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.

STAR 3 randomized controlled trial to compare sensor-augmented insulin pump therapy with multiple daily injections in the treatment of type 1 diabetes: research design, methods, and baseline characteristics of enrolled subjects.

BACKGROUND: Sensor-augmented pump therapy (SAPT) integrates real-time continuous glucose monitoring (RT-CGM) with continuous subcutaneous insulin infusion (CSII) and offers an alternative to multiple daily injections (MDI). Previous studies provide evidence that SAPT may improve clinical outcomes among people with type 1 diabetes. Sensor-Augmented Pump Therapy for A1c Reduction (STAR) 3 is a multicenter randomized controlled trial comparing the efficacy of SAPT to that of MDI in subjects with type 1 diabetes. METHODS: Subjects were randomized to either continue with MDI or transition to SAPT for 1 year. Subjects in the MDI cohort were allowed to transition to SAPT for 6 months after completion of the study. SAPT subjects who completed the study were also allowed to continue for 6 months. The primary end point was the difference between treatment groups in change in hemoglobin A1c (HbA1c) percentage from baseline to 1 year of treatment. Secondary end points included percentage of subjects with HbA1c < or =7% and without severe hypoglycemia, as well as area under the curve of time spent in normal glycemic ranges. Tertiary end points include percentage of subjects with HbA1c < or =7%, key safety end points, user satisfaction, and responses on standardized assessments. RESULTS: A total of 495 subjects were enrolled, and the baseline characteristics similar between the SAPT and MDI groups. Study completion is anticipated in June 2010. CONCLUSIONS: Results of this randomized controlled trial should help establish whether an integrated RT-CGM and CSII system benefits patients with type 1 diabetes more than MDI.

Accelerating self-consistent field convergence with the augmented Roothaan-Hall energy function.

Based on Pulay's direct inversion iterative subspace (DIIS) approach, we present a method to accelerate self-consistent field (SCF) convergence. In this method, the quadratic augmented Roothaan-Hall (ARH) energy function, proposed recently by Høst and co-workers [J. Chem. Phys. 129, 124106 (2008)], is used as the object of minimization for obtaining the linear coefficients of Fock matrices within DIIS. This differs from the traditional DIIS of Pulay, which uses an object function derived from the commutator of the density and Fock matrices. Our results show that the present algorithm, abbreviated ADIIS, is more robust and efficient than the energy-DIIS (EDIIS) approach. In particular, several examples demonstrate that the combination of ADIIS and DIIS ("ADIIS+DIIS") is highly reliable and efficient in accelerating SCF convergence.

Scaling limits of a model for selection at two scales

The dynamics of a population undergoing selection is a central topic in evolutionary biology. This question is particularly intriguing in the case where selective forces act in opposing directions at two population scales. For example, a fast-replicating virus strain outcompetes slower-replicating strains at the within-host scale. However, if the fast-replicating strain causes host morbidity and is less frequently transmitted, it can be outcompeted by slower-replicating strains at the between-host scale. Here we consider a stochastic ball-and-urn process which models this type of phenomenon. We prove the weak convergence of this process under two natural scalings. The first scaling leads to a deterministic nonlinear integro-partial differential equation on the interval $[0,1]$ with dependence on a single parameter, $\lambda$. We show that the fixed points of this differential equation are Beta distributions and that their stability depends on $\lambda$ and the behavior of the initial data around $1$. The second scaling leads to a measure-valued Fleming-Viot process, an infinite dimensional stochastic process that is frequently associated with a population genetics.