W-GCV for Tikhonov Regularization Sample Clauses

W-GCV for Tikhonov Regularization. The standard GCV method is a popular parameter choice method used in a variety of applications; however, as illustrated in Section 2.3.2, the method may not perform well for certain classes of problems. Other studies in statis- tical nonparametric modeling and function approximation noted that in prac- tical applications, GCV occasionally chose Tikhonov parameters too small, thereby under-smoothing the solution [28, 46, 89, 106, 127]. To circumvent this problem, these papers use a concept that we call weighted-GCV. In con- trast, we observed over-smoothing difficulties when using GCV in Lanczos- hybrid methods, which motivated us to use a different range of weights in the W-GCV method. Instead of the Tikhonov GCV function defined in (2.7), we consider the weighted-GCV function G (ω, λ) = n||(I − AA†λ)b||2 . (2.17) A, b . † Σ2 trace(I − ωAAλ) Notice the function’s dependency on a new parameter ω in the denominator trace term. Choosing ω = 1 gives the standard GCV function (2.7). If we choose ω > 1, we obtain smoother solutions, while ω < 1 results in less smooth solutions. The obvious question here is how to choose a good value for ω. To our knowledge, in all work using W-GCV, only experimental ap- proaches are used to choose ω. For smoothing spline applications, Xxx and Gu empirically found that standard GCV consistently produced regulariza- tion parameters that were too small, while choosing ω in the range of 1.2-1.4 worked well [89]. In our problems, though, the GCV regularization parame- ter is chosen too large; thus, we seek a parameter ω in the range 0 < ω ≤ 1. In addition, rather than using a user-defined parameter choice for ω as in previous papers, we propose a more automated, adaptive approach that is also versatile and can be used on a variety of problems.