A regularization algorithm for inverse Sturm–Liouville problem on the star-shaped graph

We consider an inverse Sturm–Liouville problem on a star-shaped metric graph, where the aim is to reconstruct edge potentials from spectral data. The inverse map from spectral data to the potentials is highly ill-conditioned, so classical spectral-mapping iterations suffer from severe noise amplification and spurious oscillations. To stabilize the reconstruction, we propose a regularized spectral-mapping method that combines spectral filtering, damping of the nonlinear fixed-point iteration, spatial Tikhonov smoothing with Neumann boundary conditions, and projection onto an affine constraint set encoding endpoint and integral information. The method admits a unified operator formulation that allows a transparent analysis of stability and convergence. Under mild assumptions on the filtered update, we prove that the regularized iteration is locally contractive in the L²-norm, which guarantees existence and uniqueness of a fixed point and linear convergence of the iteration. Furthermore, with noise-dependent parameter choice and a discrepancy principle, we establish that the method defines a convergent regularization. The reconstructed potentials converge to the exact solution as the noise level tends to zero. Numerical experiments on star graphs with multiple edges confirm the theoretical results and demonstrate that the proposed regularization significantly improves robustness and accuracy compared with the unregularized spectral-mapping approach.