Selecting the corner in the L-curve approach to Tikhonov regularization

Area of Science:

• Numerical analysis
• Inverse problems
• Regularization techniques

Background:

• Tikhonov regularization is a key method for solving ill-posed inverse problems.
• Selecting an appropriate regularization parameter is crucial for stable and accurate solutions.
• The L-curve method is a popular technique for parameter selection, but its corner identification can be challenging.

Purpose of the Study:

• To evaluate the performance of two L-curve corner selection methods: maximum curvature and minimum product.
• To compare these methods against an empirical Composite REsidual and Smoothing Operator (CRESO) approach.
• To assess performance under varying noise conditions, specifically correlated geometry noise versus Gaussian measurement noise.

Main Methods:

• Computer simulations were employed to test the two L-curve corner selection strategies.
• The maximum curvature method identifies the corner as the point of greatest curvature.
• The minimum product method identifies the corner by minimizing the product of the curve's abcissa and ordinate.

Main Results:

• Both maximum curvature and minimum product methods yielded significantly better regularization parameters than the CRESO approach.
• The minimum product method demonstrated superior performance, especially when correlated geometry noise was dominant over Gaussian noise.
• The regularization parameter selected by the minimum product method was found to be identical to that obtained using an established zero-crossing empirical approach.

Conclusions:

• The maximum curvature and minimum product L-curve corner selection methods offer robust and effective strategies for Tikhonov regularization.
• The minimum product method is particularly advantageous in scenarios with significant correlated noise, providing reliable parameter estimation.
• These findings suggest practical improvements for solving inverse problems in the presence of complex noise environments.