arXiv subjects

Joe Klobusicky

Publications and source records attributed to Joe Klobusicky.

4 recordsLinked to original sources

arXiv2022

Renewal-scaled solutions of the Kolmogorov forward equation for residual times

Let $N(τ)$ be a renewal process for independent holding times $\{X_i\}_{k \ge 0}$ ,where $\{X_k\}_{k\ge 1}$ are identically distributed with density $p(x)$. If the associated residual time $R(τ)$ has a density $u(x,τ)$, its Kolmogorov forward equation is given by \begin{equation*} \partial_τu(x,τ)-\partial_x u(x,τ) = p(x)u(0,τ), \quad x,τ\in [0, \infty), \end{equation*} with an initial holding time density $u(x,0)=u_0(x)$. We derive a measure-valued solution formula for the density of residual times after an expected number of renewals occur. Solutions under this time scale are then shown to evolve continuously in the space of measures with the weak topology for a wide variety of holding times.

math.PR↗

arXiv2021

Two-dimensional grain boundary networks: stochastic particle models and kinetic limits

We study kinetic theories for isotropic, two-dimensional grain boundary networks which evolve by curvature flow. The number densities $f_s(x,t)$ for $s$-sided grains, $s =1,2,\ldots$, of area $x$ at time $t$, are modeled by kinetic equations of the form $\partial_t f_s + v_s \partial_x f_s =j_s$. The velocity $v_s$ is given by the Mullins-von Neumann rule and the flux $j_s$ is determined by the topological transitions caused by the vanishing of grains and their edges. The foundations of such kinetic models are examined through simpler particle models for the evolution of grain size, as well as purely topological models for the evolution of trivalent maps. These models are used to characterize the parameter space for the flux $j_s$. Several kinetic models in the literature, as well as a new kinetic model, are simulated and compared with direct numerical simulations of mean curvature flow on a network. Existence and uniqueness of mild solutions to the kinetic equations with continuous initial data is established.

math.AP↗

arXiv2020

Convergence of backpropagation with momentum for network architectures with skip connections

We study a class of deep neural networks with networks that form a directed acyclic graph (DAG). For backpropagation defined by gradient descent with adaptive momentum, we show weights converge for a large class of nonlinear activation functions. The proof generalizes the results of Wu et al. (2008) who showed convergence for a feed forward network with one hidden layer. For an example of the effectiveness of DAG architectures, we describe an example of compression through an autoencoder, and compare against sequential feed forward networks under several metrics.

cs.CV↗

arXiv2017

Concentration inequalities for a removal-driven thinning process

We prove exponential concentration estimates and a strong law of large numbers for a particle system that is the simplest representative of a general class of models for 2D grain boundary coarsening. The system consists of $n$ particles in $(0,\infty)$ that move at unit speed to the left. Each time a particle hits the boundary point $0$, it is removed from the system along with a second particle chosen uniformly from the particles in $(0,\infty)$. Under the assumption that the initial empirical measure of the particle system converges weakly to a measure with density $f_0(x) \in L^1_+(0,\infty)$, the empirical measure of the particle system at time $t$ is shown to converge to the measure with density $f(x,t)$, where $f$ is the unique solution to the kinetic equation with nonlinear boundary coupling $$\partial_t f (x,t) - \partial_x f(x,t) = -\frac{f(0,t)}{\int_0^\infty f(y,t)\, dy} f(x,t), \quad 0<x < \infty, $$ and initial condition $f(x,0)=f_0(x)$. The proof relies on a concentration inequality for an urn model studied by Pittel, and Maurey's concentration inequality for Lipschitz functions on the permutation group.

math.PR↗