Braxton Osting

Mean curvature, threshold dynamics, and phase field theory on finite graphs

In the continuum, close connections exist between mean curvature flow, the Allen-Cahn (AC) partial differential equation, and the Merriman-Bence-Osher (MBO) threshold dynamics scheme. Graph analogues of these processes have recently seen a rise in popularity as relaxations of NP-complete combinatorial problems, which demands deeper theoretical underpinnings of the graph processes. The aim of this paper is to introduce these graph processes in the light of their continuum counterparts, provide some background, prove the first results connecting them, illustrate these processes with examples and identify open questions for future study.We derive a graph curvature from the graph cut function, the natural graph counterpart of total variation (perimeter). This derivation and the resulting curvature definition differ from those in earlier literature, where the continuum mean curvature is simply discretized, and bears many similarities to the continuum nonlocal curvature or nonlocal means formulation. This new graph curvature is not only relevant for graph MBO dynamics, but also appears in the variational formulation of a discrete time graph mean curvature flow.We prove estimates showing that the dynamics are trivial for both MBO and AC evolutions if the parameters (the time-step and diffuse interface scale, respectively) are sufficiently small (a phenomenon known as “freezing” or “pinning”) and also that the dynamics for MBO are nontrivial if the time step is large enough. These bounds are in terms of graph quantities such as the spectrum of the graph Laplacian and the graph curvature. Adapting a Lyapunov functional for the continuum MBO scheme to graphs, we prove that the graph MBO scheme converges to a stationary state in a finite number of iterations. Variations on this scheme have recently become popular in the literature as ways to minimize (continuum) nonlocal total variation.

Minimal Dirichlet Energy Partitions for Graphs

Motivated by a geometric problem, we introduce a new nonconvex graph partitioning objective where the optimality criterion is given by the sum of the Dirichlet eigenvalues of the partition components. A relaxed formulation is identified and a novel rearrangement algorithm is proposed, which we show is strictly decreasing and converges in a finite number of iterations to a local minimum of the relaxed objective function. Our method is applied to several clustering problems on graphs constructed from synthetic data, MNIST handwritten digits, and manifold discretizations. The model has a semisupervised extension and provides a natural representative for the clusters as well.

Optimization of spectral functions of Dirichlet-Laplacian eigenvalues

We consider the shape optimization of spectral functions of Dirichlet-Laplacian eigenvalues over the set of star-shaped, symmetric, bounded planar regions with smooth boundary. The regions are represented using Fourier-cosine coefficients and the optimization problem is solved numerically using a quasi-Newton method. The method is applied to maximizing two particular nonsmooth spectral functions: the ratio of the nth to first eigenvalues and the ratio of the nth eigenvalue gap to first eigenvalue, both of which are generalizations of the Payne-Polya-Weinberger ratio. The optimal values and attaining regions for n=<13 are presented and interpreted as a study of the range of the Dirichlet-Laplacian eigenvalues. For both spectral functions and each n, the optimal attaining region has multiplicity two nth eigenvalue.

Modeling of kinematic diffraction from a thin silicon film illuminated by a coherent, focused X-ray nanobeam

A rigorous model of a diffraction experiment utilizing a coherent, monochromatic, X-ray beam, focused by a Fresnel zone plate onto a thin, perfect, single-crystal layer is presented. In this model, first the coherent wave emanating from an ideal zone plate equipped with a direct-beam stop and order-sorting aperture is computed. Then, diffraction of the focused wavefront by a thin silicon film positioned at the primary focal spot is calculated. This diffracted wavefront is propagated to the detector position, and the intensity distribution at the detector plane is extracted. The predictions of this model agree quite well with experimental data measured at the Center for Nanoscale Materials nanoprobe instrument at Sector 26 of the Advanced Photon Source.

Long-Lived Scattering Resonances and Bragg Structures

We consider a system governed by the wave equation with index of refraction

Diffraction on the Two-Dimensional Square Lattice

n({\bf x})

Kirchhoff's Laws as a Finite Volume Method for the Planar Maxwell Equations

, taken to be variable within a bounded region

Maximization of Laplace−Beltrami eigenvalues on closed Riemannian surfaces

\Omega\subset \mathbb R^d

Analysis of Crowdsourced Sampling Strategies for HodgeRank with Sparse Random Graphs

and constant in

Emergence of Periodic Structure from Maximizing the Lifetime of a Bound State Coupled to Radiation

\mathbb R^d \setminus \Omega