Formulation and optimization of the energy-based blended quasicontinuum method

We formulate an energy-based atomistic-to-continuum coupling method based on blending the quasicontinuum method for the simulation of crystal defects. We utilize theoretical results from Ortner and Van Koten (manuscript) to derive optimal choices of approximation parameters (blending function and finite element grid) for microcrack and di-vacancy test problems and confirm our analytical predictions in numerical tests

Deliberative Agenda Setting: Piloting Reform of Direct Democracy in California

Can the people deliberate to set the agenda for direct democracy in large scale states? How might such an institution work? The 2011 California Deliberative Poll piloted a solution to this problem helping to produce proposals that went to the ballot and also to the legislature. The paper reports on how this pilot worked and what it suggests about a possible institution to solve the deliberative agenda setting problem. The legislative proposal passed the legislature but the ballot proposition (Prop 31) failed. However, we show that the proposals actually deliberated on by the people might well have passed if not encumbered by additional elements not deliberated on by the public that drew opposition. The paper ends with an outline of how the process of deliberative agenda setting for the initiative might work, vetting proposals once every two years that could get on the ballot for a greatly reduced cost in signature collections. Adding deliberation to the agenda setting process would allow for a thoughtful and informed public will formation to determine the agenda for direct democracy

A mathematical formalization of the parallel replica dynamics

The purpose of this article is to lay the mathematical foundations of a well known numerical approach in computational statistical physics and molecular dynamics, namely the parallel replica dynamics introduced by A.F. Voter. The aim of the approach is to efficiently generate a coarse-grained evolution (in terms of state-to-state dynamics) of a given stochastic process. The approach formally consists in concurrently considering several realizations of the stochastic process, and tracking among the realizations that which, the soonest, undergoes an important transition. Using specific properties of the dynamics generated, a computational speed-up is obtained. In the best cases, this speed-up approaches the number of realizations considered. By drawing connections with the theory of Markov processes and, in particular, exploiting the notion of quasi-stationary distribution, we provide a mathematical setting appropriate for assessing theoretically the performance of the approach, and possibly improving it

Development of an Optimization-Based Atomistic-to-Continuum Coupling Method

Atomistic-to-Continuum (AtC) coupling methods are a novel means of computing the properties of a discrete crystal structure, such as those containing defects, that combine the accuracy of an atomistic (fully discrete) model with the efficiency of a continuum model. In this note we extend the optimization-based AtC, formulated in arXiv:1304.4976 for linear, one-dimensional problems to multi-dimensional settings and arbitrary interatomic potentials. We conjecture optimal error estimates for the multidimensional AtC, outline an implementation procedure, and provide numerical results to corroborate the conjecture for a 1D Lennard-Jones system with next-nearest neighbor interactions.Comment: 12 pages, 3 figure