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A common approach for minimizing a smooth nonlinear function is to employ finite-difference approximations to the gradient. While this can be easily performed when no error is present within the function evaluations, when the function is noisy, the optimal choice requires information about the noise level and higher-order derivatives of the function, which is often unavailable. Given the noise level of the function, we propose a bisection search for finding a finite-difference interval for any finite-difference scheme that balances the truncation error, which arises from the error in the Taylor series approximation, and the measurement error, which results from noise in the function evaluation. Our procedure produces reliable estimates of the finite-difference interval at low cost without explicitly approximating higher-order derivatives. We show its numerical reliability and accuracy on a set of test problems. When combined with limited memory BFGS, we obtain a robust method for minimizing noisy black-box functions, as illustrated on a subset of unconstrained CUTEst problems with synthetically added noise. [ABSTRACT FROM AUTHOR]

The article offers information on the gold-catalyzed synthesis of α-substituted vinyl sulfones. Topics discussed include synthesis of α-substituted vinyl sulfones from alkynes and sulfinic acids, application of the vinyl sulfones in biological and pharmaceutical sciences and screening of gold catalysts.

Chronic obstructive pulmonary disease (COPD) causes a high disease burden among the elderly worldwide. In Taiwan, the long-term temporal trend of COPD mortality is declining, but the geographical disparity of the disease is not yet known. Nationwide COPD age-adjusted mortality at the township level during 1999–2007 is used for elucidating the geographical distribution of the disease. With an ordinary least squares (OLS) model and geographically weighted regression (GWR), the ecologic risk factors such as smoking rate, area deprivation index, tuberculosis exposure, percentage of aborigines, density of health care facilities, air pollution and altitude are all considered in both models to evaluate their effects on mortality. Global and local Moran’s I are used for examining their spatial autocorrelation and identifying clusters. During the study period, the COPD age-adjusted mortality rates in males declined from 26.83 to 19.67 per 100,000 population, and those in females declined from 8.98 to 5.70 per 100,000 population. Overall, males’ COPD mortality rate was around three times higher than females’. In the results of GWR, the median coefficients of smoking rate, the percentage of aborigines, PM10 and the altitude are positively correlated with COPD mortality in males and females. The median value of density of health care facilities is negatively correlated with COPD mortality. The overall adjusted R-squares are about 20% higher in the GWR model than in the OLS model. The local Moran’s I of the GWR’s residuals reflected the consistent high-high cluster in southern Taiwan. The findings indicate that geographical disparities in COPD mortality exist. Future epidemiological investigation is required to understand the specific risk factors within the clustering areas. [ABSTRACT FROM AUTHOR]