The L-curve Method for Parameter Selection in Inverse Problem Regularization

The L-curve method represents a fundamental approach for parameter selection in inverse problem regularization techniques. This method enables the identification of optimal parameters to enhance the performance of regularization methods in solving inverse problems. Our implementation utilizes cross-validation to determine the L-curve's characteristic shape, where we systematically compute errors across different parameter values. The algorithm involves calculating residual norms versus solution norms for various regularization parameters, typically generating a plot that forms an "L" shape. Key functions include parameter sweep iterations, norm calculations, and curvature analysis to identify the corner point representing the optimal balance between data fitting and solution smoothness. By comparing these computational results, we determine the optimal parameter value that minimizes both approximation error and regularization error. This selected parameter is then applied to the inverse problem regularization scheme, significantly improving method accuracy, robustness, and ultimately enhancing the quality of obtained solutions. The implementation includes visualization routines to plot the L-curve and highlight the optimal parameter selection point.