Regularization theory

Starting from the continuous chain model as the dimensionally regularized theory we write the renormalization factors as... 

In the dimensionally regularized theory the coefficients A are functions only of , which have to be chosen to cancel the singularities of the bare theory occurring for e —> 0. This goal can be reached with the ansatz... 

The formal solution of the ill-posed inverse problem could result in unstable, unrealistic models. The regularization theory provides a guidance how one can overcome this difficulty. The foundations of the regularization theory were developed in numerous publications by Andrei N. Tikhonov, which were reprinted in 1999 as a special book, published by Moscow State University (Tikhonov, 1999). In this Chapter, I will present a short overview of the basic principles of the Tikhonov regularization theory, following his original monograph (Tikhonov and Arsenin, 1977). 

Following classical principles of regularization theory (Tikhonov and Arsenin, 1977 Lavrent ev et al., 1986) we can give the following definition of the well-posed problem. 

The fundamental result of the regularization theory is that this operator, il(di,a), is a regularizing operator for the problem (2. A). I do not present here the proof of this result, which requires an extensive mathematical derivation, referring interested readers to the original monograph by Tikhonov and Arsenin (1977). 

To justify this choice we should prove that sms (j ) can actually be considered as a stabilizer according to regularization theory. According to the definition given above, a nonnegative functional s (m) in some Hilbert space M is called a stabilizing functional if, for any real c > 0 from the domain of the functional s (m) values, the subset Mf, of elements m M, for which s (m) < c, is compact. 

Note that there is no global minimum of the misfit functional if the solution of the original inverse problem (5.1) is nonunique. We know also that in this case we have to apply regularization theory to solve the ill-posed inverse problem. In this section, however, we will assume that misfit functional (5.2) has a global minimum, so there is only one point at which ( (m) assumes its least value. Our main goal will be to find this point. 

We apply the Tikhonov regularization theory to solve the linear system (10.65). 

The inverse problem (10.103) is usually ill-posed, i.e., the solution can be non-unique and unstable. The conventional way of solving ill-posed inverse problems, according to regularization theory (Chapter 2), is to minimize the Tikhonov parametric functional ... 

Following the general principles of regularization theory (Chapter 2), we solve the linear inverse problems (15.9) by regularization methods, imposing additional conditions on the class of inverse models with corresponding stabilizing functionals. 

In this appendix we will discuss the mathematical tools necessary for developing the regularization theory of inverse problem solutions. They are based on the methods of functional analysis, which employ the ideas of functional spaces. Thus we should start our discussion by introducing the basic definitions and notations from functional analysis. Before doing so, I remind the reader of the basic properties of the simplest and, at the same time, the most fundamental mathematical space - Euclidean space. 

The step of first taking the limit f —> 0 for d < 4 to arrive at a finite unrenormalized theory without cut-off is known as dimensionai regularization in contrast to the cut-off regularization of the discrete chain model. The breakdown of the dimensionally regularized theory as d — 4 shows up in pole terms v. These terms have to be absorbed into the renormal-... 

The standard technique concerning regularization theory is essentially based on three steps ... 

Figure 1. Alice s adventures in wonderland offer a useful analogy to remind the basic steps of regularization theory a) a change of coordinates, b) the introduction of a fictitious time, c) the preservation of energy.

Basics of regularization theory We define the Jacobi constant as... 

Basics of regularization theory whose determinant takes the form... 

Acknowledgements The author is deeply greatful to Paola Celletti for hand-drawing the pictures concerning Alice s adventures in wonderland , whose aim was to help the reader to remind easily the basics of regularization theory. 

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