Computational time requirement

To estimate the computational time required in a Gaussian elimination procedure we need to evaluate the number of arithmetic operations during the forward reduction and back substitution processes. Obviously multiplication and division take much longer time than addition and subtraction and hence the total time required for the latter operations, especially in large systems of equations, is relatively small and can be ignored. Let us consider a system of simultaneous algebraic equations, the representative calculation for forward reduction at stage is expressed as... 

Figure 2 A particle i interacts mainly with particles that are within the cutoff radius The neighbor list contains only those particles that are within a sphere of radius rj > Particles outside this sphere will not contribute to the force or energy affecting particleThe use of a neighbor list that is periodically updated during the simulation reduces the computer time required in calculating pairwise interactions.

The LSDA approach requires simultaneous self-consistent solutions of the Schrbdinger and Poisson equations. This was accomplished using the Layer Korringa-Kohn-Rostoker technique which has many useful features for calculations of properties of layered systems. It is, for example, one of only a few electronic structure techniques that can treat non-periodic infinite systems. It also has the virtue that the computational time required for a calculation scales linearly with the number of different layers, not as the third power as most other techniques. 

The advantages of the simple approach outlined above are the limited number of water molecules needed in the simulation and the well-defined water structure. The major drawback is that, owing to the periodicity, this water structure fits best on a (111) or (lll)-like surface, e.g., (211). There are at least two other approximations to model the water interaction. One is to include a large number of water molecules and apply molecular dynamics to determine a structure for the water and include this water arrangement in the simulations [Filhol and Neurock, 2006]. The drawbacks of this approach are the computational time required and the results sensitivity to the water structure. 

Quantum mechanical approaches, such as those applied by Merz [84], use traditional quantum mechanics to calculate free energies of binding. Though relatively successful, the computational time required to calculate the energies make this approach impractical to large data sets. 

This iterative procedure depends linearly on the number of fragments and on the size of the target macromolecule M, as long as the parent molecules Mk are confined to some limited size. The storage of the information on the macromolecular basis set has relatively small computer memory requirements. The computation of the macromolecular electron density from this basis set information and the final macromolecular density matrix P(K) obtained from the finite iterative process (56) can rely on relation (32). As a consequence of the sparsity macromolecular density matrix P(AT), the computational task has linear computer time requirement with respect to the number of fragments, hence, with respect to the size of the target macromolecule M. 

Another result that is not evident in Fig. 33 concerns the computational times required for gathering the statistical steady-state values of various quantities (such as the slip velocity shown in Fig. 33) at comparable grid resolutions, the computational time required to solve the filtered equations is much smaller than that for the microscopic equations. This can be attributed to the fact that the structures obtained in the solution of the filtered equations are comparatively coarser than those for the microscopic TFM equations. 

The computing time required to evaluate Equation 4.19 in a Newton-Raphson iteration increases with the cube of the number of equations considered (Dongarra et al., 1979). The numerical solution to Equations 4.3 1.6, therefore, can be found most rapidly by reserving from the iteration any of these equations that can be solved linearly. There are four cases in which equations can be reserved ... 

In some cases, the characteristic velocity can cause difficulties in solution, owing to the presence of an implicit equation. In this the appropriate value of L or Gn satisfying the value of hn generated by the differential material balance equation must be found by root finding algorithms increasing computation time required. 

The disadvantage of these NLP algorithms is the large amount of computation time required relative to the successive linearisation algorithm. Nevertheless, their range of application is wider, and they are able to manage nonlinear objective functions, equality and inequality constraints, and bounds on variables. 

Since this treatment is parametrized to mimic the results of ab initio calculations, it is not surprising to find that it gives equally inaccurate estimates of heats of atomization. The errors in bond lengths and bond angles are also greater, while force constants are in error by a factor of two or three. While the computation time required is much less than for the ab initio methods,... 

Physical and Structural Aspects.—Perhaps the most significant theoretical paper comes from an application of pseudopotential SCF methods to PX3 molecules.1 This method neglects core orbitals, and hence shortens computer-time requirements, in comparison with more conventional SCF calculations. The results1 are really most encouraging, and compare favourably with standard SCF results, in relation to experimental values for bond angles and lengths, and for dipole moments. 

In tearing, the objective is to wind up with less computation time required to solve the torn system compared with the time required to solve the entire block of equations simultaneously. However, the criteria for evaluating the effectiveness of the tearing are by no means so well defined as those for partitioning, where the objective is clearly to obtain the smallest possible subsystems of irreducible equations. There is no general method for determining the time needed to effect a solution of a set of equations it is necessary to consider the particular equations involved. Any feasible method of tearing, then, must be based on criteria that are related to the solution time. Some of the more obvious criteria are ... 

The major drawback for employing the Car-Parrinello approach in dynamics simulations is that since a variational wavefunction is required, the electronic energy should in principle be minimized before the forces on the atoms are calculated. This greatly increases the amount of computer time required at each step of the simulation. Furthermore, the energies calculated with the electronic structure methods currently used in this approach are not exceptionally accurate. For example, it is well established that potential energy barriers, which are of importance to chemical reactivity, often require sophisticated methods to be accurately determined. Nonetheless, the Tirst-principles calculation of the forces during the dynamics is an appealing idea, and will continue to be developed as computer resources expand. 

Notice that four derivative evaluations are required per ODE at each time step. Thus the computer time required to run Euler with a step size of 0.05 would be about the same as the time required to run Runge-Kutta with a step size of 0.2,... 

The core of the Evidence theory lies in the combination of belief assignments. As already described, several belief combination rules exist, each of them corresponding to an interpretation of the conflict between bba. As a consequence, a rule should be first chosen according to the interpretation of conflict appropriate to the final objective. In addition, some mathematical considerations should be also accounted for. Indeed, only the Dempster s and Smets rules are associative. This means that for other combination rules e.g., Yager s or Dubois and Trade s rules), we have (i) either to perform a simultaneous combination of all available bba or (ii) to determine the sequence of combinations appropriate to the final objective. The first choice is satisfactory since it does not need to perform any assumption on the combination order. However, it implies to compute a great number of intersections of focal elements and is thus difficult to apply for more than seven bba (due to the computation time required). 

In a VOC-NO mixture containing many different organics, the number of reactions becomes unmanageable for application in models used to describe an air basin or region. Thus the amount of computer time required for numerical integration of the rate equations associated with the thousands of individual species found in ambient air is prohibitive. Furthermore, even as computing power increases, in practice, the kinetics and mechanisms required as input are not all known. 

In molecular orbital calculations, the molecular orbitals are represented as a linear combination of atomic orbital functions (Xi). A variety of different mathematical functions may be used to represent these atomic orbital functions. If a very sophisticated mathematical function is used, then the resulting answer is higher in quality, providing very accurate energies and geometries for the drug molecule being studied however, such calculations may be extremely expensive in terms of computer time required. If a... 

In both methods the effects of triple excitations can be estimated. These effects can be significant, if quantitative accuracy is the goal of the calculations. However, performing a calculation at either the CCSD(T) or QCISD(T) level of theory comes at the cost of substantially increasing the computer time required, beyond that consumed by a CCSD or QCISD calculation. 

When available, fundamental process models are preferred. For many complex processes such as composite manufacturing in general and autoclave curing in particular, however, these models are often not available. This lack of availability is due to an inadequate understanding of the complex events that take place during the process. A fundamental process model is occasionally available, but it is still unsuitable for on-line model predictive control application due to the extensive computing time required to solve the model s equations. This lack of... 

A significant problem with stellar nucleosynthesis models is that they do not adequately describe the physical state of the star. To decrease the amount of computer time required to... 

Finally, there are a number of entirely mundane (but still very worthwhile ) steps that can be taken to reduce the total computer time required for a MD simulation. As a single example, note that any force on a particle derived from a force-field non-bonded energy term is induced by some other particle (i.e., the potential is pairwise). Newton s Third Law tells us that... 

---