Iterative hybrid methods

Basically, two types of approaches are developed here iterative (optimization-based) approaches like the one by Sippl et al. [101] and direct approaches like the one by Kabsch [102, 103], based on Lagrange multipliers. Unfortunately, the much expedient direct methods may fail to produce a sufficiently accurate solution on some degenerate cases. Redington [104] suggested a hybrid method with an improved version of the iterative approach, which requires the computation of only two 3x3 matrix multiplications in the inner loop of the optimization. 

We noted earlier that conventional iterative methods often contain adjustable parameters, the values of which had to be chosen with care, since they would influence the efficiency and success of the calculation. We have not escaped from these parameters by using an EA, and a set of variables is emerging whose values will affect the course of the calculation. There is the population size - how should that be chosen At what rate should individuals be mutated Should child solutions be selected deterministically or stochastically - or using some hybrid method ... 

In Section 14.6, we examine the relationship between coupled-cluster theory and perturbation theory. After a perturbation analysis of the coupled-cluster models, we consider various hybrid methods, in which perturbational corrections are applied (iteratively and noniteratively) within the framework of coupled-cluster theory. In particular, we apply a triples correction to the CCSD energy, arriving at the highly successful CX SD(T) approximation to the FCl electronic energy. 

In the RISM-SCF theory, the ab initio MO calculation and the RISM equation must be solved in a self-consistent manner. In other words, solvent distribution around the solute molecule is determined by the electronic structure of the solute, and the electronic structure of the solute is affected by the surrounding solvent distribution. To achieve this, the iteration cycle is repeated until mutual convergence between the ab initio MO calculation, which provides the partial charge on the solute, and the RISM equation, which provides PCFs, is attained. As mentioned above, it is noteworthy that the computational cost is drastically reduced by the analytic expression of the theory, compared with the QM/MM method. This means that the combination with more sophisticated (and computationally more expensive) methods such as CASPT2 and MCQDPT can be realized with moderate computational costs. Moreover, solvent distribution at the molecular level can be obtained simultaneously. These are remarkable merits compared with other hybrid-type methods. 

Computational fluid dynamics involves the analysis of fluid flow and related phenomena such as heat and/or mass transfer, mixing, and chemical reaction using numerical solution methods. Usually the domain of interest is divided into a large number of control volumes (or computational cells or elements) which have a relatively small size in comparison with the macroscopic volume of the domain of interest. For each control volume a discrete representation of the relevant conservation equations is made after which an iterative solution procedure is invoked to obtain the solution of the nonlinear equations. Due to the advent of high-speed digital computers and the availability of powerful numerical algorithms the CFD approach has become feasible. CFD can be seen as a hybrid branch of mechanics and mathematics. CFD is based on the conservation laws for mass, momentum, and (thermal) energy, which can be expressed as follows ... 

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