Constrained regularization method

S.W. Provencher, A constrained regularization method for inverting data represented by linear algebraic or integral equations, Comput. Phys. Commun. 27 (1982) 213-227. 

Although the methods previously discussed have shown some promise in solving the Fredholm equation, two recent methods were chosen for intensive study and comparison with our GEX function fit method. These methods were the constrained regularization method, and the polynomial subdistribution method. 

For both the subdistribution and the GEX fit methods a Marquardt algorithm for constrained non-linear regression was used to minimize the sum of squares error (.10). The FORTRAN program CONTIN was used for the constrained regularization method. All computations were performed on a Harris H-800 super mini computer. 

FIGURE 11.7 (a) Affinity distributions obtained at different ionic strengths from experimental data for purified peat humic acid (PPHA) (b) deconvolution of the distribution obtained for PPHA with Gaussian functions. (Data from Milne. C.J. et al.. Environ. Sci. TechnoL, 37, 958-971, 2003 Reprinted from J. Colloid Interface Sci., 336, Orsetti, S. et al., Application of a constrained regularization method to extraction of affinity distributions Proton and metal binding to humic substances, 377-387. Copyright 2009, with permission from Elsevier.)... 

Orsetti, S., E. M. Andrade, and F. V. Molina. 2009. Application of a constrained regularization method to extraction of affinity distributions Proton and metal binding to humic substances. Journal of Colloid and Interface Science 336, no. 2 377-387. 

Provencher, S. W., A Constrained Regularization Method for Inverting Data Represented by Linear Algebraie Integral Equations, Comput. Phys. Commun., 1982, 27, 213-227. 

Shih WC, Bechtel KL, Feld MS. Constrained regularization hybrid method for multivariate calibration. Analytical Chemistry 2007, 79, 234-239. 

In the current implementation we employ the Tikhonov regularization method (Press et al., 1992. This approach is similar to the constrained density fitting algorithm of Misquitta and Stone (Misquitta and Stone, 2006. Here, the redundant basis set contributions are penalized by minimizing Eseit + resulting... 

There are many ways to solve this nonlinear equality-constrained optimization problem. One method is the quadratic model approximation [44]. The maximum entropy method can provide the least biased estimates on the basis of available information about the problems. The proven objectivity of the method relies on the fact that the solution is dictated purely by the experimental data and the nature of the problem, not by an arbitrarily introduced differential operator such as the ones may be found in the regularization method. 

These difficulties can be circumvented by using the adaptive biasing force (ABF) method of Darve, Pohorille, and coworkers [18, 28, 29], which is based on unconstrained molecular dynamics simulations. This is a very efficient approach which begins by establishing a simple formula to calculate d,4/d from regular molecular dynamics in which is not constrained. This derivative represents the mean force acting on . Therefore if we remove this force from the system we obtain... 

If a flexible, but constrained, modeling technique is used for the entire polymer, the shapes depicted as feasible should represent all the conformations that are actually allowed for regular helices. The difficulty with these methods is, however, that we do not yet know precisely the limits of monomeric flexibility. That point is the subject of some other current research (9). 

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