Andrea L. Bertozzi

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Image inpainting involves filling in part of an image or video using information from the surrounding area. Applications include the restoration of damaged photographs and movies and the removal of selected objects. In this paper, we introduce a class of automated methods for digital inpainting. The approach uses ideas from classical fluid dynamics to(More)
Understanding collective properties of driven particle systems is significant for naturally occurring aggregates and because the knowledge gained can be used as building blocks for the design of artificial ones. We model self-propelling biological or artificial individuals interacting through pairwise attractive and repulsive forces. For the first time, we(More)
Abstract Motivated by empirical observations of spatio-temporal clusters of crime across a wide variety of urban settings, we present a model to study the emergence, dynamics, and steady state properties of crime hotspots. We focus on a two-dimensional lattice model for residential burglary, where each site is characterized by a dynamic attractiveness(More)
We consider the fourth order degenerate diiusion equation h t = ?r (f(h)rh) in one space dimension. This equation, derived from a `lubrication approximation', models surface tension dominated motion of thin viscous lms and spreading droplets 14]. The equation with f(h) = jhj also models a thin neck of uid in the Hele-Shaw cell 9, 10, 22]. In such problems(More)
We consider the multidimensional aggregation equation ut − ∇· (u∇K ∗ u) = 0 in which the radially symmetric attractive interaction kernel has a mild singularity at the origin (Lipschitz or better). In the case of bounded initial data, finite time singularity has been proved for kernels with a Lipschitz point at the origin (Bertozzi and Laurent 2007 Commun.(More)
This paper is a republication of an MMS paper [A. L. Bertozzi and A. Flenner, Multiscale Model. Simul., 10 (2012), pp. 1090–1118] describing a new class of algorithms for classification of high dimensional data. These methods combine ideas from spectral methods on graphs with nonlinear edge/region detection methods traditionally used in the PDE-based(More)
Equations of the type ht + (h2 − h)x = − (hhxxx)x arise in the context of thin liquid films driven by the competing effects of a thermally induced surface tension gradient and gravity. In this paper, we focus on the interaction between the fourth order regularization and the nonconvex flux. Jump initial data, from a moderately thick film to a thin precurser(More)
We construct a continuum model for the motion of biological organisms experiencing social interactions and study its pattern-forming behavior. The model takes the form of a conservation law in two spatial dimensions. The social interactions are modeled in the velocity term, which is nonlocal in the population density and includes a parameter that controls(More)
We extend a recently introduced method for numerically solving partial differential equations on implicit surfaces (Bertalmı́o, Cheng, Osher, and Sapiro 2001) to fourth order PDEs including the CahnHilliard equation and a lubrication model for curved surfaces. By representing a surface in N as the level set of a smooth function, φ we compute the PDE using(More)