Thin film flow dynamics on fiber nets

Abstract

We analyze existence and qualitative behavior of non-negative weak solutions for fourth order degenerate parabolic equations on graph domains with Kirchhoff's boundary conditions at the inner nodes and Neumann boundary conditions at the boundary nodes. The problem is originated from industrial constructions of spray coated meshes which are used in water collection and in oil-water separation processes. For a certain range of parameter values we prove convergence toward a constant steady state that corresponds to the uniform distribution of coating on a fiber net.

A Continuous-Time Analysis of Distributed Stochastic Gradient

“…A significant difficulty in a continuous-time analysis of SGD is formulating an accurate stochastic differential equation (SDE) model. Recent works have proved rigorously (Feng, Li, & Liu, 2018;Hu, Li, Li, & Liu, 2017;Li, Tai, & E, 2018) that the sequence of values x(kη) generated by the SDE…”

“…The assumption of state-independence can be justified in several ways. Theoretical analyses have demonstrated that the specific form of positive semi-definite B does not affect the O(η) weak accuracy of the approximating SDE (2) for SGD (Feng et al, 2018;Hu et al, 2017;Li et al, 2018), though it does affect the constant 9 . For relevance to general nonconvex optimization, we can assume that all agents have arrived sufficiently close to a minimum of the loss function that it can be approximately represented as a quadratic, and that the noise covariance is approximately constant (Mandt et al, 2016(Mandt et al, , 2017.…”