RUI: Inverse Spectral Problems in One and Two Dimensions

Murray State University, Murray KY

Investigators

Abstract

This work investigates the numerical and analytic solution of inverse spectral problems in one and two dimensions. In one dimension, the appearance of an eigenparameter in the boundary condition of a Sturm-Liouville problem causes a loss of self-adjointness. Although uniqueness of the inverse problem has been established, there are no constructive schemes available that lend themselves to numerical computation. This work (with William Rundell) develops and analyzes two constructive schemes involving to recover the potential in this type of problem. In two dimensions, the eigenvalues of particular membranes are used to find an approximation to a function representing the nonhomogeneity in the boundary value problem governing the elastic membrane. Projection of the boundary value problem and its coefficients onto appropriate vector spaces leads to a matrix inverse problem, which is solved using optimization techniques. This work will consider various domains and investigate the recovery of multiple coefficients. Theoretical questions regarding circular domains will also be investigated. In particular, the recovery of a radial density using techniques from differential geometry and the properties of the radial spectrum of a vibrating circular membrane will be investigated. There are many situations in which it is not practical to measure an object's properties directly. Doctors do not perform surgery to determine the size of a brain tumor prior to a patient's treatment. An engineer does not dismantle an airplane to determine the level of corrosion in its wing. Instead external measurements of an object are made and used to determine the internal properties of the object. This research focuses on the use of vibrational information to determine physical parameters of an object. If these parameters are known, the vibration is modeled mathematically by a boundary value problem. If the parameters are not known, but the vibration is known, then the problem to be solved is an inverse boundary value problem - also known as an inverse spectral problem. This project develops several constructive algorithms for the solution of this type of problem. It is important to realize that while mathematical inverse problems often have multiple solutions, their physical counterpart may not. Choosing the "correct" solution is also addressed in this work.