arXiv subjects

David P. Bourne

Publications and source records attributed to David P. Bourne.

11 recordsLinked to original sources

arXiv2026

Semi-discrete convex order and Laguerre tessellation fitting

Laguerre tessellations offer an efficient way to parameterize a large class of convex partitions of Euclidean space using only a set of points and scalar weights. For this reason, they have become popular in computational geometry, imaging and numerical analysis, both as a modeling and a discretization tool. In this paper we study the problem of reconstructing a Laguerre tessellation with prescribed cell volumes from the barycenters of its cells. We establish a geometric interpretation of this problem in terms of the set of discrete measures dominated in convex order by an absolutely continuous measure. In particular, we show that the reconstruction problem can be solved approximately by computing a Wasserstein projection onto this set. More generally, our method can also be applied to fit a Laguerre tessellation to an arbitrary set of barycenters. We give a concrete application of this in materials science, of fitting a Laguerre tessellation to an electron backscatter diffraction (EBSD) image of a steel.

math.OC↗

arXiv2026

Semi-discrete optimal transport techniques for the compressible semi-geostrophic equations

We prove existence of weak solutions of the 3D compressible semi-geostrophic (SG) equations with compactly supported measure-valued initial data. These equations model large-scale atmospheric flows. Our proof uses a particle discretisation and semi-discrete optimal transport techniques. We show that, if the initial data is a discrete measure, then the compressible SG equations admit a unique, twice continuously differentiable, energy-conserving and global-in-time solution. In general, by discretising the initial measure by particles and sending the number of particles to infinity, we show that for any compactly supported initial measure there exists a global-in-time solution of the compressible SG equations that is Lipschitz in time. This significantly generalises the original results due to Cullen and Maroofi (2003), and it provides the theoretical foundation for the design of numerical schemes using semi-discrete optimal transport to solve the 3D compressible SG equations.

math.AP↗

arXiv2026

Polynomial diagrams for microstructure modelling

We formulate a framework of polynomial diagrams, which are a generalisation of power diagrams (PDs) and anisotropic power diagrams (APDs) allowing for boundaries between cells to be algebraic curves of a prescribed degree. We show that they arise naturally from rephrasing PDs (APDs) as first-degree (second-degree) instances of linear parametrised minimisation diagrams. We also develop an efficient GPU-accelerated framework for fitting polynomial diagrams to image data using Legendre polynomials and by maximising a regularised concave objective function adapted from classical logistic regression literature. A largely self-contained analysis of the optimisation algorithm is also provided, including identification of scale and gauge invariances and the limiting objective function as the regularisation parameter vanishes. We apply the algorithm to fit polynomial diagrams to electron backscatter diffraction images of steel.

math.OC↗

arXiv2025

Semi-discrete unbalanced optimal transport and quantization

In this paper we study the class of optimal entropy-transport problems introduced by Liero, Mielke and Savaré in Inventiones Mathematicae 211 in 2018. This class of unbalanced transport metrics allows for transport between measures of different total mass, unlike classical optimal transport where both measures must have the same total mass. In particular, we develop the theory for the important subclass of semi-discrete unbalanced transport problems, where one of the measures is diffuse (absolutely continuous with respect to the Lebesgue measure) and the other is discrete (a sum of Dirac masses). We characterize the optimal solutions and show they can be written in terms of generalized Laguerre diagrams. We use this to develop an efficient method for solving the semi-discrete unbalanced transport problem numerically. As an application we study the unbalanced quantization problem, where one looks for the best approximation of a diffuse measure by a discrete measure with respect to an unbalanced transport metric. We prove a type of crystallization result in two dimensions -- optimality of a locally triangular lattice with spatially varying density -- and compute the asymptotic quantization error as the number of Dirac masses tends to infinity.

math.OC↗

arXiv2025

A definition of the background state of the atmosphere using optimal transport

The dynamics of atmospheric disturbances are often described in terms of displacements of air parcels relative to their locations in a notional background state. Modified Lagrangian Mean (MLM) states have been proposed by M. E. McIntyre using the Lagrangian conserved variables potential vorticity and potential temperature to label air parcels, thus avoiding the need to calculate trajectories explicitly. Methven and Berrisford further defined a zonally symmetric MLM state for global atmospheric flow in terms of mass in zonal angular momentum ($z$) and potential temperature ($θ$) coordinates. We prove that for any snapshot of an atmospheric flow in a single hemisphere, there exists a unique energy-minimising MLM state in geophysical coordinates (latitude and pressure). Since the state is an energy minimum, it is suitable for quantification of finite amplitude disturbances and examining atmospheric instability. This state is obtained by solving a free surface problem, which we frame as the minimisation of an optimal transport cost over a class of source measures. The solution consists of a source measure, encoding surface pressure, and an optimal transport map, connecting the distribution of mass in geophysical coordinates to the known distribution of mass in $(z, θ)$. We show that this problem reduces to an optimal transport problem with a known source measure, which has a numerically feasible discretisation. Additionally, our results hold for a large class of cost functions, and generalise analogous results on free surface variants of the semi-geostrophic equations.

math.OC↗

arXiv2025

Inverting Laguerre tessellations: Recovering tessellations from the volumes and centroids of their cells using optimal transport

In this paper we study an inverse problem in convex geometry, inspired by a problem in materials science. Firstly, we consider the question of whether a Laguerre tessellation (a partition by convex polytopes) can be recovered from only the volumes and centroids of its cells. We show that this problem has a unique solution and give a constructive way of computing it using optimal transport theory and convex optimisation. Secondly, we consider the problem of fitting a Laguerre tessellation to synthetic volume and centroid data. Given some target volumes and centroids, we seek a Laguerre tessellation such that the difference between the volumes and centroids of its cells and the target volumes and centroids is minimised. For an appropriate objective function and suitable data, we prove that local minimisers of this problem can be constructed using convex optimisation. We also illustrate our results numerically. There is great interest in the computational materials science community in fitting Laguerre tessellations to electron backscatter diffraction (EBSD) and x-ray diffraction images of polycrystalline materials. As an application of our results we fit a 2D Laguerre tessellation to an EBSD image of steel.

math.OC↗

arXiv2024

Anisotropic power diagrams for polycrystal modelling: efficient generation of curved grains via optimal transport

The microstructure of metals and foams can be effectively modelled with anisotropic power diagrams (APDs), which provide control over the shape of individual grains. One major obstacle to the wider adoption of APDs is the computational cost that is associated with their generation. We propose a novel approach to generate APDs with prescribed statistical properties, including fine control over the size of individual grains. To this end, we rely on fast optimal transport algorithms that stream well on Graphics Processing Units (GPU) and handle non-uniform, anisotropic distance functions. This allows us to find large APDs that best fit experimental data and generate synthetic high-resolution microstructures in (tens of) seconds. This unlocks their use for computational homogenisation, which is especially relevant to machine learning methods that require the generation of large collections of representative microstructures as training data. The paper is accompanied by a Python library, PyAPD, which is freely available at: www.github.com/mbuze/PyAPD.

cond-mat.mtrl-sci↗

arXiv2022

A new implementation of the geometric method for solving the Eady slice equations

We present a new implementation of the geometric method of Cullen & Purser (1984) for solving the semi-geostrophic Eady slice equations which model large scale atmospheric flows and frontogenesis. The geometric method is a Lagrangian discretisation, where the PDE is approximated by a particle system. An important property of the discretisation is that it is energy conserving. We restate the geometric method in the language of semi-discrete optimal transport theory and exploit this to develop a fast implementation that combines the latest results from numerical optimal transport theory with a novel adaptive time-stepping scheme. Our results enable a controlled comparison between the Eady-Boussinesq vertical slice equations and their semi-geostrophic approximation. We provide further evidence that weak solutions of the Eady-Boussinesq vertical slice equations converge to weak solutions of the semi-geostrophic Eady slice equations as the Rossby number tends to zero.

math.NA↗

arXiv2021

Semi-discrete optimal transport methods for the semi-geostrophic equations

We give a new and constructive proof of the existence of global-in-time weak solutions of the 3-dimensional incompressible semi-geostrophic equations (SG) in geostrophic coordinates, for arbitrary initial measures with compact support. This new proof, based on semi-discrete optimal transport techniques, works by characterising discrete solutions of SG in geostrophic coordinates in terms of trajectories satisfying an ordinary differential equation. It is advantageous in its simplicity and its explicit relation to Eulerian coordinates through the use of Laguerre tessellations. Using our method, we obtain improved time-regularity for a large class of discrete initial measures, and we compute explicitly two discrete solutions. The method naturally gives rise to an efficient numerical method, which we illustrate by presenting simulations of a 2-dimensional semi-geostrophic flow in geostrophic coordinates generated using a numerical solver for the semi-discrete optimal transport problem coupled with an ordinary differential equation solver.

math.AP↗

arXiv2021

Asymptotic optimality of the triangular lattice for a class of optimal location problems

We prove an asymptotic crystallization result in two dimensions for a class of nonlocal particle systems. To be precise, we consider the best approximation with respect to the 2-Wasserstein metric of a given absolutely continuous probability measure $f \mathrm{d}x$ by a discrete probability measure $\sum_i m_i δ_{z_i}$, subject to a constraint on the particle sizes $m_i$. The locations $z_i$ of the particles, their sizes $m_i$, and the number of particles are all unknowns of the problem. We study a one-parameter family of constraints. This is an example of an optimal location problem (or an optimal sampling or quantization problem) and it has applications in economics, signal compression, and numerical integration. We establish the asymptotic minimum value of the (rescaled) approximation error as the number of particles goes to infinity. In particular, we show that for the constrained best approximation of the Lebesgue measure by a discrete measure, the discrete measure whose support is a triangular lattice is asymptotically optimal. In addition, we prove an analogous result for a problem where the constraint is replaced by a penalization. These results can also be viewed as the asymptotic optimality of the hexagonal tiling for an optimal partitioning problem. They generalise the crystallization result of Bourne, Peletier and Theil (Communications in Mathematical Physics, 2014) from a single particle system to a class of particle systems, and prove a case of a conjecture by Bouchitté, Jimenez and Mahadevan (Journal de Mathématiques Pures et Appliquées, 2011). Finally, we prove a crystallization result which states that optimal configurations with energy close to that of a triangular lattice are geometrically close to a triangular lattice.

math.OC↗

arXiv2015

Energy Bounds for a Compressed Elastic Film on a Substrate

We study pattern formation in a compressed elastic film which delaminates from a substrate. Our key tool is the determination of rigorous upper and lower bounds on the minimum value of a suitable energy functional. The energy consists of two parts, describing the two main physical effects. The first part represents the elastic energy of the film, which is approximated using the von Kármán plate theory. The second part represents the fracture or delamination energy, which is approximated using the Griffith model of fracture. A simpler model containing the first term alone was previously studied with similar methods by several authors, assuming that the delaminated region is fixed. We include the fracture term, transforming the elastic minimization into a free-boundary problem, and opening the way for patterns which result from the interplay of elasticity and delamination. After rescaling, the energy depends on only two parameters: the rescaled film thickness, $σ$, and a measure of the bonding strength between the film and substrate, $γ$. We prove upper bounds on the minimum energy of the form $σ^a γ^b$ and find that there are four different parameter regimes corresponding to different values of $a$ and $b$ and to different folding patterns of the film. In some cases the upper bounds are attained by self-similar folding patterns as observed in experiments. Moreover, for two of the four parameter regimes we prove matching, optimal lower bounds.

math.AP↗