[Paper Review] Q-curve and area rules for choosing heuristic parameter in Tikhonov regularization

Toomas Raus, Uno Hämarik|arXiv (Cornell University)|Sep 6, 2018

Numerical methods in inverse problems27 references3 citations

TL;DR

This paper proposes the Q-curve and area rule to improve heuristic regularization parameter selection in Tikhonov regularization when noise level is unknown. By analyzing local minimizers of the quasi-optimality function and using a novel Q-curve—plotting modified discrepancy vs. quasi-optimality function—it identifies robust parameters via maximal area polygons. The method outperforms prior heuristic rules and matches performance of noise-level-dependent rules like discrepancy and monotone error rule.

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 choice of the proper local minimizer we propose to construct the Q-curve which is the analogue of the L-curve, but on 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 area rule we choose for the regularization parameter such local minimizer of the quasi-optimality function for which the area of 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 parameter chosen heuristically. Numerical experiments on 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 last two rules use the exact noise level.

Motivation & Objective

• Address the challenge of selecting regularization parameters in Tikhonov regularization when the noise level of data is unknown.
• Overcome limitations of existing heuristic rules, such as the quasi-optimality criterion, which may fail in certain ill-posed problems.
• Develop a reliable method to choose among local minimizers of the quasi-optimality function to ensure convergence and stability.
• Provide a posteriori error estimates to validate the reliability of heuristically chosen parameters.
• Improve upon previous heuristic rules by introducing the Q-curve and area rule for enhanced parameter selection accuracy.

Proposed method

• Propose the Q-curve as an analogue of the L-curve, with modified discrepancy on the x-axis and the quasi-optimality function ψ_Q(α) on the y-axis.
• Define the area rule to select the regularization parameter as the local minimizer of ψ_Q(α) that maximizes the area of a polygon formed with specific maximum points on the Q-curve.
• Use the set L_min of local minimizers of ψ_Q(α) as candidate parameters, proving at least one is pseudooptimal (i.e., yields error within a constant factor of the minimal error).
• Introduce a posteriori error estimate (20) that bounds the error of the heuristic solution using T(α_H, α*) and T1(α_H), enabling validation of parameter reliability.
• Propose variants using ψ_QD(α) or d_ME(α) instead of d_MD(α) to improve performance for nonsmooth or smooth solutions.
• Implement and test the rules on a comprehensive set of benchmark test problems from the literature, including those from [19, 5, 1, 10, 22, 39].

Experimental results

Research questions

• RQ1 Can local minimizers of the quasi-optimality function ψ_Q(α) serve as reliable candidates for regularization parameters when the noise level is unknown?
• RQ2 How can the Q-curve be constructed to improve selection among local minimizers of ψ_Q(α) compared to traditional L-curve or heuristic rules?
• RQ3 Does the area rule—selecting the local minimizer that maximizes the area of a polygon on the Q-curve—yield more accurate and stable regularization parameters?
• RQ4 Can a posteriori error estimates be used to validate the reliability of a heuristically chosen parameter, especially when the true noise level is unknown?
• RQ5 How does the performance of the proposed Q-curve and area rule compare to established rules like discrepancy principle and monotone error rule when the exact noise level is known?

Key findings

• At least one local minimizer in the set L_min of ψ_Q(α) is pseudooptimal, meaning its error is within a constant factor of the minimal possible error.
• The area rule on the Q-curve significantly outperforms previous heuristic rules in numerical experiments on a broad set of test problems.
• The proposed rules achieve accuracy comparable to the discrepancy principle and monotone error rule when these rules use the exact noise level.
• In 73% of cases on Set 1, the conditions b ≤ 2 and T1(α_H) ≤ 9 were satisfied, indicating reliable parameter choice; in 61% of cases, T1(α_H) ≤ 4.
• The method is robust: even in cases where the quasi-optimality criterion fails, the area rule on the Q-curve identifies a reliable parameter.
• The a posteriori error estimate (20) allows validation of heuristic parameters, with T1(α_H) ≤ 9 satisfied in 97% of cases on Set 1, supporting confidence in parameter reliability.