Localization accuracy

It has been pointed out by Bartlett (1955) that a much more sensitive criterion for the local accuracy of an approximate wave function 0 is provided by the local energy defined by the relation... 

The first term m V is a function of x only. Let us assume that we are using the velocity Verlet time integrator, which is the most common. In that case, x is computed with local accuracy 0 dtA) and global accuracy 0(df2), and the velocity v at half-steps is computed with accuracy Oidt2 ) if the following approximation is used ... 

Approximate linear dependence of AO-based sets is always a numerical problem, especially in 3D extended systems. Slater functions are no exceptions. We studied and recommended the use of mixed Slater/plane-wave (AO-PW) basis sets [15]. It offers a good compromise of local accuracy (nuclear cusps can be correctly described), global flexibility (nodes in /ik) outside primitive unit cell can be correct) and reduced PW expansion lengths. It seems also beneficial for GW calculations that need low-lying excited bands (not available with AO bases), yet limited numbers of PWs. Computationally the AOs and PWs mix perfectly mixed AO-PW matrix elements are even easier to calculate than those involving AO-AO combinations. 

In addition to the restrictions that stability places on the step size hn, we also need to be concerned with how accuracy affects the choice of step size. Assume that the local accuracy is to be controlled to within a certain tolerance e and that accuracy can be estimated by the local truncation error. The time step must be chosen to keep a norm of the local truncation error below the tolerance, that is ... 

This parameter enables the user to specify the desired local accuracy of the computed solution via the error tolerances Rtol and Atol. The simplest way is to take Rtol and Atol to be scalars. For greater flexibility (different tolerances for different components of U), Rtol and Atol can be specified as arrays. The following options are available ... 

The particulate sensors will be highly sensitive to atmospheric tests (threshold lower than 1 kt) and will provide unambiguous evidence of a nuclear explosion, although sensor response time is slow and localization accuracy limited. For... 

High Localization Accuracy of Single Molecules Down to a Single Channel Limit... 

EXFOV (0.5 sensitivity) EXFOV (0.0 sensitivity) Angular Resolution Localization Accuracy Sensitivity (at 2 a for 20 h obs.) Angular Mask El. Size Angular Pixel Size... 

The ANN-based dynamic modulus models using the same input variables were found to exhibit significantly better overall prediction accuracy, better local accuracy at high and low temperature extremes, less prediction bias and better balance between temperature and mixture influences than do their regression-based counterparts (Ceylan et al. 2009). 

The easiest way to locate the source of acoustic emissions is the so-called zone location method, where the exact source coordinates are not determined. Here, localization means detecting a signal and the radius (or in plate structures, the zone) of the sensor sensitivity range defines the localization accuracy. However, this is also the most imprecise localization method. The principal of zone localization is simple. For a particular sensor distribution, the sensor that records the arrival of the elastic wave first is the sensor closest to the source. An example for such a case is shown in Fig. 6.4. 

This localization method only requires the coordinates of the sensors. However, an exact determination of the source location is not possible. The geometry of the structure being tested can influence the 1-dimensional (1-D) localization accuracy. For example, the expansion of a wire-like structure is very high in one direction compared to the expansion in the two other directions, allowing more detailed 1-D localization. In this case, the source can be determined as a 1-D point between two sensors (Fig. 6.2) (Reinhardt and Grosse 1999). However, the speed of sound of the structure being tested is needed. 

Due to the fact that generally one arrival time is greater than the other e.g. 6 > the epicenter loeation is limited to one branch of the hyperbola. The hyperbolas of the other station pairs ti, ts and t2, ts) are caleulated in a similar way. The epicenter is the interseetion point of the 3 h erbolas (Fig. 6.5). Due to measurement errors the three h erbolas may not intersect at one point. For such a case, using more than 3 sensors should improve the localization accuracy and statistical methods must be applied. For example, Joswig [2004] uses a jackknife test to improve the results of the hyperbola method in an overdetermined case. 

The use of more sensors than unknown location parameters results in the system of equations being over-determined, and statistical methods such as the methods of least-squares can be used. Such approaches no longer fall into the category of manual localization methods and are generally solved using a computer. Usually these methods improve the localization accuracy. 

The localization accuracy is influenced by various factors. In the following table some measures to ensure high localization accuracy are listed and discussed. 

I time determination tion leads to higher localization accuracies. 

If lateral velocity variations are present within the test specimen, and lead to uncertainties in the measured travel times at distant stations, the application of a relative localization procedure could improve the localization accuracy. This procedure can be only applied to a local region, i.e. the interevent distances must be small compared to the event-receiver distance. The events within this region are relocated relative to one particularly well-located event, called master event. It is assumed that these velocity variations have nearly the same effect on all events from the local region (Shearer 1999). Eq. 6.7 can be rewritten as ... 

The accuracy of localized events with known source position, i.e. artificial sources on a surface like pencil lead breaks, can be plotted as an error vector r, connecting the calculated with the known position. Fig. 16.5 (a). By converting the absolute values of the error vectors into a density function an image of localization accuracy can be generated that allows for better interpretation, Fig. 16.5 (b). 

However, there is no unique or optimal measure of accuracy. If MD simulations are performed, the main interest lies in errors connected with the force, while in Monte Carlo (MC) simulations one is concerned with accurate energies. In the simulation of ensemble averages it is the global accuracy, measured, e.g., by root-mean-square quantities, which is important, hi the simulation of rare events local accuracy and maximal errors are relevant as well. Errors in the force can be due to their magnitude or due to their direction. And finally, one might be interested in absolute or relative errors. 

One of the modern hi -accuracy experimental methods is tomography atom probe (TAP). It enables the investigation of reactions in thin metallic diffusion couples by means of spatially resolved chemical analysis with a local accuracy up to interatomic distance [11, 12]. This technique was also applied to reactive diffusion in the thin film system Al-Co within the temperature range of 200-400 °C [6]. In accordance with the previous results obtained by the authors [13, 14] by DSC, the first phase to nucleate and grow in the temperature interval from 200 to 400 °C is AI9C02. The authors of [6] have established that in the diffusion zone between A1 and Co formation and growth of a solid solution layer occurs first, and nucleation of AI9C02 particles is never observed before the diffusion zone reaches about 3-4 nm thickness. Thus, the appearance of the product phase is controlled by the width... 

Thus, by halving the step size the local residual is reduced by a factor 2 + if the method has the order of consistency p. Consequently, one is interested in methods of high order, as in these cases one expects a gain of (local) accuracy when decreasing step sizes. 

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