Tikhonov and GCV in Hybrid Methods Sample Clauses

Tikhonov and GCV in Hybrid Methods. At the kth iteration, we consider using Tikhonov regularization to regularize projected problem (2.12). Thus, we would like to solve k X X X X xx X 0 f = argmin f − , x = Y f . (2.14) k f λkI 0 k k k Notice that the difference between (2.13) and (2.14) is that the regularization parameter, λk, is allowed to change per iteration here. We consider the GCV function to select these parameters: G (λ) = kǁ(I − BkB†k,λ )βe1 2 ǁ . Bk, βe1 . † Σ2 trace(I − BkBk,λ) Σ Σ Notice the dependence on matrix Bk and right hand side vector βe1. If we define the SVD of the (k + 1) × k matrix Bk as Bk = Pk ∆k QT , (2.15) k 0T k .Σ then GB , βe1 (λ) can be written as kβ2 G = x k i=1 λ2 . Σ δ2 + λ2 ΣPT e1 Σ(i)Σ2 + .ΣPT e1 Σ(k+1)Σ2Σ , (2.16) Bk, βe1 (λ) . k Σ2 λ2 δ2 + λ2 k k i where ΣPT e Σ(j) denotes the jth component of the vector PT e , and δ ith largest singular value of Bk (i.e., the ith diagonal element of ∆k). is the