178 research outputs found
Focusing Revisited: an MN-dynamics Approach
The nonlinear Schr{\"o}dinger (NLS) equation is a ubiquitous example of an
envelope wave equation for conservative, dispersive systems. We revisit here
the problem of self-similar focusing of waves in the case of the focusing NLS
equation through the prism of a dynamic renormalization technique (MN dynamics)
that factors out self-similarity and yields a bifurcation view of the onset of
focusing. As a result, identifying the focusing self-similar solution becomes a
steady state problem. The discretized steady states are subsequently obtained
and their linear stability is numerically examined. The calculations are
performed in the setting of variable index of refraction, in which the onset of
focusing appears as a supercritical bifurcation of a novel type of mixed
Hamiltonian-dissipative dynamical system (reminiscent, to some extent, of a
pitchfork bifurcation).Comment: 6 pages, 2 figure
Microscopic/stochastic timesteppers and coarse control: a kinetic Monte Carlo example
Coarse timesteppers provide a bridge between microscopic / stochastic system
descriptions and macroscopic tasks such as coarse stability/bifurcation
computations. Exploiting this computational enabling technology, we present a
framework for designing observers and controllers based on microscopic
simulations, that can be used for their coarse control. The proposed
methodology provides a bridge between traditional numerical analysis and
control theory on the one hand and microscopic simulation on the other
Coarse-graining the dynamics of network evolution: the rise and fall of a networked society
We explore a systematic approach to studying the dynamics of evolving
networks at a coarse-grained, system level. We emphasize the importance of
finding good observables (network properties) in terms of which coarse grained
models can be developed. We illustrate our approach through a particular social
network model: the "rise and fall" of a networked society [1]: we implement our
low-dimensional description computationally using the equation-free approach
and show how it can be used to (a) accelerate simulations and (b) extract
system-level stability/bifurcation information from the detailed dynamic model.
We discuss other system-level tasks that can be enabled through such a
computer-assisted coarse graining approach.Comment: 18 pages, 11 figure
Coarse Brownian Dynamics for Nematic Liquid Crystals: Bifurcation Diagrams via Stochastic Simulation
We demonstrate how time-integration of stochastic differential equations
(i.e. Brownian dynamics simulations) can be combined with continuum numerical
bifurcation analysis techniques to analyze the dynamics of liquid crystalline
polymers (LCPs). Sidestepping the necessity of obtaining explicit closures, the
approach analyzes the (unavailable in closed form) coarse macroscopic
equations, estimating the necessary quantities through appropriately
initialized, short bursts of Brownian dynamics simulation. Through this
approach, both stable and unstable branches of the equilibrium bifurcation
diagram are obtained for the Doi model of LCPs and their coarse stability is
estimated. Additional macroscopic computational tasks enabled through this
approach, such as coarse projective integration and coarse stabilizing
controller design, are also demonstrated
Data-based analysis, modelling and forecasting of the COVID-19 outbreak
Since the first suspected case of coronavirus disease-2019 (COVID-19) on December 1st, 2019, in Wuhan, Hubei Province, China, a total of 40,235 confirmed cases and 909 deaths have been reported in China up to February 10, 2020, evoking fear locally and internationally. Here, based on the publicly available epidemiological data for Hubei, China from January 11 to February 10, 2020, we provide estimates of the main epidemiological parameters. In particular, we provide an estimation of the case fatality and case recovery ratios, along with their 90% confidence intervals as the outbreak evolves. On the basis of a Susceptible-Infectious-Recovered-Dead (SIDR) model, we provide estimations of the basic reproduction number (R0), and the per day infection mortality and recovery rates. By calibrating the parameters of the SIRD model to the reported data, we also attempt to forecast the evolution of the outbreak at the epicenter three weeks ahead, i.e. until February 29. As the number of infected individuals, especially of those with asymptomatic or mild courses, is suspected to be much higher than the official numbers, which can be considered only as a subset of the actual numbers of infected and recovered cases in the total population, we have repeated the calculations under a second scenario that considers twenty times the number of confirmed infected cases and forty times the number of recovered, leaving the number of deaths unchanged. Based on the reported data, the expected value of R0 as computed considering the period from the 11th of January until the 18th of January, using the official counts of confirmed cases was found to be ~4.6, while the one computed under the second scenario was found to be ~3.2. Thus, based on the SIRD simulations, the estimated average value of R0 was found to be ~2.6 based on confirmed cases and ~2 based on the second scenario. Our forecasting flashes a note of caution for the presently unfolding outbreak in China. Based on the official counts for confirmed cases, the simulations suggest that the cumulative number of infected could reach 180,000 (with a lower bound of 45,000) by February 29. Regarding the number of deaths, simulations forecast that on the basis of the up to the 10th of February reported data, the death toll might exceed 2,700 (as a lower bound) by February 29. Our analysis further reveals a significant decline of the case fatality ratio from January 26 to which various factors may have contributed, such as the severe control measures taken in Hubei, China (e.g. quarantine and hospitalization of infected individuals), but mainly because of the fact that the actual cumulative numbers of infected and recovered cases in the population most likely are much higher than the reported ones. Thus, in a scenario where we have taken twenty times the confirmed number of infected and forty times the confirmed number of recovered cases, the case fatality ratio is around ~0.15% in the total population. Importantly, based on this scenario, simulations suggest a slow down of the outbreak in Hubei at the end of February
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