Convergence of numerical time-averaging and stationary measures via Poisson equations

Numerical approximation of the long time behavior of a stochastic di.erential equation (SDE) is considered. Error estimates for time-averaging estimators are obtained and then used to show that the stationary behavior of the numerical method converges to that of the SDE. The error analysis is based on using an associated Poisson equation for the underlying SDE. The main advantages of this approach are its simplicity and universality. It works equally well for a range of explicit and implicit schemes, including those with simple simulation of random variables, and for hypoelliptic SDEs. To simplify the exposition, we consider only the case where the state space of the SDE is a torus, and we study only smooth test functions. However, we anticipate that the approach can be applied more widely. An analogy between our approach and Stein's method is indicated. Some practical implications of the results are discussed. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

Comprehensive search for new phases and compounds in binary alloy systems based on platinum-group metals, using a computational first-principles approach

We report a comprehensive study of the binary systems of the platinum-group metals with the transition metals, using high-throughput first-principles calculations. These computations predict stability of new compounds in 28 binary systems where no compounds have been reported in the literature experimentally and a few dozen of as-yet unreported compounds in additional systems. Our calculations also identify stable structures at compound compositions that have been previously reported without detailed structural data and indicate that some experimentally reported compounds may actually be unstable at low temperatures. With these results, we construct enhanced structure maps for the binary alloys of platinum-group metals. These maps are much more complete, systematic, and predictive than those based on empirical results alone.