Tim Brereton

Fitting Laguerre tessellation approximations to tomographic image data

The analysis of polycrystalline materials benefits greatly from accurate quantitative descriptions of their grain structures. Laguerre tessellations approximate such grain structures very well. However, it is a quite challenging problem to fit a Laguerre tessellation to tomographic data, as a high-dimensional optimization problem with many local minima must be solved. In this paper, we formulate a version of this optimization problem that can be solved quickly using the cross-entropy method, a robust stochastic optimization technique that can avoid becoming trapped in local minima. We demonstrate the effectiveness of our approach by applying it to both artificially generated and experimentally produced tomographic data.

Inverting Laguerre tessellations

A Laguerre tessellation is a generalization of a Voronoi tessellation where the proximity between points is measured via a power distance rather than the Euclidean distance. Laguerre tessellations have found significant applications in materials science, providing improved modeling of (poly)crystalline microstructures and grain growth. There exist efficient algorithms to construct Laguerre tessellations from given sets of weighted generator points, similar to methods used for Voronoi tessellations. The purpose of this paper is to provide theory and methodology for the inverse construction; that is, to recover the weighted generator points from a given Laguerre tessellation. We show that, unlike the Voronoi case, the inverse problem is in general non-unique: different weighted generator points can create the same tessellation. To recover pertinent generator points, we formulate the inversion problem as a multimodal optimization problem and apply the cross-entropy method to solve it.

A General Framework for Consistent Estimation of Charge Transport Properties via Random Walks in Random Environments

A general framework is proposed for the study of the charge transport properties of materials via random walks in random environments (RWRE). The material of interest is modeled by a random environment, and the charge carrier is modeled by a random walker. The framework combines a model for the fast generation of random environments that realistically mimic materials morphology with an algorithm for efficient estimation of key properties of the resulting random walk. The model of the environment makes use of tools from spatial statistics and the theory of random geometric graphs. More precisely, the disordered medium is represented by a random spatial graph with directed edge weights, where the edge weights represent the transition rates of a Markov jump process (MJP) modeling the motion of the random walker. This MJP is a multiscale stochastic process. In the long term, it explores all vertices of the random graph model. In the short term, however, it becomes trapped in small subsets of the state space a...

Stochastic modeling and predictive simulations for the microstructure of organic semiconductor films processed with different spin coating velocities

A parametric stochastic model of the morphology of thin polymer:fullerene films is developed. This model uses a number of tools from stochastic geometry and spatial statistics. The fullerene-rich phase is represented by random closed sets and the polymer-rich phase is given by their complement. The model has three stages. First, a point pattern is used to model the locations of fullerene-rich domains. Second, domains are formed at these points. Third, the domains are rearranged to ensure a realistic configuration. The model is fitted to polymer:fullerene films produced using seven different spin coating velocities and validated using a variety of morphological characteristics. The model is then used to simulate morphologies corresponding to spin velocities for which no empirical data exists. The viability of this approach is demonstrated using cross-validation.

Efficient simulation of charge transport in deep-trap media

This paper introduces a new approach to Monte Carlo estimation of the velocity of charge carriers drift-diffusing in a random medium. The random medium is modeled by a 1-dimensional lattice and the position of the charge carrier is modeled by a Markov jump process, whose state space is the set of lattice points. The transition rates of the Markov jump process are determined by the underlying energy landscape of the random medium. This energy landscape is modeled by a Gaussian process and contains regions of relatively low energy, in which charge carriers quickly become stuck. As a result, the state space is not adequately explored by the standard algorithms and the velocity of the charge carrier is poorly estimated. In addition, the conventional Monte Carlo estimators have very high variances. Our approach aims to reduce the number of simulation steps that are spent in the low energy problem regions. We do this by identifying the problem regions via a stochastic watershed algorithm. We then use a coarsened state space model, where the problem regions are treated as single states. In this way, we are able to simulate a semi-Markov process on the coarsened state space. This results in estimators that are unbiased and have considerably lower variance than the crude Monte Carlo alternatives.

Fitting mixture importance sampling distributions via improved cross-entropy

In some rare-event settings, exponentially twisted distributions perform very badly. One solution to this problem is to use mixture distributions. However, it is difficult to select a good mixture distribution for importance sampling. We here introduce a simple adaptive method for choosing good mixture importance sampling distributions.

3D reconstruction of grains in polycrystalline materials using a tessellation model with curved grain boundaries

A compact and tractable representation of the grain structure of a material is an extremely valuable tool when carrying out an empirical analysis of the material’s microstructure. Tessellations have proven to be very good choices for such representations. Most widely used tessellation models have convex cells with planar boundaries. Recently, however, a new tessellation model — called the generalised balanced power diagram (GBPD) — has been developed that is very flexible and can incorporate features such as curved boundaries and non-convexity of cells. In order to use a GBPD to describe the grain structure observed in empirical image data, the parameters of the model must be chosen appropriately. This typically involves solving a difficult optimisation problem. In this paper, we describe a method for fitting GBPDs to tomographic image data. This method uses simulated annealing to solve a suitably chosen optimisation problem. We then apply this method to both artificial data and experimental 3D electron backscatter diffraction (3D EBSD) data obtained in order to study the properties of fine-grained materials with superplastic behaviour. The 3D EBSD data required new alignment and segmentation procedures, which we also briefly describe. Our numerical experiments demonstrate the effectiveness of the simulated annealing approach (compared to heuristic fitting methods) and show that GBPDs are able to describe the structures of polycrystalline materials very well.

A critical exponent for shortest-path scaling in continuum percolation

We carry out Monte Carlo experiments to study the scaling behavior of shortest path lengths in continuum percolation. These studies suggest that the critical exponent governing this scaling is the same for both continuum and lattice percolation. We use splitting, a technique that has not yet been fully exploited in the physics literature, to increase the speed of our simulations. This technique can also be applied to other models where clusters are grown sequentially.

Percolation and convergence properties of graphs related to minimal spanning forests

Lyons, Peres and Schramm have shown that minimal spanning forests on randomly weighted lattices exhibit a critical geometry in the sense that adding or deleting only a small number of edges results in a radical change of percolation properties. We show that these results can be extended to a Euclidean setting by considering families of stationary superand subgraphs that approximate the Euclidean minimal spanning forest arbitrarily closely, but whose percolation properties differ decisively from those of the minimal spanning forest. Since these families can be seen as generalizations of the relative neighborhood graph and the nearest-neighbor graph, respectively, our results also provide a new perspective on known percolation results from literature. We argue that the rates at which the approximating families converge to the minimal spanning forest are closely related to certain geometric characteristics of clusters in critical continuum percolation, and we show that convergence occurs at a polynomial rate.

Stochastic Models of Charge Transport in Disordered Media

Charge transport in disordered materials, such as organic and amorphous inorganic semiconductors, is often modeled in a stochastic framework. The microstructure of the disordered material is interpreted as a realization of a stochastic model and, given a realization of this model, the charge transport process itself is treated as a random process. In this paper, we give an introduction to this combined stochastic modeling approach. We first describe the basic physics underlying charge transport in disordered materials. Then, we discuss stochastic models of the material and the charge transport process. In organic semiconductors, charge transport is modeled either by a continuous-time random walk in a random environment or an interacting particle system in a random environment. In amorphous inorganic semiconductors, charge transport is modeled by a continuous-time random walk in a deterministic environment. In the organic semiconductor case, the resulting stochastic models need to be solved using numerical methods. As such, we discuss Monte Carlo methods for estimating charge transport properties. In particular, we discuss a recently developed method, Aggregate Monte Carlo, which can be used to significantly speed up Monte Carlo simulations. Finally, we discuss the problem of modeling recombination in organic semiconductors.