Relative computational efficiency

If we define computational efficiency as the number of steps required to complete the optimization, then random search will be more efficient than explicit gradient calculations when 2T n. Using the simple approximate estimate T,(n Y) that includes only the first term in the expansion of the exponential, the condition 2T, n becomes 

The approximate expression for T(n P) with the exponential not expanded does not admit a convenient explicit solution of the inequality 2T n however, numerical results illustrated in Figs 7 and 8 show that the inequality remmns satisfied also for large n. 

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Matrix elements of the form (Da H Db/ assume a particularly simple form in this special representation, the necessary cofactors being given by products of the elements of rfAB. The relative merits of the two methods have been compared in the literature.77-78 However, too few calculations of any size have been performed by either method so far for one to make any useful statement about their relative computation efficiencies. 

In order to achieve that an environmental fate model is successfully applied in a screening level risk assessment and ultimately incorporated into the decisionmaking tools, the model should have computational efficiency and modest data input. Moreover, the model should incorporate all relevant compartments and all sources of contamination and should consider the most important mechanisms of fate and transport. Although spatial models describe the environment more accurately, such models are difficult to apply because they require a large amount of input data (e.g., detailed terrain parameters, meteorological data, turbulence characteristics and other related parameters). Therefore, MCMs are more practical, especially for long-term environmental impact evaluation, because of their modest data requirements and relatively simple yet comprehensive model structure. In addition, MCMs are also widely used for the comparative risk assessment of new and existing chemicals [28-33]. 

Computational efficiency remains a central question. For a given amount of computer time, how good are nonequilibrium estimates of free energy differences compared to estimates from equilibrium methods Overall, evidence is mounting that nonequilibrium methods are less efficient than equilibrium methods [13, 20, 38], However, new approaches have been suggested that use long time steps [25]. For relatively slow transformations, it has been shown [20] that for a given amount of computer simulation time, one obtains more-accurate results for few slow transformations than for many fast transformations. At the other extreme, i.e., in the limit of... 

Nonrelativistic quantum chemistry has been discussed so far. But transition metal (starting already from the first row) and actinide compounds cannot be studied theoretically without a detailed account of relativity. Thus, the multiconfigurational method needs to be extended to the relativistic regime. Can this be done with enough accuracy for chemical applications without using the four-component Dirac theory Much work has also been done in recent years to develop a reliable and computationally efficient four-component quantum chemistry.25,26 Nowadays it can be combined, for example, with the CC approach for electron correlation. The problem is that an extension to multiconfigurational... 

In the two previous sections, we have discussed the use of the basic SCF-Xa-SW method. The advantage of this method is its computational efficiency relative to other ab initio methods which allows larger clusters to be handled. 

The principal drawback of the DFT method is that it is computationally intensive relative to the classical adsorption models, although it is still much less compute-intensive than full Monte Carlo molecular simulation. A semianalytic adsorption model that retains computational efficiency while accounting for gas-solid potential interactions in micropores was originally proposed by Horvath and Kawazoe [12], In the Horvath-Kawazoe or HK method, a pore filling correlation is obtained by calculating the mean heat of adsorption (/> required to transfer an adsorbate molecule from the gas phase to the condensed phase in a slit pore of width // ... 

A serious drawback of DPD is the absence of a drag force between the central particle and the second one orbiting about the first particle. The dissipative force Fd representing the dot product of differential velocities between interacting particles and their relative position vector is then equal to zero. This relative motion may produce a net drag only when many particles are participating at the same time (Espanol 1998). This cumulative effect requires more particles to be involved and reduces the computational efficiency of the DPD method. 

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