Regularization terms

The well-known maximum entropy method (MEM) can be implemented thanks to a non-quadratic regularization term which is the so-called negen-tropy ... 

Solving such a myopic deconvolution problem is much more difficult because its solution is highly non-linear with respect to the data. In effect, whatever are the expressions of the regularization terms, the criterion to minimize is no longer quadratic with respect to the parameters (due to the first likelihood term). Nevertheless, a much more important point to care of is that unless enough constraints are set by the regularization terms, the problem may not have a unique solution. 

The first term measures the difference between the data and the fit, KF. The second term is a Tikhonov regularization and its amplitude is controlled by the parameter a. The effect of this regularization term is to select a solution with a small 2-norm 11 F 2 and as a result a solution that is smooth and without sharp spikes. However, it may cause a bias to the result. When a is chosen such that the two terms are comparable, the bias is minimized and the result is stable in the presence of noise. When a is much smaller, the resulting spectrum F can become unstable. 

Again within the Matsubara technique one still should do the replacement lu -> tjn - 2mnT, -i f - T"=-oo/We dropped an infinite constant term in (14). However expression (14) still contains a divergent contribution. To remove the regular term that does not depend on the closeness to the critical point we find the temperature derivative of (14) (entropy per unit volume) ... 

The term is usually more complicated than given in Eq. (3.52) and will take into account non-regular terms such that... 

By hypothesis, the regular term Ehph(Z) is a (smooth) monotonic function without... 

The performance index J consists of two terms. The first, 7expt, is the data-fitting term, which measures the difference between experimentally observed and calculated values of the data. The second,. /re , is a regularization term and is used to impose any a priori knowledge, such as smoothness constraints, on the functional form of permeability distribution obtained as a solution. These terms take the following forms ... 

We now assume a > 0 so the temperature dependence of eq. (100) is more important than the regular term (cc t) that is also present. Here we used the fact that the singular part of short range correlations has the same singularity as the internal energy f7sing = (H sing = - oc The... 

A variety of techniques is nowadays available for the solution of inverse problems [26,27], However, one common approach relies on the minimization of an objective function that generally involves the squared difference between measured and estimated variables, like the least-squares norm, as well as some kind of regularization term. Despite the fact that the minimization of the least-squares norm is indiscriminately used, it only yields maximum likelihood estimates if the following statistical hypotheses are valid the errors in the measured variables are additive, uncorrelated, normally distributed, with zero mean and known constant standard-deviation only the measured variables appearing in the objective function contain errors and there is no prior information regarding the values and uncertainties of the unknown parameters. 

The first and the second term are regular terms whose dependence in S and k2 is in accordance with the scaling predictions. The last term is an additional term proportional to S. Let us set... 

The last term is a regular term and we may drop it. Thus, taking into account (14.6.36) and the equality z = 6Si/2(2jt) 3/2, we find that the singular part of C at the tricritical point is given by... 

The perturbation expansion (12) accordingly involves the expansion of a time-dependent phase containing information about the overall energy level shift o1 the system and lead to the appearance of time divergent terms. Such seculaj (as opposed to regular) terms may appear for time-dependent perturbation as well. Formally, they do not present any difficulties since they do not contribute to expectation values, but, as carefully analyzed by Langhoff et al. [10], thej lead to a rather cumbersome formalism. 

For the solution of the PDE models of the columns, a Galerkin method on finite elements is used for the liquid phase and orthogonal collocation for the solid phase. The switching of the node equations is considered explicitly, that is, a full hybrid plant model is used. The objective function F is the sum of the costs incurred for each cycle (e.g., the desorbent consumption) and a regularizing term that is added in order to smooth the input sequence in order to avoid high fluctuations of the inputs from cycle to cyde. The first equality constraint represents the plant model... 

For a Coulomb potential the cut-off function / varies from 0 to 1 as illustrated in Fig. 7. The iterative solution of (103) provides regular terms that finally lead to a regular expansion of the Bloch and des Cloizeaux effective Hamiltonians. The first terms are ... 

Therefore, in the function h, both the diverging term of order x and the dangerous regular term of order are subtracted by /oo(xo). We see that this is not a coincidence it is intimately related to the general form of the functions g and h (Equations 238 and 239) and it applies to (A ) and (ft ) (and actually also to all other physical quantities). Having applied Equations 240 and 241 to Equations 238 and 262, Duplantier has found for (/ ) (see Equation 260)... 

The regular part of the asymptotic expansion does not generally satisfy the boundary condition (6.2). In the terminology of the paper of Vishik and Lyusternik [27], the regular terms of the asymptotics introduce a discrepancy into the boundary condition. The purpose of the boundary layer functions Il,(p, is to compensate for this discrepancy. The equality (6.5) shows that the boundary layer functions together with the regular terms must satisfy the boundary condition (6.2). 

Here u k>Hk are the regular terms of the asymptotics, 0 and 0 are transition layer functions, and 11. are boundary functions. The variables p, are defined in an analogous way to that in Section VI.B. The variables t, 6 are introduced as follows. First we introduce new (local) coordinates (r, 0) in the vicinity of the curve F similarly to the introduction of local coordinates (r, ) in the vicinity of dft (see Section VI.B) 0 plays the same role as (we assume that 0 0 Itt), r is the distance from given point M to the curve Fq along the normal to Fq with a plus sign if M E ft, and with a minus sign if Af E ft. ... 

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