Dejan Slepčev

Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations

In this paper we provide a well-posedness theory for weak measure solutions of the Cauchy problem for a family of nonlocal interaction equations. These equations are continuum models for interacting particle systems with attractive/repulsive pairwise interaction potentials. The main phenomenon of interest is that, even with smooth initial data, the solutions can concentrate mass in finite time. We develop an existence theory that enables one to go beyond the blow-up time in classical norms and allows for solutions to form atomic parts of the measure in finite time. The weak measure solutions are shown to be unique and exist globally in time. Moreover, in the case of sufficiently attractive potentials, we show the finite-time total collapse of the solution onto a single point for compactly supported initial measures. Our approach is based on the theory of gradient flows in the space of probability measures endowed with the Wasserstein metric. In addition to classical tools, we exploit the stability of the flow with respect to the transportation distance to greatly simplify many problems by reducing them to questions about particle approximations.

A Linear Optimal Transportation Framework for Quantifying and Visualizing Variations in Sets of Images

Transportation-based metrics for comparing images have long been applied to analyze images, especially where one can interpret the pixel intensities (or derived quantities) as a distribution of ‘mass’ that can be transported without strict geometric constraints. Here we describe a new transportation-based framework for analyzing sets of images. More specifically, we describe a new transportation-related distance between pairs of images, which we denote as linear optimal transportation (LOT). The LOT can be used directly on pixel intensities, and is based on a linearized version of the Kantorovich-Wasserstein metric (an optimal transportation distance, as is the earth mover’s distance). The new framework is especially well suited for computing all pairwise distances for a large database of images efficiently, and thus it can be used for pattern recognition in sets of images. In addition, the new LOT framework also allows for an isometric linear embedding, greatly facilitating the ability to visualize discriminant information in different classes of images. We demonstrate the application of the framework to several tasks such as discriminating nuclear chromatin patterns in cancer cells, decoding differences in facial expressions, galaxy morphologies, as well as sub cellular protein distributions.

An Optimal Transportation Approach for Nuclear Structure-Based Pathology

Nuclear morphology and structure as visualized from histopathology microscopy images can yield important diagnostic clues in some benign and malignant tissue lesions. Precise quantitative information about nuclear structure and morphology, however, is currently not available for many diagnostic challenges. This is due, in part, to the lack of methods to quantify these differences from image data. We describe a method to characterize and contrast the distribution of nuclear structure in different tissue classes (normal, benign, cancer, etc.). The approach is based on quantifying chromatin morphology in different groups of cells using the optimal transportation (Kantorovich-Wasserstein) metric in combination with the Fisher discriminant analysis and multidimensional scaling techniques. We show that the optimal transportation metric is able to measure relevant biological information as it enables automatic determination of the class (e.g., normal versus cancer) of a set of nuclei. We show that the classification accuracies obtained using this metric are, on average, as good or better than those obtained utilizing a set of previously described numerical features. We apply our methods to two diagnostic challenges for surgical pathology: one in the liver and one in the thyroid. Results automatically computed using this technique show potentially biologically relevant differences in nuclear structure in liver and thyroid cancers.

Coarsening Rates for a Droplet Model: Rigorous Upper Bounds

Certain liquids on solid substrates form a configuration of droplets connected by a precursor layer. This configuration coarsens: The average droplet size grows while the number of droplets decreases and the characteristic distance between them increases. We study this type of coarsening behavior in a model given by an evolution equation for the film height on an n-dimensional substrate. Heuristic arguments based on the asymptotic analysis of Glasner and Witelski [Phys. Rev. E, 67 (2003), p. 016302, Phys. D., 209 (2005), pp. 80-104] and numerical simulations suggest a statistically self-similar behavior characterized by a single exponent which determines the coarsening rate. In this paper, we establish rigorously an upper bound on the coarsening rate in a time-averaged sense. We use the fact that the evolution is a gradient flow, i.e., a steepest descent in an energy landscape. Coarse information on the geometry of the energy landscape serves to obtain coarse information on the dynamics. This robust metho...

Continuum Limit of Total Variation on Point Clouds

We consider point clouds obtained as random samples of a measure on a Euclidean domain. A graph representing the point cloud is obtained by assigning weights to edges based on the distance between the points they connect. Our goal is to develop mathematical tools needed to study the consistency, as the number of available data points increases, of graph-based machine learning algorithms for tasks such as clustering. In particular, we study when the cut capacity, and more generally total variation, on these graphs is a good approximation of the perimeter (total variation) in the continuum setting. We address this question in the setting of Γ-convergence. We obtain almost optimal conditions on the scaling, as the number of points increases, of the size of the neighborhood over which the points are connected by an edge for the Γ-convergence to hold. Taking of the limit is enabled by a transportation based metric which allows us to suitably compare functionals defined on different point clouds.We consider point clouds obtained as random samples of a measure on a Euclidean domain. A graph representing the point cloud is obtained by assigning weights to edges based on the distance between the points they connect. Our goal is to develop mathematical tools needed to study the consistency, as the number of available data points increases, of graph-based machine learning algorithms for tasks such as clustering. In particular, we study when the cut capacity, and more generally total variation, on these graphs is a good approximation of the perimeter (total variation) in the continuum setting. We address this question in the setting of Γ-convergence. We obtain almost optimal conditions on the scaling, as the number of points increases, of the size of the neighborhood over which the points are connected by an edge for the Γ-convergence to hold. Taking of the limit is enabled by a transportation based metric which allows us to suitably compare functionals defined on different point clouds.

Ostwald ripening of droplets: The role of migration

A configuration of near-equilibrium liquid droplets sitting on a precursor film which wets the entire substrate can coarsen in time by two different mechanisms: collapse or collision of droplets. The collapse mechanism, i.e., a larger droplet grows at the expense of a smaller one by mass exchange through the precursor film, is also known as Ostwald ripening. As was shown by K. B. Glasner and T. P. Witelski (‘Collision versus collapse of droplets in coarsening of dewetting thin films’, Phys. D 209(1–4), 2005, 80–104) in case of a one-dimensional substrate, the migration of droplets may interfere with Ostwald ripening: The configuration can coarsen by collision rather than by collapse. We study the role of migration in case of a two-dimensional substrate for a whole range of mobilities. We characterize the velocity of a single droplet immersed into an environment with constant flux field far away. This allows us to describe the dynamics of a droplet configuration on a two-dimensional substrate by a system of ODEs. In particular, we find by heuristic arguments that collision can be a relevant coarsening mechanism.

Existence of Ground States of Nonlocal-Interaction Energies

We investigate which nonlocal-interaction energies have a ground state (global minimizer). We consider this question over the space of probability measures and establish a sharp condition for the existence of ground states. We show that this condition is closely related to the notion of stability (i.e.

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