Speed of convergence

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). 

For problem (51) one can derive an a priori estimate of the same type as estimate (48) for problem (44). However, in this case such an estimate fails to provide with a quite reliable idea on the speed of convergence of... 

Estimate (52) indicates an effective reduction of the speed of convergence of scheme (51) on the non-equidistant grid ui), in comparison with scheme (44) on the equidistant grid. However, as we have stated above, if the error of approximation is evaluated not in the grid norm of the spaces C or Ljj but in a specially constructed negative norm ( ip then the error of approximation on any non-equidistant grid will be of the same order 0 h ). Namely, the negative norm... 

It has been shown [14] for both types ofbasis sets (1.1) and (1.2) that a given set of dimension n can be regarded as a member B of a family of basis sets that in the limit n oo become complete both in the ordinary sense and with respect to a norm in the Sobolev space - which is the condition for the eigenvalues and eigenfunctions of a Hamiltonian to converge to the exact ones. However, as to the speed of convergence the two basis sets (1.1) and (1.2) differ fundamentally. 

In the second experiment (Figure 9.14b), the ES was initialized by a feasible initial population that consisted of the EV-solution and other randomly generated feasible solutions. Here, the ES converges faster than with infeasible initialization. Although the ES is robust against infeasible initialization, a feasible initialization is recommended to improve speed of convergence. 

The distribution pi is constructed in such a way that the grids concentrate more and more in the region where the integrand is large. The construction of pi involves a positive parameter (3 that controls the speed of convergence to a stable configuration. In most cases we chose (3 = 0.5. However, we may even choose [3 = 0 (no change in p) which is useful in some cases. 

As for all iterative methods, the following questions are posed. What are the conditions of convergence What is the speed of convergence When should computations be stopped ... 

Iteration is stopped when the relative contributions of v. to u.. is lower than a pre-defined threshold e. The speed of convergence of the protocol depends on the values of and e. At the end of the protocol, all the free energy barriers... 

Since in the Floquet representation the Hamiltonian K defined on the enlarged Hilbert space is time-independent, the analysis of the effect of perturbations (like, e.g., transition probabilities) can be done by stationary perturbation theory, instead of the usual time-dependent one. Here we will present a formulation of stationary perturbation theory based on the iteration of unitary transformations (called contact transformations or KAM transformations) constructed such that the form of the Hamiltonian gets simplified. It is referred to as the KAM technique. The results are not very different from the ones of Rayleigh-Schrodinger perturbation theory, but conceptually and in terms of speed of convergence they have some advantages. 

Since the efficiency of multidimensional minimization schemes is dependent on the number of degrees of rotational freedom, the separation of "major" from "minor" torsional rotations should dramatically enhance the speed of convergence of these routines. Of course, "normal" rotational methods may be applied simply by specifying no "minor" or secondary rotation parameters. 

The set of 2C simultaneous equations is nonlinear and fairly complex since it involves calculating fugacities and enthalpies, themselves nonlinear functions of the temperature, pressure, and composition. The equations may be solved simultaneously or by some iterative method, tn general, the computational methods depend on which two variables are selected as the independent variables. Although in principle any two independent variables may be fixed, the problem complexity may vary from case to case. It is found, for instance, that a solution is more readily reached if P and Trather than P and Q are the independent variables. Since most of these calculations are carried out on computers, the solution methods should be designed for speed of convergence and reliability. Several methods have been proposed for handling the different types of flash calculations, some of which are discussed herewith. 

With a reasonable starting geometry, speed of convergence is similar to that obtained for minima. 

Our remarks so far concern the speeding of convergence in iterative solutions of the SSOZ equation with various closures, and are applicable to any site-site pair potentials. Additional considerations arise when some of the sites carry charges. Special techniques then have to be adopted to overcome difficulties associated with the long range of the Coulomb potential. The method conventionally used for this purpose is the renormalization introduced by Hirata and Rossky, which is analogous to Allnatt s method for electrolytes. This has been combined with the Gillan method by Morriss and Monson and was later used to obtain solutions for... 

This BFGS update is only slightly more complicated than the DFP update, and this very slight extra computation is more than compensated for by the extra speed of convergence that is observed in practice. 

The choice will depend on the complexity of the rate equation, the speed of convergence and the capabilities of the available computers. Unfortunately even the best of... 

The number of iterations should reflect the usual trade-off between accuracy and computation cost. However, there are other techniques to increase the speed of convergence. There is always some error in the simulation estimate due to sample variability. As the number of replications increases, the estimate converges to the true value at a speed proportional to the square root of the number of replications. More replications bring about more precise estimates but take longer to estimate. In fast-moving markets, or with complex securities, speed may be more important than accuracy. 

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