Contact    Resume/CV    Research    Teaching

I have a wide variety of research interests, including but not limited to image processing, computational physics, biological swarming, pattern formation in particle self-assembly, machine learning and the analysis of nonlinear partial differential equations.  Abstracts from recent papers along with links to the full articles are posted below.

J. H. von Brecht and A. L. Bertozzi.  Well-Posedness Theory for Aggregation Sheets.  Commun. Math. Phys, 2012.    PDF

In this paper, we consider distribution solutions to the aggregation equation $\rho_{t} + \mathrm{div}(\rho \mathbf{u} ) = 0, \; \mathbf{u} = -\nabla V * \rho$ in $\Rd$ where the density $\rho$ concentrates on a co-dimension one manifold. We show that an evolution equation for the manifold itself completely determines the dynamics of such solutions. We refer to such solutions aggregation sheets. When the equation for the sheet is linearly well-posed, we show that the fully non-linear evolution is also well-posed locally in time for the class of bi-Lipschitz surfaces.  Moreover, we show that if the initial sheet is $C^{1}$ then the solution itself remains $C^{1}$ as long as it remains Lipschitz. Lastly, we provide conditions on the kernel $g(s) = -\frac{\rd V}{\rd s}$ that guarantee the solution remains a bi-Lipschitz surface globally in time, and construct explicit solutions that either collapse or blow up in finite time when these conditions fail.

J. H. von Brecht and D. Uminsky.  Linearized Inverse Statistical Mechanics.  J. Nonlin. Sci, pages 1–25. 10.1007/s00332-012-9132-7. 2012.    PDF

The classical inverse statistical mechanics question involves inferring properties of pairwise interaction potentials from exhibited ground states. For patterns that concentrate near a sphere, the ground states can range from platonic solids for small numbers of particles to large systems of particles exhibiting very complex structures. In this setting, our previous work allows us to infer that the linear instabilities of the pairwise potential accurately characterize the resulting nonlinear ground states. Potentials with a small number of spherical harmonic instabilities may produce very complex patterns as a result. This leads naturally to the linearized inverse statistical mechanics question: given a finite set of unstable modes, can we construct a potential that possesses precisely these linear instabilities? If so, this would allow for the design of potentials with arbitrarily intricate spherical symmetries in the ground state. In this paper, we solve our linearized inverse problem in full, and present a wide variety of designed ground states.

J. H. von Brecht, D. Uminsky, T. Kolokolnikov and A.L. Bertozzi.  Predicting Pattern Formation in Particle Interactions.  Math. Models and Meth. in Appl. Sci., Suppl. 1, 2012.    PDF

Large systems of particles interacting pairwise in $d$-dimensions give rise to extraordinarily rich patterns. These patterns generally occur in two types. On one hand, the particles may concentrate on a co-dimension one manifold such as a sphere (in 3D) or a ring (in 2D). Localized, space-filling, co-dimension zero patterns can occur as well. In this paper, we utilize a dynamical systems approach to predict such behaviors in a given system of particles. More specifically, we develop a non-local linear stability analysis for particles uniformly distributed on a  $d-1$ sphere. Remarkably, the linear theory accurately characterizes the patterns in the ground states from the instabilities in the pairwise potential. This aspect of the theory then allows us to address the issue of inverse statistical mechanics in self-assembly: given a ground state exhibiting certain instabilities, we construct a potential that corresponds to such a pattern.

J. Bedrossian, J. H. von Brecht, S. Zhu, E. Sifakis and J. M. Teran.  A second order virtual node method for elliptic problems with interfaces and irregular domains.  J. Comp. Phys, 2010.    PDF

We present a second order accurate, geometrically flexible and easy to implement method for solving the variable coefficient Poisson equation with interfacial discontinuities or on irregular domains, handling both cases with the same approach. We discretize the equations using an embedded approach on a uniform Cartesian grid employing virtual nodes at interfaces and boundaries. A variational method is used to define numerical stencils near these special virtual nodes and a Lagrange multiplier approach is used to enforce jump conditions and Dirichlet boundary conditions. Our combination of these two aspects yields a symmetric positive definite discretization. In the general case, we obtain the standard 5-point stencil away from the interface. For the specific case of interface problems with continuous coefficients, we present a discontinuity removal technique that admits the use of the standard 5-point finite difference stencil everywhere in the domain. Numerical experiments indicate second order accuracy in $L^{\infty}$.

J. von Brecht, T. Kolokolnikov, A. L. Bertozzi and H. Sun.  Swarming on Random Graphs.  J. Stat. Phys, 2013.    PDF

X. Bresson, T. Laurent, D. Uminsky and J. H. von Brecht.  Convergence and Energy Landscape for Cheeger Cut Clustering. NIPS, 2012.    PDF

This paper provides both theoretical and algorithmic results for the $\ell_1$-relaxation of the Cheeger cut problem. The $\ell_2$-relaxation, known as spectral clustering, only loosely relates to the Cheeger cut; however, it is convex and leads to a simple optimization problem. The $\ell_1$-relaxation, in contrast, is non-convex but is provably equivalent to the original problem. The $\ell_1$-relaxation therefore trades convexity for exactness, yielding improved clustering results at the cost of a more challenging optimization. The first challenge is understanding convergence of algorithms. This paper provides the first complete proof of convergence for algorithms that minimize the $\ell_1$-relaxation. The second challenge entails comprehending the $\ell_1$-energy landscape, i.e. the set of possible points to which an algorithm might converge. We show that $\ell_1$-algorithms can get trapped in local minima that are not globally optimal and we provide a classification theorem to interpret these local minima. This classification gives meaning to these suboptimal solutions and helps to explain, in terms of graph structure, when the $\ell_1$-relaxation provides the solution of the original Cheeger cut problem.