Multidimensional interpolation

F.G. Horowitz, P. Hornby, D. Bone, and M. Craig (1996) Fast multidimensional interpolations. 26th Proceedings of the Application of Computers and Operations Research in the Mineral Industry (APGOM26), R.V. Ramani (ed.), Soc. Mining, Metall., and Explor. (SME), Littelton, Colorado, USA, 53-56. 

J. W. Downing, J. Michi, J. Cizek, and J. Paldus, Multidimensional interpolation by polynomial roots, Chem. Phys. Lett. 67 377 (1979). 

The second approach is based on the multidimensional interpolation approach (MIA) developed by Smolentsev, Soldatov and co-workers as well as Minuit XANes (MXAN) by Benfatto and Della Longa and co-workers... 

Such an approximation is the result of a natural generalization of homogeneous conservative schemes from Chapter 3 for one-dimensional equations to the multidimensional case. These schemes can be obtained by means of the integro-interpolational method without any difficulties. 

CVT approach is particularly attractive due to the limited amount of potential energy and Hessian information that is required to perform the calculations. Direct dynamics with CVT thus offers an efficient and cost-effective methodology. Furthermore, several theoretical reviews60,61 have indicated that CVT plus multidimensional semi-classical tunneling approximations yield accurate rate constants not only for gas-phase reactions but also for chemisorption and diffusion on metals. Computationally, it is expensive if these Hessians are to be calculated at an accurate level of ab initio molecular orbital theory. Several approaches have been proposed to reduce this computational demand. One approach is to estimate rate constants and tunneling contributions by using Interpolated CVT when the available accurate ab initio electronic structure information is very limited.62 Another way is to carry out CVT calculations with multidimensional semi-classical tunneling approximations. 

Structural interpolation takes place in a multidimensional space. Each structural dimension refers to a site which may be a primary site or a a>site S (interaction site). The cosites augment the number of directions taken into account to evaluate surround ings of each predictable structure by experimental structures in hyperstructure HS ... 

A large number of explicit numerical advection algorithms were described and evaluated for the use in atmospheric transport and chemistry models by Rood [162], and Dabdub and Seinfeld [32]. A requirement in air pollution simulations is to calculate the transport of pollutants in a strictly conservative manner. For this purpose, the flux integral method has been a popular procedure for constructing an explicit single step forward in time conservative control volume update of the unsteady multidimensional convection-diffusion equation. The second order moments (SOM) [164, 148], Bott [14, 15], and UTOPIA (Uniformly Third-Order Polynomial Interpolation Algorithm) [112] schemes are all derived based on the flux integral concept. 

Ewald summation presented above calls for the calculation of AP terms for each of the periodic boxes, a computationally demanding requirement for large biomolecular systems. Recently, Darden et al. proposed an N log N method, called particle mesh Ewald (PME), which incorporates a spherical cutoff R. This method uses lookup tables to calculate the direa space sum and its derivatives. The reciprocal sum is implemented by means of multidimensional piecewise interpolation methods, which permit the calculation of this sum and its first derivative at predefined grids with fast Fourier transform methods. The overhead for this calculation in comparison to Coulomb interactions ranges from 16 to 84% of computer time, depending on the reciprocal sum grid size and the order of polynomial used in calculating this sum. 

Habermann, C. Kindermann, F. 2007. Multidimensional Spline Interpolation Theory and Applications. Computational Economics 30(2) 153-169. 

A very efficient method is to precalculate steady-state solutions and save the results in look-up tables. Interpolation in multidimensional look-up tables is most often much faster than iterative solutions of non-linear equations. It is also possible to use look-up tables for initial-value differential equations using time as one variable in the look-up table. 

In applications to a wide range of experimental results, one needs an efficient theoretical tool allowing for quick comparison of experiment and theory, perhaps followed by adjustment of theoretical parameters to experiment. With this goal in mind, the early formulation of the SACM included a simple empirical representation of the main features of the electronic potential by a very few adjustable parameters. Furthermore, the complicated calculation of adiabatic channels by a solution of the multidimensional clamped -q- rovibrational Schrbdinger equation, which is an exceedingly demanding task even today for larger than triatomic systems, was completely circumvented by a simple channel interpolation procedure. We shall present here a very brief description of this empirical approach for simple bond fission reactions. 

Also, a method has been introduced in which a multidimensional potential energy surface is represented by a linear combination of products of one-dimensional functions. This procedure can be used to formulate analytical representations of PESs and to interpolate between ab initio energy points. 

INTERPOLATION OF MULTIDIMENSIONAL POTENTIAL SURFACES BY POLYNOMIAL ROOTS... 

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