The chaotic behavior of fixed-point methods for steady-state process simulation is studied. It is shown that direct substitution and Newton's method exhibit all of the rich structure of chaos (period doubling, aperiodicity, fractal basin boundaries, and related properties) on simple process examples. These examples include finding roots to the SoaveRedlich-Kwong and Underwood equations, dew point and flash calculations for heterogeneous mixtures, and a simple process flowsheet.For single variable problems, it is shown that direct substitution follows a classical period-doubling route to chaos. On the other hand, the chaotic behavior of direct substitution and Newton's method on multivariable problems is considerably more complex, and can give the appearance that no organized route to chaos is followed. For example, for the dew point problems, truncated period doubling, odd periodic cycles, and chaotic behavior can be observed, within which are embedded narrow regions of global convergence. Many numerical results and geometric illustrations are presented.

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Vapor−liquid equilibria for refrigerant mixtures modeled by an equation of state are studied. Phase behavior calculated by the Soave−Redlich−Kwong (SRK) equation with a single adjustable binary interaction parameter is compared with experimental data for binary refrigerant mixtures, two with a supercritical component and one that exhibits azeotropic behavior. It is shown that the SRK equation gives an adequate description of the phase envelope for binary refrigerant systems. The complex domain trust region methods of Lucia and co-workers (Lucia and Xu, 1992; Lucia et al., 1993) are applied to fixed vapor, isothermal flash model equations, with particular attention to root finding and root assignment at the equation of state (EOS) level of the calculations, and convergence in the retrograde and azeotropic regions of the phase diagram. Rules for assigning roots to the vapor and liquid phases in the case where all roots to the EOS are complex-valued are proposed and shown to yield correct results, even in retrograde regions. Convergence of the flash model equations is also studied. It is shown that the complex domain trust region algorithms outperform Newton's method in singular regions of the phase diagram (i.e., at near azeotropic conditions and in the retrograde loop), primarily due to the eigenvalue−eigenvector decomposition strategy given in Sridhar and Lucia (1995). A variety of geometric figures are used to illustrate salient points.

There have been increased concerns that the use of statins, one of the most commonly prescribed drugs for treating coronary artery disease, is potentially associated with the increased risk of new-onset type II diabetes (T2D). However, because existing clinical studies with limited sample sizes often suffer from selection bias issues, there is no robust evidence supporting as to whether and what kind of populations are indeed vulnerable for developing T2D after taking statins. In this case study, building on the biobank and electronic health record data in the Partner Health System, we introduce a new data analysis pipeline from a biological perspective and a novel statistical methodology that address the limitations in existing studies to: (i) systematically examine heterogeneous treatment effects of stain use on T2D risk, (ii) uncover which patient subgroup is most vulnerable to T2D after taking statins, and (iii) assess the replicability and statistical significance of the most vulnerable subgroup via bootstrap calibration. Our proposed bootstrap calibration approach delivers asymptotically sharp confidence intervals and debiased estimates for the treatment effect of the most vulnerable subgroup in the presence of possibly high-dimensional covariates. By implementing our proposed approach, we find that females with high T2D genetic risk at baseline are indeed at high risk of developing T2D due to statin use, which provides evidences to support future clinical decisions with respect to statin use.