Research

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Preprints on CVGMT
Preprints on Heriot-Watt Research Portal
Preprints on the arXiv

Research Interests

My research areas are optimal transport theory, the calculus of variations, and applied analysis. In the last few years one of my main interests has been semi-discrete optimal transport theory and its applications to materials science (geometric modelling of polycrystalline microstructures), weather modelling (the semi-geostrophic equations), pattern formation (crystallization), and optimal quantization.

Selected Publications

Bourne, D.P., Schmitzer, B. & Wirth, B. (2024) Semi-discrete unbalanced optimal transport and quantization, arXiv:1808.01962. PDF
Bourne, D.P., Pearce, M. & Roper, S.M. (2024) Inverting Laguerre tessellations: Recovering tessellations from the volumes and centroids of their cells using optimal transport, arXiv:2406.00871. PDF software
Buze, M., Feydy, J., Roper, S.M., Sedighiani K. & Bourne, D.P. (2024) Anisotropic power diagrams for polycrystal modelling: Efficient generation of curved grains via optimal transport, Computational Materials Science 245, 113317. PDF software
Bourne, D.P., Pearce, M. & Roper, S.M. (2023) Geometric modelling of polycrystalline materials: Laguerre tessellations and periodic semi-discrete optimal transport, Mechanics Research Communications 127, 104023. PDF software
Egan, C.P., Bourne, D.P., Cotter, C.J., Cullen, M.J.P., Pelloni, B., Roper, S.M. & Wilkinson, M. (2022) A new implementation of the geometric method for solving the Eady slice equations, Journal of Computational Physics 469, 111542. PDF software
Bourne, D.P., Egan, C.P., Pelloni, B. & Wilkinson, M. (2022) Semi-discrete optimal transport methods for the semi-geostrophic equations, Calculus of Variations and Partial Differential Equations, 61, 39. PDF
Bourne, D.P. & Cristoferi, R. (2021) Asymptotic optimality of the triangular lattice for a class of optimal location problems, Communications in Mathematical Physics, 387, 1549-1602. PDF
Bourne, D.P., Kok, P.J.J., Roper, S.M. & Spanjer, W.D.T. (2020) Laguerre tessellations and polycrystalline microstructures: A fast algorithm for generating grains of given volumes, Philosophical Magazine, 100, 2677-2707. PDF software
Bourne, D.P., Cushing, D., Liu, S., Münch F. & Peyerimhoff, N. (2018) Ollivier-Ricci idleness functions of graphs. SIAM Journal on Discrete Mathematics, 32, 1408-1424. PDF
Bourne, D.P., Conti, S. & Müller, S. (2017) Energy bounds for a compressed elastic film on a substrate. Journal of Nonlinear Science, 27, 453-494. PDF
Bourne, D.P. & Roper, S.M. (2015) Centroidal power diagrams, Lloyd's algorithm and applications to optimal location problems. SIAM Journal on Numerical Analysis, 53, 2545-2569. PDF
Bourne, D.P., Peletier, M.A. & Roper, S.M. (2014) Hexagonal patterns in a simplified model for block copolymers. SIAM Journal on Applied Mathematics, 74, 1315-1337. PDF
Bourne, D.P., Peletier, M.A. & Theil, F. (2014) Optimality of the triangular lattice for a particle system with Wasserstein interaction. Communications in Mathematical Physics, 329, 117-140. PDF

Full list

Grants

EPSRC grant EP/V00204X/1 Mathematical Theory of Polycrystalline Materials, £576,848, 1 April 2021 - 31 March 2025, Co-I (with John Ball PI and Lucia Scardia Co-I)
EPSRC First grant EP/R013527/1, EP/R013527/2 Designer Microstructure via Optimal Transport Theory, £101,152, 1 Jan 2018 - 31 May 2020, PI

Supervision

Postdocs:

Maciej Buze (co-supervised with John Ball, Lucia Scardia & Steven Roper, funded by EP/V00204X/1, Feb 2023 -)
Charlie Egan (co-supervised with Beatrice Pelloni, funded by a Maxwell Institute Research Fellowship, Nov 2022 - Oct 2023)
Myrto Galanopoulou (co-supervised with John Ball & Lucia Scardia, funded by EP/V00204X/1, Sept 2021 - Oct 2023)
Riccardo Cristoferi (funded by Heriot-Watt University and EP/R013527/2, Oct 2018 - July 2020)
David Cushing (co-supervised with Norbert Peyerimhoff, funded by an EPSRC Doctoral Prize, Durham University, Jan 2017 - May 2018)

PhD students:

Théo Lavier (co-supervised with Beatrice Pelloni, funded by the Maxwell Institute Graduate School in Mathematical Modelling, Analysis and Computation, Sept 2021 -)
Mason Pearce (co-supervised with Steven Roper, funded by the Maxwell Institute Graduate School in Mathematical Modelling, Analysis and Computation, Sept 2019 - July 2024)
Charlie Egan (co-supervised with Beatrice Pelloni and Mark Wilkinson, funded by the Maxwell Institute Graduate School in Analysis & its Applications, Sept 2018 - Oct 2022)

Collaboration with Tata Steel

I have an ongoing collaboration with Tata Steel Europe. In particular I work with the Plasticity & Tribology Knowldge Group based in IJmuiden, which is part of the Application & Engineering (R&D) department. You can find out more about our work here:

Buze, M., Feydy, J., Roper, S.M., Sedighiani K. & Bourne, D.P. (2024) Anisotropic power diagrams for polycrystal modelling: efficient generation of curved grains via optimal transport, arXiv:2403.03571. PDF software
Bourne, D.P., Pearce, M. & Roper, S.M. (2023) Geometric modelling of polycrystalline materials: Laguerre tessellations and periodic semi-discrete optimal transport, Mechanics Research Communications 127, 104023. PDF software
Bourne, D.P., Kok, P.J.J., Roper, S.M. & Spanjer, W.D.T. (2020) Laguerre tessellations and polycrystalline microstructures: A fast algorithm for generating grains of given volumes, Philosophical Magazine, 100, 2677-2707. PDF software
Watch my seminar on an application of optimal transport theory in the steel industry at the workshop Mathematical design of new materials, 11-15 March 2019: