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.

Analysis of an asymptotic preserving scheme for linear kinetic equations in the diffusion limit

We present a mathematical analysis of the asymptotic preserving scheme proposed in [M. Lemou and L. Mieussens, SIAM J. Sci. Comput., 31 (2008), pp. 334-368] for linear transport equations in kinetic and diffusive regimes. We prove that the scheme is uniformly stable and accurate with respect to the mean free path of the particles. This property is satisfied under an explicitly given CFL condition. This condition tends to a parabolic CFL condition for small mean free paths and is close to a convection CFL condition for large mean free paths. Our analysis is based on very simple energy estimates. © 2010 Society for Industrial and Applied Mathematics.

Preoperative predictors of extended hospital length of stay following total knee arthroplasty.

The purpose of this study was to identify the preoperative predictors of hospital length of stay after primary total knee arthroplasty in a patient population reflecting current trends toward shorter hospitalization and using readily obtainable factors that do not require scoring systems. A single-center, multi-surgeon retrospective chart review of two hundred and sixty consecutive patients who underwent primary total knee arthroplasty was performed. The mean length of stay was 3.0 days. Among the different variables studied, increasing comorbidities, lack of adequate assistance at home, and bilateral surgery were the only multivariable significant predictors of longer length of stay. The study was adequately powered for statistical analyses and the concordance index of the multivariable logistic regression model was 0.815.

A population model of folate-mediated one-carbon metabolism.

BACKGROUND: Previous mathematical models for hepatic and tissue one-carbon metabolism have been combined and extended to include a blood plasma compartment. We use this model to study how the concentrations of metabolites that can be measured in the plasma are related to their respective intracellular concentrations. METHODS: The model consists of a set of ordinary differential equations, one for each metabolite in each compartment, and kinetic equations for metabolism and for transport between compartments. The model was validated by comparison to a variety of experimental data such as the methionine load test and variation in folate intake. We further extended this model by introducing random and systematic variation in enzyme activity. OUTCOMES AND CONCLUSIONS: A database of 10,000 virtual individuals was generated, each with a quantitatively different one-carbon metabolism. Our population has distributions of folate and homocysteine in the plasma and tissues that are similar to those found in the NHANES data. The model reproduces many other sets of clinical data. We show that tissue and plasma folate is highly correlated, but liver and plasma folate much less so. Oxidative stress increases the plasma S-adenosylmethionine/S-adenosylhomocysteine (SAM/SAH) ratio. We show that many relationships among variables are nonlinear and in many cases we provide explanations. Sampling of subpopulations produces dramatically different apparent associations among variables. The model can be used to simulate populations with polymorphisms in genes for folate metabolism and variations in dietary input.

Enhancing the Hodgkin-Huxley Equations: Simulations Based on the First Publication in the Biophysical Journal.

The experiments in the Cole and Moore article in the first issue of the Biophysical Journal provided the first independent experimental confirmation of the Hodgkin-Huxley (HH) equations. A log-log plot of the K current versus time showed that raising the HH variable n to the sixth power provided the best fit to the data. Subsequent simulations using n(6) and setting the resting potential at the in vivo value simplifies the HH equations by eliminating the leakage term. Our article also reported that the K current in response to a depolarizing step to ENa was delayed if the step was preceded by a hyperpolarization. While the interpretation of this phenomenon in the article was flawed, subsequent simulations show that the effect completely arises from the original HH equations.