Abstract
We consider choice of the regularization parameter in Tikhonov method if the noise level of the data is unknown. One of the best rules for the heuristic parameter choice is the quasi-optimality criterion where the parameter is chosen as the global minimizer of the quasi-optimality function. In some problems this rule fails. We prove that one of the local minimizers of the quasi-optimality function is always a good regularization parameter. For the choice of the proper local minimizer we propose to construct the Q-curve which is the analogue of the L-curve, but on the x-axis we use modified discrepancy instead of discrepancy and on the y-axis the quasi-optimality function instead of the norm of the approximate solution. In the area rule we choose for the regularization parameter such local minimizer of the quasi-optimality function for which the area of the polygon, connecting on Q-curve this minimum point with certain maximum points, is maximal. We also provide a posteriori error estimates of the approximate solution, which allows to check the reliability of the parameter chosen heuristically. Numerical experiments on an extensive set of test problems confirm that the proposed rules give much better results than previous heuristic rules. Results of proposed rules are comparable with results of the discrepancy principle and the monotone error rule, if the last two rules use the exact noise level.
Highlights
Let A ∈ L( H, F ) be a linear bounded operator between real Hilbert spaces H, F
The results of the numerical experiments for test set 1 (n = 100 ) for the Triangle area rule (TA-rule) and some other rules are given in Tables 4 and 5. These results show that the TA-rule works well in all these test problems, the accuracy is comparable with δ-rules, but previous heuristic rules fail in some problems
For the problems which do not need regularization we can improve the performance of the TA-rule searching the proper local minimizer smaller or equal than αHQ := max{αHR, αQ }, where αHR, αQ are the global minimizers of functions ψHR (α) and ψQ (α), respectively on the interval [max (α N, λmin ), α0 ]
Summary
Let A ∈ L( H, F ) be a linear bounded operator between real Hilbert spaces H, F. We consider choice of the regularization parameter if the noise level for k f − f ∗ k is unknown. The parameter choice rules which do not use the noise level information are called heuristic rules. For finding proper local minimizer of the function ψQ (α), we propose the area rules on the Q-curve. The idea of proposed rules is that we form, for every minimizer of the function ψQ (α), a certain function which approximates the error of the approximate solution and has one minimizer; we choose for the regularization parameter such local minimizer of ψQ (α), for which the area of the polygon, connecting this minimum point with certain maximum points, is maximal. These algorithms are illustrated by the results of numerical experiments
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