Fig. 1. Singular values of A₀ decaying without gap towards 0, where the data A₀ comes from a discrete ill-posed problem. Singular values of the noisy data A level off at the noise level.

Fig. 2. The true solution x₀ is chosen to be the discretization of a smooth function. However, the LS and TLS solutions x computed from the noisy data are influenced by the noise and are far from being reasonable approximations of the underlying true solution x₀.

Fig. 3. Several truncated TLS and SVD solutions x_k, together with the true solution x₀, for the noisy simulation data described in Section 2.2. For both TTLS and TSVD solutions, the best truncation level is k=12, minimizing x_k-x₀/x₀ to the value of 1e-3.

Fig. 4. The filter factors associated with the TTLS solutions for a representative range of k values between 1 and n=30. The interrupted horizontal lines mark the 0 and 1 levels. The numerical example is the same noisy example as in Section 2.2.

Fig. 5. The left plot shows the GCV function in the TTLS and the classical TSVD cases for the example in Section 2.2. The right plot compares the effective number of parameters in the TTLS case and in the TSVD case. Note that is approximately equal to k for a certain range of k values.

Fig. 6. Comparison between the two GCV criteria, corresponding to the TTLS and the TSVD methods. For each test problem (from 1 to 8) we use 100 random noise realizations. Each plot shows the ratio between the value of the truncation level k^GCV chosen by GCV and the best k^best value minimizing the distance of the truncated solution to the true solution. Plotted values equal to 1 are thus optimal. The four plots correspond to: (top) noisy A and b, (bottom) noiseless A, noisy b; (left) Ax≈b solved with truncated total least squares; (right) Ax≈b solved with truncated singular value decomposition.