Regularization algorithm

Experimental considerations Frequently a numerical inverse Laplace transformation according to a regularization algorithm (CONTEST) suggested by Provencher [48,49] is employed to obtain G(T). In practice the determination of the distribution function G(T) is non-trivial, especially in the case of bimodal and M-modal distributions, and needs careful consideration [50]. Figure 10 shows an autocorrelation function for an aqueous polyelectrolyte solution of a low concentration (c = 0.005 g/L) at a scattering vector of q — 8.31 x 106 m-1 [44]. 

In other words, any regularization algorithm is based on the approximation of the noncontinuous inverse operator A by the family of continuous inverse operators (d) that depend on the regularization parameter a. The regularization must be such that, as a vanishes, the operators in the family should approach the exact inverse operator A. ... 

A central point of this book is the application of so-called regularizing algorithms for the solution of ill-posed inverse geophysical problems. These algorithms can u.se a priori geological and geophysical information about the earth s subsurface to reduce the ambiguity and increase the stability of the solution. 

A.N. Tikhonov, A.V. Concharskii, V.V. Stepanov and A.C. Yagola (eds.). Regularizing Algorithms and a Priori Information, Nauka, Moscow, 1983 (in Russian). 

P. Dewilde and E. Deprettere. Architectural synthesis of large, nearly regular algorithms design trajectory and environment. Annales des telecommunications, 46, number 1-2, pages 48-59, Jan-Feb 1991. 

Because the pseudo-inverse filter is chosen from the class of additive filters, the regularization can be done without taking into account the noise, (n). At the end of this procedure the noise is transformed to the output of the pseudo-inverse filter (long dashed lines on Fig. 1). The regularization criteria F(a,a) has to fulfill the next conditions (i) leading to an additive filter algorithm, (ii) having the asymptotic property a, —> a, for K,M... 

Several numerical procedures for EADF evaluation have also been proposed. Morrison and Ross [19] developed the so-called CAEDMON (Computed Adsorption Energy Distribution in the Monolayer) method. Adamson and Ling [20] proposed an iterative approximation that needs no a priori assumptions. Later, House and Jaycock [21] improved that method and proposed the so-called HILDA (Heterogeneity Investigation at Loughborough by a Distribution Analysis) algorithm. Stanley et al. [22,23] presented two regularization methods as well as the method of expectation maximalization. 

At the cost of deriving specialized algorithms, multiplicative methods can be generalized to other expressions of the penalty to account for different noise statistics (Lanteri et al., 2001) and can even be used to explicitly account for regularization (Lanteri et al., 2002). 

When started with a smooth image, iterative maximum likelihood algorithms can achieve some level of regularization by early stopping of the iterations before convergence (see e.g. Lanteri et al., 1999). In this case, the regularized solution is not the maximum fikelihood one and it also depends on the initial solution and the number of performed iterations. A better solution is to explicitly account for additional regularization constraints in the penalty criterion. This is explained in the next section. 

Benzene has often been used as a test system for vibrational calculations using a variety of different electronic structure algorithms. The molecule exhibits regular hexagonal planar symmetry with six carbon atoms joined by a bonds and six remaining p-orbitals which overlap to form a delocalised n electron over all six carbon atoms. Table 1 shows comparisons of several different methods for benzene. 

---