The Tikhonov regularization method

Different modifications of least square solutions of linear inverse problems have resulted from the straightforward minimization of the corresponding misfit functionals. However, all these solutions have many limitations and are very sensitive to small variations of the observed data. An obvious limitation occurs when the inverse matrices (A A) or (A W A) do not exist. However, even when the inverse matrices exist, they can still be ill-conditioned (become nearly singular). In this case our solution would be extremely unstable and unrealistic. To overcome these difficulties we have to apply regularization methods. 

Let us consider first the general approach based on the Tikhonov regularization technique (Tikhonov and Arsenin, 1977). The corresponding parametric functional can be introduced in the following form  

In a majority of practical applications we assume that Wm = I, but it also can be chosen arbitrarily (for example as a matrix of first or second order finite-difference differentiation to obtain a smooth solution). We will discuss some specific choices of Wm later. 

According to the basic principles of the regularization method, we have to find a quasi-solution of the inverse problem as the model ma that minimizes the parametric functional 

The regularization parameter a is determined from the misfit condition  

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Alvares et. al. [141] successfully applied a method known as the Tikhonov Regularization method and L-curve criterion to generate data in close accord with the Malvern software. 

Our goal is to find the parameters inverse problem. The inverse problem is ill-posed. To solve it, we use the Tikhonov regularization method and minimize the parametric functional with the appropriate stabilizer s cr) ... 

The Tikhonov Regularization Method. We shall dwell briefly on the mathematical aspect of solving the first-kind Fredholm equation by the Tikhonov regularization method, referring basically to the works that deal with the use of this method in the EXAFS spectroscopy [41-44]. The integral equation (92) can be presented in the operator form... 

Experience in using the iterative procedure for solving the inverse problem by the Tikhonov regularization method in EXAFS spectroscopy shows that the obtained solution can be improved by performing a few iterations. 

The Tikhonov regularization method as applied to solving the inverse problem in the SEFS method has the advantage that it allows one to take into account oscillations of two types (oscillations determined by different wave numbers) in the kernel of the integral operator. 

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... 

As was shown previously [9] Eq. (1) is reduced to the Fredholm integral equation of the first kind, which yields function /(P) after solution via the Tikhonov regularization method. This inverse problem was solved on the basis of an algorithm from [9]. As a result, the function of the distribution over kinetic heterogeneity in /(lnP)-lnM coordinates with each maximum related to the functioning of AC of one type was obtained. 

The previous method may not lead to a unique, physically meaningful spectrum, because the value of G, depends on those of A, and N and contains choices, especially as N increases, it can lead to negative values of G, which are generally thought to be physically unreahstic. To select the true solution from the set of possible solution, Honerkamp and Weese [8, 9] introduced an additional criterion by means of the Tikhonov regularization method. They propose the use of the following condition in place of Eq. (10) ... 

According to the conventional Tikhonov regularization method, we substitute for the solution of the linear inverse problem (10.16) a minimization of the corresponding parametric functional with, for example, a minimum norm stabilizer ... 

In general, problems having solutions that vary radically or discon-tinuously for small input changes are said to be ill-posed. Deconvolution is an example of such a problem. Tikhonov was one of the earliest workers to deal with ill-posed problems in a mathematically precise way. He developed the approach of regularization (Tikhonov, 1963 Tikhonov and Arsenin, 1977) that has been applied to deconvolution by a number of workers. See, for example, papers by Abbiss et al (1983), Chambless and Broadway (1981), Nashed (1981), and Bertero et al. (1978). Some of the methods that we have previously described fall within the context of regularization (e.g., the method of Phillips and Twomey, discussed in Section V of Chapter 3). Amplitude bounds, such as positivity, are frequently used as key elements of regularization methods. 

Besides Tikhonov regularization, there are numerous other regularization methods with properties appropriate to distinct problems [42, 53,73], For example, an iterated form of Tikhonov regularization was proposed in 1955 [77], Other situations include using different norms instead of the Euclidean norm in Equation 5.25 to obtain variable-selected models [53, 79, 80] and different basis sets such as wavelets [81],... 

Tikhonov, A.N., Solution of incorrectly formulated problems and the regularization method, Soviet Math. Dokl., 4, 1035-1038, 1963. 

In the middle of the XX Century the Russian mathematician Andrei N. Tikhonov developed the foundations of the theory of ill-posed problem solutions. He introduced a regularization method in the solution of an inverse problem which was based on an approximation of an ill-posed problem by a number of well-posed problems. In this book we will systematically study the principles of ill-posed inverse problem solution. 

To justify this approach we will examine more carefully the properties of all three functionals involved in the regularization method the Tikhonov parametric functional and the stabilizing and misfit functionals. 

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. 

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