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Stabilizing functionals revisited

The main role of the stabilizing functional (a stabilizer) is to select the appropriate class of models for inverse problem solution. The examples listed above show that there are several common choices for a stabilizer. One is based on the least squares criterion, or, in other words, on the Lo norm for functions describing model parameters  [Pg.45]

In the last formula we assume that the function m (r), describing model parameters, is given within a t hree-dimensional domain 1, and r is a radius-vector of an observation point. The conventional argument in support of the norm (2.51) comes from statistics and is based on an ivssumption that the least sciuarcs image is the best over the ent ire ensemble of all ))ossible images (sec, for example, Chapter. 3). [Pg.45]

Another stabilizer uses a minimum norm of difference between a selected model and some a pn.ori model m j,r  [Pg.45]

The minimum norm criterion (2.51), as applied to the gradient of the model parameters Vm, brings us to a maximum smoothness stabilizing functional  [Pg.45]

In some cases, one can use the minimum norm of the Laplacian of model parameters [Pg.46]


Here we revisit the VB resonance model in formamide by taking all six resonance structures into account. Such a study allows us to compare the individual contributions from resonance structures 1-6 to the resonance effect in formamide. For comparison, the isoelectronic systems vinylamine and formamidine are also investigated to gain insights into the trends of resonance stabilization. A 6-31G(d) basis set is employed in the calculations, and the orbitals in the VB functions are self-consistently determined for each resonance structure, but restricted to be atomic orbitals. The structural weights of the six resonance structures are listed in Table 2. [Pg.168]

In this paper we have examined commercial equilibrium catalysts (Beat) to look for correlations between zeolite and matrix area stability of various catalyst families as a function of sodium. We have performed density separation of Feat samples to measure the sodium distribution as a function of catalyst age. Finally, we have revisited the issue of whether fresh catalyst sodium is equivalent to feedstock sodium in its influence on zeolite stability, catalytic activity and selectivity. [Pg.160]


See other pages where Stabilizing functionals revisited is mentioned: [Pg.45]    [Pg.45]    [Pg.469]    [Pg.35]    [Pg.190]    [Pg.331]    [Pg.65]    [Pg.794]    [Pg.105]    [Pg.414]    [Pg.345]    [Pg.174]   


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