Tsili, Antonia 
ORCID: 0000-0001-6959-5338
(2024)
Geometric Optimisation of Crystal Structures: Foundations and Algorithmic Approaches.
PhD thesis, University of Liverpool.
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Abstract
The focus of this thesis is to explore the geometric optimisation of crystal structures through an algorithmic lens. As a high-level description, this optimisation problem constitutes the arranging of atoms in the continuous space defined by a convex shape. Continuous optimisation has been rapidly expanding as a field in recent years, especially due to its important role in machine learning, however, the examination of the algorithmic aspect of the geometric optimisation of crystals has been more static. This work aims to change that by establishing the foundations required for understanding the mathematical problem, benchmarking staple algorithmic approaches, offering an intuitive implementation that can be easily used for experimentation and endeavouring to propose new methods and research directions. We first want to create a comprehensive framework with detailed documentation that facilitates testing new algorithmic recipes in the setting of the geometric optimisation of crystals. This is addressed in three ways; we provide the essential interdisciplinary background and all mathematical derivations in detail, we implement the formulae elaborated on and create a benchmarking set of data and experiments using our implementation. We conduct experiments using popular first order methods and provide a thorough analysis on the results, showing the significant impact that step size selection can have on the whole optimisation process. We also propose a new scoring tool which we use to assess the employed algorithmic recipes, thus creating a complete benchmark case that can be used as a reference. A long standing practice in the context of the geometric optimisation of crystals has been to switch from Conjugate Gradient to Broyden-Fletcher-Goldfarb-Shanno (BFGS), however, there are no formally recorded results to support the choice of this hybrid approach. We run experiments using this hybrid method on our benchmarking dataset and prove that switching is much more beneficial than employing a pure first or second order algorithm, assessing the results with our proposed scoring tool. The latter argument is underpinned by results from preliminary tests run with our adaptation of a recent second order algorithm, newly introduced in the context of the application that we study. This algorithm, which is based on cubic regularisation, reaches its maximum potential as a method for the geometric optimisation of crystals when combined with Conjugate Gradient. While the first produces radical minimisation updates with just a few Hessian calculations, it hits a threshold where descent efforts do not make enough progress, at which point Conjugate Gradient can take over and locate the local minimum efficiently.
| Item Type: | Thesis (PhD) |
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| Uncontrolled Keywords: | continuous optimization, crystal structure prediction, gradient methods, local minimum, structural relaxation |
| Divisions: | Faculty of Science and Engineering Faculty of Science and Engineering > School of Electrical Engineering, Electronics and Computer Science |
| Depositing User: | Symplectic Admin |
| Date Deposited: | 21 Jan 2025 10:36 |
| Last Modified: | 03 Feb 2025 20:09 |
| DOI: | 10.17638/03187183 |
| Supervisors: |
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| URI: | https://livrepository.liverpool.ac.uk/id/eprint/3187183 |
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