Constraining parameter

Use a pseudo reaction coordinate with one parameter constrained using successively smaller steps for the constrained parameter until the desired accuracy is reached. 

In parameter estimation we are occasionally faced with an additional complication. Besides the minimization of the objective function (a weighted sum of errors) the mathematical model of the physical process includes a set of constrains that must also be satisfied. In general these are either equality or inequality constraints. In order to avoid unnecessary complications in the presentation of the material, constrained parameter estimation is presented exclusively in Chapter 9. 

The above constrained parameter estimation problem becomes much more challenging if the location where the constraint must be satisfied, (xo,yo), is not known a priori. This situation arises naturally in the estimation of binary interaction parameters in cubic equations of state (see Chapter 14). Furthermore, the above development can be readily extended to several constraints by introducing an equal number of Lagrange multipliers. 

Most of the constrained parameter estimation problems belong to this case. Based on scientific considerations, we arrive quite often at constraints that the parameters of the mathematical model should satisfy. Most of the time these are of the form,... 

Historically, this has been the most constrained parameter, particularly for confocal laser scanning microscopes that require spatially coherent sources and so have been typically limited to a few discrete excitation wavelengths, traditionally obtained from gas lasers. Convenient tunable continuous wave (c.w.) excitation for wide-held microscopy was widely available from filtered lamp sources but, for time domain FLIM, the only ultrafast light sources covering the visible spectrum were c.w. mode-locked dye lasers before the advent of ultrafast Ti Sapphire lasers. 

For comparison with our results in table 3, which presents values of 20 adjusted parameters with 15 parameters constrained to define the rotational g factor, Dulick et alii [115] required also 20 adjusted parameters, with a constrained parameter T> for the equilibrium binding energy for a function of potential energy having a modified Morse form. The latter parameter is specified as... 

Table 6. Coefficients of radial functions and other molecular parameters of NaCl X 2, depending on extent of data set and constrained parameters...

A set 6c of actively constrained parameters, which currently lie at a limit of Eq. (6.4-2). Nonzero entries i or —i in the vector LBOX identify any such parameters and the active limit (upper or lower) for each. 

Should A be singular, this method still gives a basis inverse, namely, the inverse of the submatrix of A in which the pivots were found. A version of this algorithm using strictly positive, diagonal pivots is used for constrained parameter estimation in Subroutine GREGPLUS, as described in Sections 6.4 and 7.3 there the determinant of the basis submatrix is obtained directly as the product of the pivotal divisors. Basis inverse matrices also prove useful in Sections 6.6 and 7.5 for interval estimation of parameters and related functions. 

These two relationships can be easily programmed in GSAS and in the majority of Rietveld software codes by using a constraint apparatus, which was briefly discussed above (see section 7.3.3 and Eq. 7.9). Since the constraints affect only the shifts that are determined during every least squares refinement cycle but not the values of the related parameters, the latter should be synchronized manually prior to imposing constraints. For example, in our case when the computed shift for g-v2a is 0.02, then the new values of the constrained parameters (Eqs. 7.9, 7.11 and 7.12) will be calculated as follows ... 

Another example of synchronizing constrained parameters can be given considering the coordinates of VI and 02 - 05 atoms in this structure. All of them are located in the 6(c) sites, with the coordinates of the independent atom xxz. Hence, when entering the respective coordinate parameters it is necessary to ensure that y = -x. 

SOME PARAMETERS WERE LOG OR LOGIT TRANSFORMED TO CONSTRAIN PARAMETERS... 

The problem of evolution of stars to their explosion and subsequent interaction with the circumstellar medium has many time scales (ranging from tens of milliseconds to tens of thousands of years) and macroscopic length scales (from dimensions effectively that of a white dwarf to that of a supernova remnant, i.e from few thousand kilometers to many tens of light years). The physics of supernova explosions is complex and diverse and in many parts, the explosion mechanism is still an unsolved problem. Even the constraining parameters and ingredients which makes the SN explode are still controversial (see e.g. the discussion in [96]). It is possible that the identification of the key... 

Heiser WJ, Kroonenberg PM, Dimensionwise fitting in PARAFAC-CANDECOMP with missing data and constrained parameters, Internal Report PRM 97-01, University of Leiden, The Netherlands, 1997. 

RI is a derived parameter with an error summarized from those of the components (baseline, intensity, FWHM - is estimated at ca. 10% relative %) constrained parameters... 

The resulting density matrix is a function of the constraining parameters p = p(Ai, A2). Because P is time-odd p (A 1, 2) has a time-odd part representing a net collective current. 

---