Projective and Coarse Projective Integration for Problems with Continuous Symmetries

Temporal integration of equations possessing continuous symmetries (e.g. systems with translational invariance associated with traveling solutions and scale invariance associated with self-similar solutions) in a ``co-evolving'' frame (i.e. a frame which is co-traveling, co-collapsing or co-exploding with the evolving solution) leads to improved accuracy because of the smaller time derivative in the new spatial frame. The slower time behavior permits the use of {\it projective} and {\it coarse projective} integration with longer projective steps in the computation of the time evolution of partial differential equations and multiscale systems, respectively. These methods are also demonstrated to be effective for systems which only approximately or asymptotically possess continuous symmetries. The ideas of projective integration in a co-evolving frame are illustrated on the one-dimensional, translationally invariant Nagumo partial differential equation (PDE). A corresponding kinetic Monte Carlo model, motivated from the Nagumo kinetics, is used to illustrate the coarse-grained method. A simple, one-dimensional diffusion problem is used to illustrate the scale invariant case. The efficiency of projective integration in the co-evolving frame for both the macroscopic diffusion PDE and for a random-walker particle based model is again demonstrated

Faster is More Different: Mean-Field Dynamics of Innovation Diffusion

Based on a recent model of paradigm shifts by Bornholdt et al., we studied mean-field opinion dynamics in an infinite population where an infinite number of ideas compete simultaneously with their values publicly known. We found that a highly innovative society is not characterized by heavy concentration in highly valued ideas: Rather, ideas are more broadly distributed in a more innovative society with faster progress, provided that the rate of adoption is constant, which suggests a positive correlation between innovation and technological disparity. Furthermore, the distribution is generally skewed in such a way that the fraction of innovators is substantially smaller than has been believed in conventional innovation-diffusion theory based on normality. Thus, the typical adoption pattern is predicted to be asymmetric with slow saturation in the ideal situation, which is compared with empirical data sets.Comment: 11 pages, 4 figure

Equation-free modeling of evolving diseases: Coarse-grained computations with individual-based models

We demonstrate how direct simulation of stochastic, individual-based models can be combined with continuum numerical analysis techniques to study the dynamics of evolving diseases. % Sidestepping the necessity of obtaining explicit population-level models, the approach analyzes the (unavailable in closed form) `coarse' macroscopic equations, estimating the necessary quantities through appropriately initialized, short `bursts' of individual-based dynamic simulation. % We illustrate this approach by analyzing a stochastic and discrete model for the evolution of disease agents caused by point mutations within individual hosts. % Building up from classical SIR and SIRS models, our example uses a one-dimensional lattice for variant space, and assumes a finite number of individuals. % Macroscopic computational tasks enabled through this approach include stationary state computation, coarse projective integration, parametric continuation and stability analysis.Comment: 16 pages, 8 figure

A Computationally Efficient Moment-Preserving Monte Carlo Electron Transport Method with Implementation in Geant4

The subject of this dissertation is a moment-preserving Monte Carlo electron transport method that is more efficient than analog or detailed Monte Carlo simulations, yet provides accuracy that is statistically indistinguishable from the detailed simulation. Moreover, the Moment-Preserving (MP) method is formulated such that it is distinctly different than Condensed History (CH) methods making the MP method free of the limitations inherent to CH and proving a viable alternative for transporting electrons. Analog, or detailed, Monte Carlo simulations of charged particle transport is computationally intensive; thus, it is impractical for routine calculations. The computational cost of analog Monte Carlo is directly attributed to the underlying charged particle physics characterized by extremely short mean free paths (mfp) and highly peaked differential cross sections (DCS). As a result, a variety of efficient, although approximate solution methods were developed over the past 60 years. The most prolific method is referred to as the Condensed History method. However, CH is widely known to suffer from inconsistencies between the underlying theory and the application of the method to real, physical problems. Therefore, it is of interest to develop an alternative method that is both efficient and accurate, but also a completely different approach to solving the charged particle transport equation that is free of the limitations inherent to CH. This approach arose from the development of a variety of reduced order physics (ROP) methods that utilize approximate representations of the collision operators. The purpose of this dissertation is the theoretical development and numerical demonstration of an alternative to CH referred to as the Moment-Preserving method. The MP method poses a transport equation with reduced order physics models characterized by less-peaked DCS with longer mfps. Utilizing pre-existing single-scatter algorithms for transporting particles, a solution to the aforementioned transport equation is obtained efficiently with analog level accuracy. The process of constructing ROP models and their properties are presented in detail. A wide variety of theoretical and applied charged particle transport problems are studied including: calculation of angular distributions and energy spectra, longitudinal and lateral distributions, energy deposition in one and two dimensions, a validation of the method for energy deposition and charge deposition calculations, and response function calculations for full three-dimensional detailed detector geometries. It is shown that the accuracy of the MP method is systematically controllable through refinement of the ROP models. In many cases, efficiency gains of two to three orders of magnitude over analog Monte Carlo are demonstrated, while maintaining analog level accuracy. That is, solutions generated sufficient ROP DCS models are statistically indistinguishable from the analog solution. To maintain analog level accuracy under strict problem conditions, small efficiency gains are realized. However, loss of efficiency under these conditions is true of all approximate methods, but the MP method remains accurate where other methods may fail. That is not to say the MP method does not suffer from limitations because the MP method will result in discrete artifacts when the problems conditions are strict. However, where limitations of the method arise, they are overcome through systematic refinement of the ROP DCS models required by the method. In addition to accuracy and efficiency results, it is shown that the MP method does not require a boundary crossing or pathlength correction algorithm, which is in great contrast to the CH method. Finally, implementation and maintenance of the MP method was found to be straightforward and requires significantly less effort than CH when measured by the number of lines of code required for each method. In particular, as compared with the class II CH method utilized in the Geant4 standard electromagnetic physics list. Ultimately, the MP is shown to be accurate, efficient, versatile, and simple to implement and maintain

Low-order modeling of micro-flier impact with thin stationary energetic targets

The impact of high-speed (500-1500 m/s), laser driven micro-fliers with thin energetic targets (10-100 &# 956;m) is being examined to characterize impact-induced heating and combustion of these materials. Aluminum fliers are propelled by a laser into a thin metallic target plate having a layer of energetic material deposited on its backside. Mass spectrometry is performed in vacuo on the energetic side to interrogate the shock-induced chemistry of the energetic material. It is important to ignite and possibly detonate the energetic material without perforation of the target plate avoiding contamination of the vacuum chamber. To guide development of these experiments, a low-order (zero-dimensional) model is formulated to estimate ballistic performance for large dimensional parameter spaces in a computationally inexpensive manner. The imaging of post-impact target coupons gives insight into deformation and failure modes of the target plate. The model accounts for both the early-time system response with 1-D shock relations and the late-time response with quasi-static strength of flat plates. The model is validated against impact data for larger scale flier-target configurations, and gives predictions for micro-scale configurations. The post-impact target plates show that the system behavior is stochastic in nature. Thus, a method for propagating input uncertainty is presented to estimate the uncertainty in model output variables, and a sensitivity study is performed to highlight dependence of the system response on input parameters. Model output is most sensitive to the ratio of flier width to diameter. Predictions are performed for energetic materials including HMX (C4H8N8O8), TNT (C7H5N3O6), and PETN (C5H8N4O12) over the initial flier velocity - flier thickness parametric plane for given target thicknesses to produce ballistic initiation maps to identify configurations for which initiation of energetic material may occur without perforation of the target plate. Because of the high critical shock initiation energy of HMX (150 J/cm2) and TNT (77 J/cm2), it is difficult to identify micro configurations that result in initiation. However, such configurations were found for PETN which has a lower critical shock energy (5.03 J/cm2). The area of the region for initiation increases with increasing target thickness for these configurations