Big Chemical Encyclopedia

Chemical substances, components, reactions, process design ...

Articles Figures Tables About

L2 norm

The Manhattan distance is obtained for r = 1 and the Euclidean distance for r = 2. In this context the Euclidean distance is also referred to as the L2-norm. [Pg.67]

In the case of r = 2 we obtain the ordinary Euclidean distance of eq. (31.75), which is also called the L2-norm. In the case of r = 1 we derive the city-block distance (also called Hamming-, taxi- or Manhattan-distance), which is also referred to as... [Pg.147]

Thus, all of the harmonics i= TkX are stable under one and the same condition a > a0. We are going to show that the stability of scheme (16a) in every harmonic known as the spectral stability implies that in the grid L2-norm with respect to the initial data y(x, 0) = 0(x). where m0(x) is a grid function defined for 0 < x < 1 and vanishing at the points x — 0 and x = l. With this aim, the general solution of problem (16a) is sought as the sum of particular solutions having the form (22) ... [Pg.309]

This particular statement lacks rigor in the comparison between interfaces with the same amount of volume. A more precise statement involves those interfaces where the integral of the squared difference are equivalent. This is called the L2 norm on functions. [Pg.607]

One way of estimating the iteration convergence error is done by way of monitoring the normalized residual, Ruer, i.e. some norm (such as the L2 norm) over the computational cells of the remainder after the numerical solution is substituted into the discretized counter part of the PDE. An example of residual monitoring is depicted in Figure A. 1. It is seen in this case that the residual of each equation reduces to machine accuracy. [Pg.170]

In most cases, the loudness of the result of mixing two signals corresponds to the RMS value of the sum, and not the peak value, so scaling by the L2 norm is appropriate. Leaving headroom for the resulting signal s crest factor (the ratio of the peak to the... [Pg.470]

Alternative characteristic functions have also been used. For example, the sum of the absolute values of the residuals (53] has been evaluated for fitting C(11 data and produced results similar to those based on the chi-square statistic 49. When one sets w, = 1, the characteristic function in Eq. (52) is known as the L2 norm. [Pg.218]

The Born scries would be a powerful tool for EM modeling if they were convergent. However, in practice the condition (9.130) does not hold, because in a general case the L2 norm of the Green s operator is bigger than 1. That is why the Born series has not found a wide application in EM modeling. [Pg.257]

This norm is called an L2 norm. It follows that the distance between two functions will be measured as... [Pg.549]

The linear space of functions, integrable on the real interval [a, 6] and equipped with the L2 norm, is called L2[a, 6] space. A linear normed space L2[a, b] is a Hilbert space, and therefore, possesses all the properties of the Hilbert space discussed above. [Pg.550]

One can improve on the resolvability of the fine structures in Figure 4 by using a mapping involving the display of the two proxy quantities E (x) and kmax(x) as a function of the horizontal axis x, where E ax is the maximum of the L2-norm of the... [Pg.198]

Nonconserved Fields and the Allen-Cahn Equation. Nonconserved order parameters (sueh as the state of order itself as introduced in eqn (12.22)) can have complex spatial distributions that evolve in time. The governing equation in this case is provided by the Allen-Cahn equation. To obtain this equation using the ideas introduced above, we hark back to the L2 norm. It is assumed that the instantaneous free energy of the system can be written down as a functional of the order parameter and its gradients. For example, in the context of the Allen-Cahn equation, the relevant free energy functional is... [Pg.679]

As a simple example to test the efficiency of the one-sided difference operators, consider a rectangular cavity with a small metallic stub on its lower base. Figure 2.5 gives the temporal evolution of the global error (L2 norm) and Table 2.2 summarizes the dispersion... [Pg.25]

Another common measure of convergence utilizes the L2 norm. This can be termed the average error and can be associated with errors in any quantity. The L2 norm is defined as... [Pg.384]

While the absolute values given by the energy or L2 norms have little value, one can construct a relative percentage error that can be more readily interpreted ... [Pg.384]

The location of the function values >) must be determined in order to provide for the match-point. This is achieved by minimizing a goal function which measures the distances between the grey levels in template and patch. The goal function to be minimized in this approach is the L2-norm of the residuals of least squares estimation. The location is described by shift parameters Ax, Ay, which are counted with respect to an initial position of gix, y), the approximation of the conjugate patch g x, y). [Pg.341]

A common approach is the method of least squares (L2 norm) which leads to root mean squared residuals (where the residual is the difference between the observed and calculated travel times). However, the use of least squares procedures requires the assumption that the distribution of the residuals is of Gaussian nature (Mendecki 1997). This is generally not true. [Pg.112]

The way the time integration is carried out is as follows we start with a homogeneous distribution of Gaussian chains p/ = p/ (where p is the mean density of the fields) and [7/=0. We solve the CN equations (to be defined below) by an iterative process in which the external potential fields are updated according to a steepest descent method. The updated external potential fields are related, by the chain density functional, to unique density fields. We calculate the intrinsic chemical potential from the set of external potential fields and density fields and update the density fields. The density fields are only accepted as an update for the next time step if the L2 norm of the residual is below some predefined upper boundary. An overview of this scheme is shown in Fig. 12. [Pg.594]

Significant wave heidhi Hrr.i- [2 5 [m] Wave direction [l2 [ ] Normative storm duration M... [Pg.358]

As the grain boundary fails, the value of S gradually decreases from an initial value close to unity to zero, point at which complete failure is acheived. The coupling between normal and tangential failure in mixed-mode situations is captured by defining S in terms of the L2-norm of the normalized displacement jump vector A, as indicated by Equation 2 and illustrated in Figure 2. [Pg.207]

A characteristic of A for ill-posed problems is that it has a very large condition number. In other words, the ill-conditioned matrix A is very near to being singular. Briefly, the condition number is defined as k A) = A IIA II or the ratio of maximum to minimum singular values measured in the l2 norm. The ideal problem conditioning occurs for orthogonal matrices which have k (A) 1, while an ill-conditioned... [Pg.402]


See other pages where L2 norm is mentioned: [Pg.273]    [Pg.181]    [Pg.166]    [Pg.44]    [Pg.238]    [Pg.243]    [Pg.256]    [Pg.257]    [Pg.449]    [Pg.451]    [Pg.459]    [Pg.461]    [Pg.496]    [Pg.179]    [Pg.330]    [Pg.107]    [Pg.91]    [Pg.372]    [Pg.113]    [Pg.318]    [Pg.352]    [Pg.31]    [Pg.163]    [Pg.23]    [Pg.23]    [Pg.23]   
See also in sourсe #XX -- [ Pg.67 ]




SEARCH



NORM

Norming

© 2024 chempedia.info