Numerical methods computational efficiency

In Section 3, the performance of the software implementation of PS electronic methods, the PS-GVB suite of ab initio electronic structure programs, is discussed. Three issues are addressed numerical precision of the calculations as compared to conventional analytical basis set methods, computational efficiency as a function of system size for the various electronic structure methods in the program, and effectiveness in addressing real chemical problems. For the last of these, we focus on conformational energy differences, calculation of which we have been intensively studying over the past several years, and thermochemistry which we have just begun to study with GVB-MP2. 

Appendix F is a case study by Hjertager et al. illustrating the above method. Such numerical methods will become more widely used in the long term. These techniques will probably remain research tools, rather than routine evaluation methods, until such time as available computing power and algorithm efficiency greatly increase. 

The main advantage of using the ASA method to obtain approximate electron densities is the very important gain in computational efficiency to compute MQSM and thus perform similarity analysis among the molecules of some molecular set. This is important for those cases where either the molecules are very numerous, or very big or a combination of both. 

Finally, FDTD may be used to model the coupling of the focal field into the PhC-waveguide, potentially with the presence of an air or glue gap. Even such a simulation procedure with adapted numerical methods for each part of the propagation requires a considerable computation time. To speed up the simulation process for system optimisation remarkably, the FDTD-simulation can be replaced by a formula for the coupling efficiency to a conventional high-index or a PhC-waveguide, ... 

An excellent resource for learning about efficient numerical methods for optimization (and many other problems) is W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C++ The Art of Scientific Computing, Cambridge University Press, Cambridge, UK, 2002. Multiple editions of this book are available with equivalent information in other computing languages. 

When these methods are unsuitable, nonlinear methods may be applied. The function local minima and overall computational efficiency. The function (u) is often expensive to compute, so maximum advantage must accrue from each evaluation of it. To this end, numerous methods have been developed. Optimization is a field of ongoing research. No one single method is best for all types of problem. Where (u) is a sum of squares, as we have expressed it, and where derivatives dQ>/dvl are available, the method of Marquardt (1963) and its variants are perhaps best. Other methods may be desirable where constraints are to be applied to the vt, or where (u) cannot be formulated as a sum of... 

In the very short time limit, q (t) will be in the reactants region if its velocity at time t = 0 is negative. Therefore the zero time limit of the reactive flux expression is just the one dimensional transition state theory estimate for the rate. This means that if one wants to study corrections to TST, all one needs to do numerically is compute the transmission coefficient k defined as the ratio of the numerator of Eq. 14 and its zero time limit. The reactive flux transmission coefficient is then just the plateau value of the average of a unidirectional thermal flux. Numerically it may be actually easier to compute the transmission coefficient than the magnitude of the one dimensional TST rate. Further refinements of the reactive flux method have been devised recently in Refs. 31,32 these allow for even more efficient determination of the reaction rate. 

Wajge et al. (1997) attempted to develop rigorous PDAE model for packed batch distillation with and without chemical reaction and used finite difference and orthogonal collocation techniques to solve such model. The main purpose of the study was to investigate the efficiencies of the numerical methods employed. The authors observed that the collocation techniques are computationally more efficient compared to the finite difference method, however the order of approximating polynomial needs to be carefully chosen to achieve a right balance between accuracy and efficiency. See the original reference for further details. 

To analyze the transport and retention of chemical contaminants in groundwater flowing through soils, experimental and theoretical studies generated several reliable models. Diverse numerical methods have been applied to solve the governing equations efficiently. Some computer models include the simulation of physical and chemical processes. 

Recently, a new version of ADSA-P has been developed [67.681 that is superior to the original program in terms of computation time and range of applicability. The new version is written in the C language (rather than FORTRAN) and utilizes more efficient and accurate numerical methods. For example, the new algorithm uses the curvature at the apex (rather than the radius of curvature), it permits an additional optimization parameter (the vertical misalignment of the camera), and it gives improved initial estimates of the apex location and shape. 

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