RelSea: A Reliable L-Curve Search Region to Improve Regularization Parameter Selection

Abstract

The selection of regularization parameters (<inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula>) using the L-curve method is a well-established strategy for addressing ill-posed inverse problems. Several techniques have been developed to identify an appropriate <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> in the neighborhood of the L-curve corner, including the <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-criterion, cubic spline, adaptive pruning, Menger curvature (MC), and <inline-formula> <tex-math notation="LaTeX">$\theta $ </tex-math></inline-formula>-curve (TC) methods. However, these approaches may present some limitations: sensitivity to L-shape distortion and secondary inflections, dependence on manual parameter tuning, loss of relevant information during curve simplification, and, in some cases, high computational complexity (CC). To overcome these drawbacks, this study introduces the Reliable Search Region (RelSea) algorithm, which defines a region of the L-curve most likely to contain an appropriate <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula>. Specifically, RelSea serves as an auxiliary scheme that enhances existing regularization parameter selection methods, thereby improving their accuracy while reducing its CC. It incorporates a modified Golden Section Search to reduce CC while supporting both continuous and discrete L-curves. Moreover, RelSea can be easily integrated into existing L-curve-based regularization parameter selection techniques. To illustrate this fact, we present the Enhanced MC Algorithm (EMA) and the Modified <inline-formula> <tex-math notation="LaTeX">$\theta $ </tex-math></inline-formula>-curve Algorithm (MOTA). By narrowing the search region through RelSea, EMA and MOTA attain a low CC <inline-formula> <tex-math notation="LaTeX">${\mathrm { O}}(log~N)$ </tex-math></inline-formula> while preserving high accuracy and robustness against L-shape distortion and secondary inflections, outperforming their respective base methods. MOTA achieves significantly lower CC compared to TC, which operates at <inline-formula> <tex-math notation="LaTeX">${\mathrm { O}}(N^{2})$ </tex-math></inline-formula>, whereas EMA retains the same CC as MC, confirming that RelSea introduces no additional computational overhead into methods that already exhibit low CC.