implementation of Lubich's Fractional Multistep
Method for Fractional Differential Equations
An important aspect of the work is an improved understanding of the
comparative diffi culty of using MPCT for obtaining fair comparisons of parallel
implementation. We present details of experimental results which are
not satisfactory, and we explain how the problems may be overcome to give
meaningful experimental results. Therefore, an important aspect of the conclusions of this work is the advice for other users of MPCT who may be planning to use the package as a prototyping tool for parallel algorithm development: by understanding how implicit multithreading operates, controls
can be put in place to allow like-for-like performance comparisons between
sequential and parallel programs.
Citation
Banks, N. E. (2015). Insights from the parallel implementation of efficient algorithms for the fractional calculus. (Doctoral dissertation). University of Chester, United Kingdom.
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University of Chester
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Thesis or dissertation
Language
en