Difficulty in using GCV in Hybrid Methods Sample Clauses

Difficulty in using GCV in Hybrid Methods. In this section we use standard GCV to choose the Tikhonov regularization parameter λk at each iteration of the Lanczos-based hybrid methods for the test problems in Section 2.1. The results are shown in Figure 2.4. In all of our examples, LSQR, which is essentially LBD with no regularization, exhibits semi-convergent behavior, as we expect. If we use “optimal” regularization parameters at each iteration (determined using knowledge of x true to make the relative error in the solution as small as possible), then Lanczos-hybrid methods would be excellent at stabilizing the regularized solution, as shown with the dashed lines. However, in realistic situations, we do not know the optimal solution, so this is impossible. On the Xxxxxxxx, Xxxx and Deriv2 problems, the performance of standard GCV, though slightly worse than optimal, is acceptable. For the other three problems, the convergence behav- ior when using standard GCV is significantly worse than when the optimal parameters are used. A major concern is the possibility that rounding errors in the computation of the matrices Wk, Yk and Bk are causing the poor behavior. Bj¨orck, Xxxxxx and Xxx Xxxxxx [9] showed that in some cases reorthogonalization may be necessary for better performance, and Xxxxxx [92] considered partial reorthogonalization. However, in our tests GCV still had difficulty even after reorthogonalization. Another option is to use a different regularization method such as TSVD or exponential filtering, but we found little to no improvement in the solution. In addition, we delayed regularization until after k > kmin to wait until Bk more fully captures the ill-conditioning of A, but that attempt proved futile as well. These phenomena are illustrated in LSQR (no regularization) Tikhonov with optimal λk Tikhonov with GCV λk relative error (Satellite) 1 LSQR (no regularization) Tikhonov with optimal λk Tikhonov with GCV λk relative error (Xxxxxxxx) 0.6 0.6 0.4 0.5 0.3 1 50 100 150 200 250 300 iteration 0 10 20 30 40 50 iteration LSQR (no regularization) Tikhonov with optimal λk Tikhonov with GCV λk LSQR (no regularization) Tikhonov with optimal λk Tikhonov with GCV λk 0.8 0.8 relative error (Xxxx) relative error (Deriv2) 0.6 0.6 0.4 0.4 0.2 0.2 0 10 20 30 40 50 iteration iteration LSQR (no regularization) Tikhonov with optimal λk Tikhonov with GCV λk LSQR (no regularization) Tikhonov with optimal λk Tikhonov with GCV λk 1 1 0.8 0.8 relative error (Baart) relative error (Heat) 0.6 0.6 0.4 0.4 0.2 ...
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