Tikhonov regularization

For instance, the smoothness regularization in Eq. (31) is quadratic with respect to the sought parameters and is a particular case of Tikhonov regularization ... 

The first term measures the difference between the data and the fit, KF. The second term is a Tikhonov regularization and its amplitude is controlled by the parameter a. The effect of this regularization term is to select a solution with a small 2-norm 11 F 2 and as a result a solution that is smooth and without sharp spikes. However, it may cause a bias to the result. When a is chosen such that the two terms are comparable, the bias is minimized and the result is stable in the presence of noise. When a is much smaller, the resulting spectrum F can become unstable. 

Instead of Fourier transforming the time traces, they are commonly analyzed with the program DeerAnalysis2008 from Jeschke et al. (2006), which is based on Tikhonov regularization and the assumption that the full Pake pattern is excited. 

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],... 

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. 

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). 

Note that this stabilizer works similarly to the maximum entropy regularization principles, considered, for example, in Smith et al. (1991), and Wernecke and D Addario (1977). However, in the framework of the Tikhonov regularization, the goal is to minimize a stabilizing functional, which justifies the minimum entropy name for this stabilizer. 

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 ... 

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 ... 

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

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 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. 

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. 

Vogt et al. have therefore made use of Tikhonov regularization to handle both of the problems of the analytical transform method, i.e. instability to... 

Eor evaluation of experimental DEER data several software packages are available [59, 61]. They cater either for data analysis based on a model of the distance distribution [97-99] or for model-free methods, e.g., Tikhonov regularization [57, 59]. Eor the model-free approach, the underlying mathematical problem is (moderately) ill-posed, i.e., quality of the analyzed data is very cmcial. Incomplete labeling of double mutants results in (1) lower signal to noise of the primary data with increasing number of completely unlabelled molecules and (2) reduced modulation depth with decreasing number of doubly labeled molecules. 

In this study, one mutant includes a pair of cysteines placed within a single helix to provide an internal distance control. Distance distributions were obtained by DEER measurements and Tikhonov regularization. Studying ASYN bound to detergent and lysophospholipid micelles, it has been shown that the inter-helical... 

Chiang YW, Borbat PP, Freed JH (2005) The determination of pair distance distributions by pulsed ESR using Tikhonov regularization. J Magn Reson 172 279-295... 

Fig. 7 Modeling a high-resolution dimeric structure, (a) Estimate of the mean distance on the example of spin label K-221R1. The primary DEER trace (P(t)A (0)), the form factor and the distance distribution obtained by Tikhonov regularization (the L curve is shown in the inset) with the software Deer Analysis are presented, (b) Fits of primary experimental DEER data black lines) by simulated data red lines) corresponding to the final structure of the NhaA dimer and a distribution of spin label conformations modeled by a rotamer library, (c) C2 symmetry axis of the dimer created with the EPR constraints, (d) Comparison between the EPR stmcture and the electron density projection to the membrane plane obtained by cryo-EM on 2D crystals, (e) Comparison between the EPR structure and the dimer modeled on the refined cryo-EM data (PDB 3EI1). Adapted from [77]...

Fig. 7 PELDOR signal analysis, (a) Time domain PELDOR signal as a function of the delay time T of the pump pulse. The dashed line shows the exponentially decaying intermolecular dipolar contribution to the signal, (b) Time domain PELDOR signal after division of the original PELDOR time domain data by the fit-function representing the intermolecular decay, (c) Fourier transform of the PELDOR time trace (b) representing the dipolar Pake-pattem. (d) Distance distribution function obtained from the PELDOR time traces (b) by Tikhonov regularization. From the last representation the distances for spin pairs A-B can be the most easily extracted...

Fig. 15 Two RNA sequences, named 7 and 12, individually fold into a hairpin conformation but form an extended duplex when mixed. PELDOR experiments for different ratios of 7 and 12 are shown together with the distance distribution functions obtained from Tikhonov regularization of the PELDOR time traces. Integration of the peaks in the distance distribution allows a precise quantification of the different RNA structures in the mixture. Figure adapted from [30]...

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