Effect of Noise on ω. We now consider how the choice of ω depends on the amount of noise in the data. In particular, we report on numerical results for the test problems described in Section 2.1 with three different noise levels: ǁεǁ2 = 0.1, 0.01, and 0.001. ǁAx trueǁ2 Thus, these problems have 10%, 1% and 0.1% noise levels respectively. Some of the results reported in previous sections for 10% noise are repeated here for comparison purposes. Recall that because standard GCV computes regularization parameters that are too large, we should choose 0 < ω ≤ 1 in W-GCV. Generally we observe that the over-smoothing caused by standard GCV is more pro- nounced for larger noise levels. Therefore large noise levels typically need smaller values of ω, while small noise levels need larger values of ω. Our next experiments were designed to see how far the “optimal” value of ω differs from the standard GCV value ω = 1. The results are shown in Table 2.2. We provide the ω values that allow W-GCV to compute near-optimal reg- ularization parameters at each iteration of the Lanczos-hybrid method. For example, in Figure 2.6 we see that for 10% noise, ω = 0.2 produces near optimal convergence behavior for the Heat problem; thus, this value appears in the first row, last column of Table 2.2. Table 2.2: Values of ω for different noise levels. This table contains the values of ω (found experimentally) that produce optimal convergence behavior of the Lanczos-hybrid method for different noise levels. Figure 2.9 shows how these values perform on the Baart and Heat examples. Noise Level Satellite ωopt Xxxxxxxx ωopt Xxxx ωopt Deriv2 ωopt Baart ωopt Heat ωopt 10 % 0.40 0.20 0.05 0.10 0.01 0.20 1% 0.50 0.40 0.05 0.20 0.05 0.40 0.1% 0.80 0.50 0.10 0.60 0.10 0.80 The results reported in Table 2.2 were found experimentally. We see clearly from this table that optimal values of ω depend on the noise level (increasing with decreasing noise level), as well as on the problem. However, more work is needed to better understand these relationships. Figure 2.9 shows how our adaptive approach to choosing ω compares to the optimal values on two of the test problems (Baart and Heat) and for three different noise levels. These two test problems are representative of the convergence behavior we observed with the other test problems. We see that if a good choice of ω can be found, W-GCV is very effective (much more so than GCV) at choosing regularization parameters and thus at stabilizing the convergence behavior, especiall...
