Hybrid Methods. As described in the previous section, direct methods such as Tikhonov reg- ularization can be computationally impractical for large problems. Further- more, iterative methods like LSQR when applied to ill-posed inverse problems exhibit inherent semi-convergence, where early iterations tend to approxi- mate spectral components corresponding to signal, while later iterations be- 100 singular values,σ(A) and σ(B ) 100 σ(B ) σ(A) singular values,σ(A) and σ(B ) 100 σ(B ) σ(A) singular values,σ(A) and σ(B ) 10−2 10−2 10−2 10−4 10−4 10−4 10−6 i 10−6 σ(B ) i 20 σ(A) i 10−6 i 100 100 100 |σ(A)−σ(B )|/σ(A) 10−5 10−5 10−5 10−10 10−10 10−10 10−15 10−15 10−15 |σ(A)−σ(B )|/σ(A) |σ(A)−σ(B )|/σ(A) 0 10 20 30 40 50 10−20 0 10 20 30 40 50 10−20 0 10 20 30 40 50 Figure 2.2: Convergence of singular values. The plots in the top row of this figure show the singular values of A, denoted as σi(A), along with the singular values of Bk, denoted as σi(Bk), for k = 10, 20, 50. The plots in the bottom row show the relative difference, |σi(A) − σi(Bk)| . σi(A) come contaminated with noise [68]. A good stopping criterion is required for computing reliable solutions. To overcome these limitations, we consider hybrid methods for large-scale ill-posed inverse problems. Previous work on hybrid methods can be divided broadly into two cate- gories: those that use iterative methods to solve the regularized problem and those that embed regularization within an iterative scheme. In this work we focus on the latter case, but first we make some remarks on the former.
Hybrid Methods. 4447. Many tariff adjustment methods currently being used combine elements of both the rate-of- return and the price-cap methods with a view to both reducing the risk borne by the service providers and providing sufficient incentives for efficiency in the operation of the infrastructure. One such hybrid method employs sliding scales for adjusting the tariffs that ensure upward adjustment when the rate of return falls below a certain threshold and downward adjustment when the rate of return exceeds a certain maximum, with no adjustment for rates of return falling between those levels. Other possible approaches to balancing the rate-of-return and price-cap methods include a review by the contracting authority of the investments made by the private partner concessionaire to ensure that they meet the criteria of usefulness in order to be taken into accountconsidered when calculating the private partnerconcessionaire’s revenue requirement. Another tariff adjustment technique that may be used to set tariffs, or more generally to monitor tariff levels, is benchmark or yardstick pricing. By comparing the various cost components of one public service provider with those of another and with international norms, the contracting authority may be able to judge whether tariff adjustments requested by the public service provider are reasonable.
