Optimal finite difference grids for direct and inverse Sturm–Liouville problems

Abstract

We study finite difference approximations of solutions of direct and inverseSturm–Liouville problems, in a finite or infinite interval on the real line. Thediscretization is done on optimal grids, with a three-point finite difference stencil.The optimal location of the grid points is calculated via a rational approximationof the Neumann-to-Dirichlet map and the latter converges exponentiallyfast. We prove that optimal grids obtained for constant coefficients areasymptotically optimal for variable coefficient direct problems. We also showthat optimal grids, together with methods of inverse spectral problemsfor Jacobi matrices, can be used for the solution of continuous inverseSturm–Liouville problems. In particular, we formulate and analyse a new inversionalgorithm, where the unknown coefficients that we image are optimallydiscretized. We prove that optimal grids provide necessary conditionsfor convergence of the discrete inverse problem and we demonstrate theeffectiveness of our imaging approach through numerical simulations.