Frechet derivative

The Lagrange multiplier p. determined by normalization, is the chemical potential [232], such that pt = dE/dN when the indicated derivative is defined. This derivation requires the locality hypothesis, that a Frechet derivative of Fs p exists as a local function (r). 

Since J2ini = N anexclusion principle for any compact system with more than two electrons. The failure of this sum rule implies that in general the assumed Frechet derivative of l s [p] cannot exist for more than two electrons, and there can be no exact Thomas-Fermi theory. 

If the hypothetical Frechet derivative vT(r) could be replaced by the operator t when acting on occupied orbital functions , there would be no contradiction. It... 

This defines a Gateaux functional derivative [26, 102], whose value depends on a direction in the function space, reducing to a Frechet derivative only if all e, are equal. Defining Tt = t + v, an explicit orbital index is not needed if Eq. (5.10) is interpreted to define a linear operator acting on orbital wave functions, TL — v = t. The elementary chain rule is valid when the functional... 

This argument shows that the locality hypothesis fails for more than two electrons because the assumed Frechet derivative must be generalized to a Gateaux derivative, equivalent in the context of OEL equations to a linear operator that acts on orbital wave functions. The conclusion is that the use by Kohn and Sham of Schrodinger s operator t is variationally correct, but no equivalent Thomas-Fermi theory exists for more than two electrons. Empirical evidence (atomic shell structure, chemical binding) supports the Kohn-Sham choice of the nonlocal kinetic energy operator, in comparison with Thomas-Fermi theory [288]. A further implication is that if an explicit approximate local density functional Exc is postulated, as in the local-density approximation (LDA) [205], the resulting Kohn-Sham theory is variation-ally correct. Typically, for Exc = f exc(p)p d3r, the density functional derivative is a Frechet derivative, the local potential function vxc = exc + p dexc/dp. 

Thomas-Fermi theory) requires a Frechet derivative for the kinetic energy, and cannot exist for more than two electrons [288],... 

This proposition has been tested in the exact-exchange limit of the implied linear-response theory [329], The TDFT exchange response kernel disagrees qualitatively with the corresponding expression in Dirac s TDHF theory [79,289]. This can be taken as evidence that an exact local exchange potential does not exist in the form of a Frechet derivative of the exchange energy functional in TDFT theory. 

Here, Df Dpf, D2J, D2cpf are the Frechet derivatives of the nonlinear operator / and D2caf (c cj), is a symmetric bilinear form. 

The accuracy of the averaged model truncated at order p9(q 0) thus depends on the truncation of the Taylor series as well as on the truncation of the perturbation expansion used in the local equation. The first error may be determined from the order pq 1 term in Eq. (23) and may be zero in many practical cases [e.g. linear or second-order kinetics, wall reaction case, or thermal and solutal dispersion problems in which / and rw(c) are linear in c] and the averaged equation may be closed exactly, i.e. higher order Frechet derivatives are zero and the Taylor expansion given by Eq. (23) terminates at some finite order (usually after the linear and quadratic terms in most applications). In such cases, the only error is the second error due to the perturbation expansion of the local equation. This error e for the local Eq. (20) truncated at 0(pq) may be expressed as... 

Equation (5.7) can be simplified using the properties of the adjoint of the Frechet derivative operator, considered in Appendix D. 

The notion of the adjoint operator makes it possible to move a linear operator of a Frechet derivative, Fm, from the left to the right-hand side of the inner product in equation (5.7) ... 

Note that the steepest descent method can be applied to the solution of the linear inverse problem as well. In this case the Frechet derivative of the linear operator A is equal to the operator itself ... 

Calculation of the first variation (Frechet derivative) of the electromagnetic field for 2-D models... 

We can find the equations for the Frechet derivative in the 3-D case by differentiating the corresponding Maxwell s equations. Let us write the general field equations (8.56) and (8.57) in the frequency domain, allowing for magnetic currents ... 

Formulae (9.59) and (9.60) play an important role in electromagnetic inversion. 9.1.7 Frechet derivative calculation using the differential method We now present another way of calculating the Frechet derivative using the differential method, proposed by McGillivray et al., 1994. 

McGillivray, P. R., and D. W. Oldenburg, 1990, Methods for calculating Frechet derivatives and sensitivities for the nonlinear inverse problem a comparative study Geophys. Prosp., 38, 499-524. 

Note that comparing formulae (10.12), (10.13) and (9.51), (9.52), we see that the expressions in the right hand sides of the Born approximations can be represented as the Frechet derivative operators calculated for the background conductivity db and the anomalous conductivity A5 ... 

From the last formula we conclude that the adjoint Frechet derivative operator Fg is given by the formula ... 

We can see that practical implementation of this algorithm requires computing the adjoint of the Frechet derivative Pg Hn corresponding forward modeling... 

In particular, considering the infinitely small domain of the conductivity perturbation, we arrive at the following formula for the Frechet derivative of the electric field ... 

The last formula provides an analytical expression for computing the Frechet derivative for the forward modeling operator. Note that, in this case, the amount of calculation for the forward modeling solution and for the Frechet derivative is equivalent to computing the Born approximation. 

Let us consider the derivation of the Frechet derivative matrix of the discrete forward operator (10.103). Noting that the model parameters are the anomalous conductivity values in the cells of the anomalous body, that matrix A is independent of the model parameters, and that B is a diagonal matrix, one can express the perturbation of the forward operator (10.103) with respect to the model parameters in the form... 

Frechet derivative matrix vMh respect to the logarithm of the total conductivity... 

The most critical part of the RCG algorithm (12.87) is computing the Frechet derivative matrix, F , or applying the adjoint Frechet derivative matrix to the weighted residual field, F WdR . We will discuss below the solution of this problem in a discrete case. 

Frechet derivative calculation using finite difference methods The problem of the Frechet derivative, or sensitivity matrix calculation for electromagnetic field has been examined in many publications and was discussed in Chapter... 

We denote by the element of the vector corresponding to the r, -th receiver position. It can be treated as an electric field, generated by an electric source Sae, located in the cell Vq. The Frechet derivative matrix, F, is formed by the components 6ej /6a. Therefore, the direct, brute force method of computing the Frechet matrix would require Nm forward modeling solutions for each inversion iteration. 

We can reduce significantly the number of the required iterations using the reciprocity principle (Madden, 1972 Rodi, 1976 Madden and Mackie, 1989 McGillivray and Oldenburg, 1990 de Lugao and Wannamaker, 1996 and de Lugao et ah, 1997) for Frechet derivative calculations. Due to reciprocity, the field is equal to the... 

Calculation of the first variation (Frechet derivative) of the vector wavefield We begin this section with an analysis of the equation for the vector wavefield variation. This equation can be derived by applying the perturbation operator to both sides of the vector Helmholtz equation (14.67), expressed in terms of the slowness function s(r). 

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 ... 

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