Differential Frechet

The Frechet differential is defined by Equation (2.6) when the norm of h is used for its size. In this case, therefore, the ratio (/i)/ /i is required to 

Dividing both sides of Equation (2.6) by h and taking its limit to zero, we get the following equivalent definition for the Frechet differential  

Placing no restriction on the form of h, i.e., the shape of the curve h x), this requirement implies that the error should decrease uniformly with h, as explained below. 

These changes are different forms of h and are shown as arrow vectors. Thus, hk i is the first change having the norm (or radius) k with a component along each of the two mutually perpendicular axes. The norm is given by 

Equation (2.3). Having the same norm k, hk,2 is the second change in a different direction. The existence of the Frechet differential at yo requires that the ratio e(/t)/ /t should decrease as any h having some norm k or less is downsized to any h having a smaller norm. 

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Davies, L., An efficient frechet differentiable high breakdown multivariate location and dispersion estimator, J. Multivar. Anal., 40, 311-327, 1992. 

Because 6G is linear in its second argument, it must be expressible as a weighted integral of s(z) the weighting function, however, may still depend on the particular point (z) in Hilbert space where the Frechet differential is evaluated. Hence, the weighting function is given by a functional G [ ] of n z), which depends parametrically on x G [ ] is called the functional derivative of G[ ]. One has... 

Substituting formulae (9.25) and (9.26) into (9.27), we find the expressions for the corresponding Frechet differentials ... 

The last two equations provide a straightforward way to calculate the Frechet differential (for definition see Appendix D) of the electromagnetic operator for forward modeling of complex three-dimensional electrical structures ... 

Note that the arguments in the expressions for the Frechet differentials, F/. //(5,6ct), consist of two parts. The first part, d, is a conductivity distribution, at which we calculate the forward modeling ojicrator variation, the Green s tensors are... 

We call these expressions the Frechet differentials as well (see Chapter 9 and Appendix D). 

Therefore, the Frechet differential of the forward modeling wavefield operator is given by the following expression ... 

Comparing formulae (15.5), (15.6) and (14.29), (14.84), we see that Born approximations can be expressed as the Frechet derivative operators (or Frechet differentials) calculated for the background square slowness s and the anomalous square slowness As ... 

In Chapter 14 we developed expressions for the Frechet differentials of the forward modeling wavefield operators (14.29) and (14.84), which we reproduce here for convenience ... 

Note that vector u in expression (15.241) represents the elastic field for the given velocities Cp and Cj, and the Green s tensor is calculated for the same Lame velocities. We will use below the following simplified notation for the Frechet differential... 

From the last formula we obtain the expression for the Frechet differential of the functional g ... 

The differentiability of different functionals used in density-functional theory (DFT) is investigated, and it is shown that the so-called Levy Lieb functional FLL[p] and the Lieb functional FL[p] are Gateaux differentiable at pure-state v-representable and ensemble v-representable densities, respectively. The conditions for the Frechet differentiability of these functionals are also discussed. The Gateaux differentiability of the Lieb functional has been demonstrated by Englisch and Englisch (Phys. Stat. Solidi 123, 711 and 124, 373 (1984)), hut the differentiability of the Levy-Lieb functional has not been shown before. 

Another form of functional differential is the Frechet differential, also termed the strong differential, which can be defined as follows ([13], p. 37), ([14], p. 451). With M being a subset of a Banach space E with the norm ll-ll, the function/ M — R is Frechet differentiable at a point x0 E M, if there exists a continuous and linear operator, L(-) E — R, such that (1)... 

The function L(h) is the Frechet differential at the point x0 in the direction h, and the operator L( ) is sometimes termed the Frechet derivative at x0. Here, however, we shall reserve the term Frechet derivative to a function equivalent to the Gateaux derivative (32). Frechet differentiability is a stronger requirement than that of Gateaux differentiability, but—with the interpretations we make—the difference is quite subtle. 

In order to demonstrate the difference further, we shall assume that the functional is Gateaux differentiable and then find out what the additional requirement is to make it Frechet differentiable. If we make the replacement A — A/z and set h = 1, the expression (34) becomes... 

If we assume that indicates some kind of singularity. 

Frechet differentiability is obviously a stronger condition than Gateaux differentiability. Therefore, if a functional is Frechet differentiable, it is also Gateaux differentiable, and the Frechet and the Gateaux derivatives are identical. 

Having considered the Gateaux differentiability of FLL[p] at PS-v-representable densities, we may now turn to the question of possible Frechet differentiability at these densities. The condition (34) is then... 

Using the same arguments as in the previous case, it seems that functionals of physical interest are also Frechet differentiable at the Zs-v-representable densities. 

The constant is here undetermined, due to the normalization constraint, J drp(r) = 0. If the functional LL [p] is Gateaux (Frechet) differentiable, then its Gateaux (Frechet) derivative will vanish at the minimum ([14], p. 460). 

The procedure we have applied does not depend on the (global) convexity of the functional, and we have been able to demonstrate the Gateaux differentiability of the Levy-Lieb functional at all PS-v-representable densities, where this functional is locally convex. It seems plausible that both these functionals are also Frechet differentiable at the same densities, although we have not been able to find a rigorous proof. 

Figure 2.3 Changes in a two-component function for a Frechet differential...

In general, we have the following condition for a Frechet differential to exist. As long as ft. decreases, the ratio (/i)/ /i, and consequently the error e h), should decrease uniformly with h, i.e., regardless of its direction. If this condition is satisfied, then at some sufficiently small and non-zero norm m of h, the error itself would become zero. Then for any h having the norm m or less, the functional change is representable by the Frechet differential, which is a linear and continuous functional of h. 

Figure 2.4 Changes in a continuous function for the Frechet differential...

The Frechet differential is based on the aforementioned condition of uniform error disappearance. This condition is too stringent to be satisfied by a large class of functionals. If we loosen that condition by preserving the direction of h during its downsizing, the resulting differential is known as the Gateaux differential. Denoted by d/(t/o h), it is a linear and continuous functional of h defined by... 

The Frechet differential of a functional is also the Gateaux differential. In turn, the Gateaux differential of a functional is also the variation. Thus, for a functional, the existence of the Frechet differential implies the existence of the Gateaux differential. In turn, the existence of the Gateaux differential implies the existence of the variation. However, there is no guarantee that the reverse relations hold. For example, a functional may have the variation but not the Gateaux differential. Using conditional statements, these relations are... 

If the variation SI y, h) is equivalent to the Frechet differential, then the above statement must hold for the variation 51 y h) substituted for d/(y h). 

Weak continuity of 51 means the continuity of 51 with respect to y for each fixed 5y. This is a relaxed requirement. Not fixing 5y would have imposed the demand for the Frechet differential instead of the variation of I. 

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