Enhancing convergence

Krylov subspace methods (such as Conjugate Gradient CG, the improved BiCGSTAB, and GMRES) in combination with preconditioners for matrix manipulations aimed at enhanced convergence, and... 

This equation implicitly accounts for the interaction between the velocities of two phases and therefore, enhances convergence rate. When the interphase interaction coefficient is zero, the above equation reduces to that for single-phase flows. It is useful to note here that the coefficient of pressure gradient term also becomes modified by the presence of the second phase. These modified coefficients should be used when... 

The computer codes used for these calculations are modifications of Dobash s program supplied by QCPE (3). The modifications principally consisted of increased dimensions to handle the large systems, and a matrix extrapolation routine incorporated into the SCF portion of the program to enhance convergence. 

One of the advantages of FEMLAB is that it differentiates the equations symbolically, including any terms that you add to the problem, and this enhances convergence. The parametric solver in FEMLAB is possible because the Newton-Raphson method is being used. Consider Eq. (F.l) when it depends upon a parameter ... 

More powerful minimization methods can also be employed. The method of conjugate gradients uses the information from earlier steps to choose a more efficient sequence of direaions to minimize along. In addition, preconditioning, which utilizes an easily invertible approximation to A, can enhance convergence. These more advanced techniques require fewer iterations at each grid... 

Modern process simulators (e.g. Aspen-Plus from AspenTech or ChemGad from Chemstations) simultaneously solve the MESH equations using algorithms based on Newton-Raphson methods (Gmehling and Brehm, 1996). However, for highly non-ideal or complex systems, modifications have been developed to enhance convergence behavior. 

The steady-state rate-based model used in this case study was similar to the model from Section 10.4.2, with the pseudohomogeneous mode for the reaction kinetics and the Maxwell-Stefan diffusion description. The model was implemented into ASPEN Custom Modeler together with another, simpler rate-based model, with effective diffusivities. The latter allows starting values for the Maxwell-Stefan-based model to be generated, thus enhancing convergence. 

To enhance convergence of free-particle sampling in centroid path integral simulations, a bisection sampling technique was used for a ring of beads by extending the original approach of Pollock and Ceperley for free particle... 

Second card FORMAT(8F10.2), control variables for the regression. This program uses a Newton-Raphson type iteration which is susceptible to convergence problems with poor initial parameter estimates. Therefore, several features are implemented which help control oscillations, prevent divergence, and determine when convergence has been achieved. These features are controlled by the parameters on this card. The default values are the result of considerable experience and are adequate for the majority of situations. However, convergence may be enhanced in some cases with user supplied values. 

A potential advantage of methods based on a series expansion of the free energy is that the convergence of the series is determined by the A dependence of the potential energy function meaning that the efficiency of the approach could be enhanced by a judicious choice of coupling scheme. 

Equation (2.106) gives rise to an implicit scheme except for 0 = 0. The application of implicit schemes for transient problems yields a set of simultaneous equations for the field unknown at the new time level n + 1. As can be seen from Equation (2.111) some of the terms in the coefficient matrix should also be evaluated at the new time level. Therefore application of the described scheme requires the use of iterative algorithms. Various techniques for enhancing the speed of convergence in these algorithms can be found in the literature (Pittman, 1989). 

Vertical air motions affect both weather and the mixing processes of importance to air pollution. Upward vertical motions can be caused by lifting over terrain, lifting over weather fronts, and convergence toward low-pressure centers. Downward vertical motions can be caused by sinking to make up for divergence near high-pressure centers. One must know whether the atmosphere enhances or suppresses these vertical motions to... 

A cursory inspection of key intermediate 8 (see Scheme 1) reveals that it possesses both vicinal and remote stereochemical relationships. To cope with the stereochemical challenge posed by this intermediate and to enhance overall efficiency, a convergent approach featuring the union of optically active intermediates 18 and 19 was adopted. Scheme 5a illustrates the synthesis of intermediate 18. Thus, oxidative cleavage of the trisubstituted olefin of (/ )-citronellic acid benzyl ester (28) with ozone, followed by oxidative workup with Jones reagent, affords a carboxylic acid which can be oxidatively decarboxylated to 29 with lead tetraacetate and copper(n) acetate. Saponification of the benzyl ester in 29 with potassium hydroxide provides an unsaturated carboxylic acid which undergoes smooth conversion to trans iodolactone 30 on treatment with iodine in acetonitrile at -15 °C (89% yield from 29).24 The diastereoselectivity of the thermodynamically controlled iodolacto-nization reaction is approximately 20 1 in favor of the more stable trans iodolactone 30. 

This considerable enhancement in redox properties may however remain chemically hidden. Several causes may converge to mask these properties. First of all electron transfer is an intermolecular act of reactivity even when thermodynamically feasible it may have to compete with very rapid intramolecular acts of deactivation (fluorescence, phosphorescence, internal conversion)99. The rate of electron transfer is given by the Rehm-Weller equation96,100... 

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