A Study For Inverse Sturm-Liouville Problems

This dissertation mainly focuses on three types of the inverse eigenvalue problem:the first one is the inverse eigenvalue problem for a regular Sturm-Liouville operator. For this inverse problem, we propose a modified Numerov’s method and establish the convergence. In addition, we discuss the convergence of a boundary value method in [3] for computing Sturm-Liouville potentials. The second one is the inverse eigenvalue problem for Sturm-Liouville operator in impedance form. For this inverse problem, we propose the descent flow methods and a finite difference method. The third is the inverse eigenvalue problem for Helmholtz equation. For this inverse problem, we propose a new method. By solving a least squares problem with the steepest descent method, we recover an piecewise constant approximation to the unknown density.First, we deal with the inverse eigenvalue problem for a regular Sturm-Liouville operator. Based on the Numerov’s method in [13], we propose a mod-ified Numerov’s method and establish its convergence. Modified Numerov’s method is to use interpolation to refine the mesh sufficiently for Numerov’s method to be effective even without the asymptotic correction technique. The mesh size is no longer constrained by the number of known eigenvalues while in [13] the number of spectral data and the mesh size are tied together, and it is pointed out in [13,16] that the option of using a finer mesh is not normally available. In addition, we discuss the convergence of a boundary value method in [3] for computing Sturm-Liouville potentials. In [3], an approximation to the unknown potential belonging to a suitable function space of finite dimension is obtained by forming an associated set of nonlinear equations and solving these with a Quasi-Newton approach. Then, we consider two convergence problems: one is the convergence of the Quasi-Newton approach and then a convergence theorem is established. The other is the convergence of the estimate of the un-known potential, provided by the exact solution of the nonlinear equation, to the true potential, and then another convergence theorem is proved. To fur- ther investigate the properties of the boundary value method, we introduce some other spaces of functions. Numerical examples confirm the theoretically predict-ed convergence properties and show stability and effectiveness of the modified Numerov’s method and those of the boundary value method.Second, we discuss the inverse Sturm-Liouville operator in impedance form. Based on different papers [26] and [72], we propose two kinds of methods. The first kind is the descent flow methods. By using a finite difference method to discretize the Sturm-Liouville operator and approximating the impedance with a function belonging to a suitable function space, we obtain a generalized matrix eigenvalue problem. Inspired by the asymptotic correction technique first stud-ied in connection with the computation of the eigenvalues of a regular Sturm-Liouville operator, we discuss the correction technique for eigenvalues of Sturm-Liouville operator in impedance form. By solving an least squares problem, we find an approximation to the impedance. In addition, we are interested in the methods for recovering the impedance by solving the matrix inverse eigenvalue problem. Therefore, we propose a finite difference method for computing the symmetric impedance function. Using the correction technique for eigenvalues of Sturm-Liouville operator in impedance form, the matrix inverse eigenvalue prob-lem is solved by solving the nonlinear equations with modified Newton’s method. Then an approximation to the impedance is obtained. We also prove the conver-gence. Effectiveness and stability is shown by the numerical experiments.Finally, we consider the two-dimensional inverse spectral problem for Helmholtz equation. Based on the method in [65], we propose a new method for recover-ing an unknown symmetric density function. By using the piecewise constant function to approximate the density function and using the Rayleigh-Ritz ap-proach to discretize the differential equation, the continuous inverse eigenvalue problem is converted to a related matrix inverse eigenvalue problem and then a least squares problem is formulated. Taking advantage of the sensitivity analysis of the eigenvalue, the solution of the least squares problem via the steepest de-scent method is discussed and then an approximation to the unknown density is recovered. Numerical results show our method can recover the unknown density effectively.