Method of Regularization

This section presents a simplified treatment of the technique of Tikhonov regularization. The idea of regularization is to convert a original ill-posed problem (which means that the error in the solution is magnified by errors in the input data) into a well-posed problem for which the error in the solution is under control. More specifically, consider the solution of the following Fredholm integral equation of the first kind. 

the ill-posed problem of a Fredholm equation of the first kind is approximated by the well-posed problem of the second kind. 

The strategy for choosing a is to minimize the functional J(u). The criterion is called the L criterion in view of the shape of the function J(u). 

For the inverse problems of chemical engineering interest, the reader is referred to Ramkrishna and coworkers (Sathyagal et al. (1985), Muralidhar and Ramkrishna (1989), Wright and Ramkrishna 

FIGURE 2.2 Total error of approximation as represented by the functional /( ). 

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One of the important corollaries to stability theory is the general method of regularization in the class of stable schemes (by changing the operators A and B) for the design of schemes of a desired quality. 

The analysis of variance and the method of regular regression are used in evaluating the results. Equation improvements are discussed. The difficulties in predicting surface tensions of polymers on the basis of bulk properties are emphasized. 

The main goal of this book is to present a detailed exposition of the methods of regularized solution of inverse problems based on the ideas of Tikhonov regularization, and to show different forms of their applications in both linear and nonlinear geophysical inversion techniques. 

In the previous sections we have seen several examples of transport problems that are amenable to analysis by the method of regular perturbation theory. As we shall see later in this book, however, most transport problems require the use of singular-perturbation methods. The high-frequency limit of flow in a tube with a periodic pressure oscillation provided one example, which was illustrative of the most common type of singular-perturbation problem involving a boundary layer near the tube wall. Here we consider another example in which there is a boundary-layer structure that we can analyze by using the method of matched asymptotic expansions. 

Stiefel, E. L. and ossler, M. R and Waldvogel, J. and Burdet, C. A. (1967) Methods of regularization for computing orbits in Celestial Mechanics, NASA Contractor Report CR-769, Washington... 

Jorg Waldvogel (1967b). The restricted elliptic three-body problem. In E. Stiefel et. al., Methods of Regularization for Computing Orbits in Celestial Mechanics, NASA Contractor Report NASA CR 769, 88-115. 

Introducing a cut-off of the Coulomb potential at short distances introduces some arbitrariness. A method of regularization that works with the unmodified Coulomb potential is to take into account the anomalous magnetic moment of the electron. The Dirac operator now reads (in dimensionless units)... 

The theory just described was not predictive because its renormaliz-ability was only conjectured, not proven. The difficulties to be overcome were largely technical, being associated with problems of choice of gauge, method of regularizing infinities, and Fadeev-Popov ghosts. When they... 

A. Tikhonov and V. Arsenin. Solution of Ill-Posed Problems. Winston, Washington, DC, 1977. This is the book in which Tikhonov describes his method of regularization for ill-posed problems. 

A dimensional regularization is one of the methods of regularization, based on the idea to use the space dimensionality d as a continuous variable. Application of the method begins with calculating the integrals at d < 2 to yield forms that are subsequently extrapolated to any d (Kholodenko and Freed, 1983). 

Each method is accompanied by at least two worked-out examples demonstrating the important steps in the construction of a solution to a given problem. Also, the method of regular perturbation, a technique that can be very helpful in estimating a solution to some nonlinear problems, is briefly introduced in this chapter. 

One-dimensional models for inviscid, incompressible, axisymmetric, armular liquid jets falling under gravity have been obtained by means of methods of regular perturbations for slender or long jets, integral formulations, Taylor s series expansions, weighted residuals, and variational principles [27, 47]. 

In books on field theory one starts with the Lagrangian in which all the parameters, coupling and masses are called bare parameters and are labelled eo, 50, "lo etc. These bare parameters are not what one measures to make the theory finite they have to be allowed to depend on a cut-off A temporarily introduced into the theory and most of them become infinite when at the end one lets A -> 00. What A is depends upon the method of regularization . Thus A may literally be a cut-off, i.e. the upper limit of some loop integration, or it may be 1/e where one uses dimensional regularization to work in 4 — e dimensions and at the end lets c 0. 

However, when we try to compute the LHS of (9.5.14) we get the difference of two divergent integrals. The diagram for Afj, p has, of coiurse, to be subjected to renormalization to make it finite and when this is done it can be arranged that (9.5.14) comes out correctly, but now (9.5.15) fails to hold It can be shown that there is no consistent method of regularization such that both (9.5.14) and (9.5.15) hold. One of the bizarre features is that the result depends upon the choice of integration variables, i.e. 

Utilizing this complex methodology, we obtain decision rules and results of activity prediction that are context-independent from the training set composition, the methods of compound structure representation, and the methods of regularity detection. 

Tikhonov, A. N. (1963). On the solution of incorrectly stated problems and a method of regularization. Dokl. Acad. Nauk, USSR, 151, 501-4. 

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