Direct-computation integral methods

The general problem, even without consideration of space-charge effects, is intrinsically non-linear. Only DC solutions for potential, concentrations, fluxes and currents are possible by exact methods in some specific cases. Time-dependence problems, such as the derivation of the impedance, require some simplifications or consideration of special cases. Diffusion-migration models were used to simulate the behaviour of electroactive films both by direct computer integration of the equations to calculate cyclic voltammo-grams, or potential distributions across the film and by Monte Carlo simulation. ... 

Numerics has been used intensively in the field of integrated optics (10) since its early days, simply due to the fact that even the basic example of the slab waveguide requires the solution of a transcendental equation in order to calculate the propagation constants of the slab- guided modes. Of course, the focus was directed to analytical methods, primarily, as long as the power of a desktop computer did allow for a few coupled equations and special functions only e.g. to describe the nonlinear directional coupler in a coupled mode theory (CMT) picture. During the years, lots of analytic and semi-analytic approaches to solve the wave equation have been developed in order... 

With the advent of molecular dynamics simulations applied to carbohydrates, one can anticipate the direct computation of more conceptually appealmg surfaces of V in 0s) space from a given U( qint,qext)) in the near future. Monte Carlo integration over (qext) and (b,x, 0h) for fix (0s) provides an alternative procedure, but one which is probably less attractive in terms of efficiency than the molecular dynamics approach. A second alternative, known as adiabatic mapping, provides an approximation to V((0s ), and applications of this method to carbohydrates have recently begun to appear. 12,13 in this approach the conformational... 

The concept of purification is well known in the linear-scaling literature for one-particle theories like Hartree-Fock and density functional theory, where it denotes the iterative process by which an arbitrary one-particle density matrix is projected onto an idempotent 1-RDM [2,59-61]. An RDM is said to be pure A-representable if it arises from the integration of an Al-particle density matrix T T, where T (the preimage) is an Al-particle wavefiinction [3-5]. Any idempotent 1-RDM is N-representable with a unique Slater-determinant preimage. Within the linear-scaling literature the 1-RDM may be directly computed with unconstrained optimization, where iterative purification imposes the A-representabUity conditions [59-61]. Recently, we have shown that these methods for computing the 1 -RDM directly... 

Computing the sensitivities is time consuming. Fortunately the direct integral approximation of the sensitivity matrix and its principal component analysis can offer almost the same information whenever the direct integral method of parameter estimation applies. 

In the case of the ultramicroelectrodes such as the disk electrode, it is necessary to integrate over the surface, and sometimes there will be unequally spaced points along the surface, as for example, in direct discretisation on an unequal grid in the example program UME DIRECT. As mentioned in Chap. 12, it is found that due to the errors in the computed concentration values, the local fluxes are so inaccurate that any integration method better than the simple trapezium method is not justified. The routine U TRAP is thus recommended here. It integrates local current densities, precalculated by using the above routine U DERIV. 

For these excitation parameters, existing spatial modes are all damped One set corresponding to the lower frequency (identified as 1 and 2 in Table 4.2) are below the neutral curve and the other set (B) corresponding to the higher frequency (identified as 3, 4 and 5 in the table) are above the neutral curve. In Fig. 4.10, the computed disturbance velocity in streamwise direction, obtained by Bromwich contour integral method is shown for the case of B (wq = 0.15). 

In Fig. 4.12, the computed disturbance velocity in streamwise direction, obtained by Bromwich contour integral method is shown for the case of the... 

A disadvantage of using Car-Parrinello path integral methods is that the molecular dynamics is used only to compute averaged properties, the simulation dynamics having no direct physical meaning. A recently developed, albeit approximate method for generating fully quantum mechanical dynamics is the ab initio centroid molecular dynamics method (Marx et al., 1999 Pavese et al., 1999). Tlie application of Car-Parrinello methods to... 

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