Regularization parameters

With this tensor structure of the kernel, 2D Laplace inversion can be performed in two steps along each dimension separately [50]. Even though such procedure is applicable when the signal-to-noise ratio is good, the resulting spectrum, however, tends to be noisy [50]. Furthermore, it is not dear how the regularization parameters should be chosen. 

Our goal is to estimate the function P(r) from the set of discrete observations Y(tj). We use a nonparametric approach, whereby we seek to estimate the function without supposing a particular functional form or parameterization. We require that our estimated function be relatively smooth, yet consistent with the measured data. These competing properties are satisfied by selecting the function that minimizes, for an appropriate value of the regularization parameter X, the performance index ... 

We use a method that implements the Unbiased Prediction Risk criterion [13] to provide a data-driven approach for the selection of the regularization parameter. The equality constraints are handled with LQ factorization [14] and an iterative method suggested by Villalobos and Wahba [15] is used to incorporate the inequality constraints [10]. The method is well suited for the relatively large-scale problem associated with analyzing each image voxel as no user intervention is required and all the voxels can be analyzed in parallel. 

Here the B-spline Bim(zf, Xj) is the ith B-spline basis function on the extended partition Xj (which contains locations of the knots in the Zj direction), and is a coefficient. We use cubic splines and sufficient numbers of uniformly spaced knots so that the estimation problem is not affected by the partition. The estimation problem now involves determining the set of B-spline coefficients that minimizes Eq. (4.1.26), subject to the state equations [Eqs. (4.1.24 and 4.1.25)], for a suitable value of the regularization parameter. At this point, the minimization problem corresponds to a nonlinear programming problem. 

Obviously, none of the three scattering experiments is able to reproduce the true molar mass distribution exactly. The two TDFRS measurements show a bi-modal structure, which must be regarded as an artifact. Otherwise, the general shape of the distribution, with the peak position around 200 kg/mol and a tail towards low M, is rather well reproduced. In principle unimodal distributions could be enforced by the choice of a larger regularization parameter. Purposely,... 

Svergun, D. I. (1992). Determination of the regularization parameter in indirect-transform methods using perceptual criteria. J. Appl. Crystallogr. 25, 495—503. 

Kilmer, M.E. and O Leary, D.P., Choosing regularization parameters in iterative methods for ill-posed problems, SIAM J. Matrix Anal. Appl., 22, 1204—1221, 2001. 

Let us now explain the main idea behind this technique. An important point is that the applicability of the integration-by-parts requires that Feynman integrals are regularized dimensionally, so that we work in continuous space-time dimension D = 4 — 2e, where e is the regularization parameter. Both, ultraviolet and infra-red divergences show up as poles in e. If dimensional regularization is adopted, one observes that the following relation ... 

Skaggs and Kabala studied a ID solute transport through a saturated homogeneous medium problem with a complex contaminant release history and assumed no prior knowledge of the release function. This closed form solution is similar to Eq. (50). Due to this similarity, Eq. (46) can be utilized to estimate the concentration release history (Cr). The accuracy of the TR method depends on the regularization parameter a. 

Here, 17 is the so-called regularization parameter. In principle, 17 is an arbitrary and positive real number. Note that the full interaction operator V is recovered from Vp in the limit 17 00. The tunneling part of V, V, is given by,... 

In the limit 17 -> ooVt tends to zero. Thus, the regularization parameter 17 effectively introduces a partitioning of V into a nonsingular, regular part Vp, and a singular part Vt. 

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. ... 

Then the regularization parameter can be determined by the misfit condition (2.49)... 

Figure 2-5 Illustration of the principle of optimal regularization parameter selection.

Figure 2-5 helps in understanding of the principle of optimal regularization parameter selection. One can see that because of the monotonic character of the function i a), there is only one point, Qq, where i(ao) = p, (A(maJ,d ) = 6. ... 

Figure 2-6 L-curve represents a simple curve for all possible a of the misfit functional, (rv), versus stabilizing functional, s(a), plotted in log-log scale. The distinct corner, separating the vertical and the horizontal branches of this curve, corresponds to the quasi-optimal value of the regularization parameter a.

L-curve analysis (Hansen, 1998) represents a simple graphical tool for qualitative selection of the quasi-optimal regularization parameter. 

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