An inner-outer iterative method for inverse Sturm-Liouville problems

In this paper, we present an inner-outer iterative method for two inverse Sturm-Liouville problems known as the symmetric and the two-spectra problems, aimed to achieve a continuous approximation of the unknown potential belonging to a suitable function space from the prescribed spectra data. To reduce the discrepancy between the matrix and differential eigenvalues, we use the optimal grid for a general reference potential and update it at each outer iteration. By discretizing the Sturm-Liouville problem over these grids, we get a series of matrix inverse eigenvalue problems. Then, a sequence of approximations of the unknown potential is obtained by employing a third-order Newton-type method as the inner iterations at each step of the outer iteration. Convergence of our method is established. Numerical experiments confirm its effectiveness.