RUI: Inverse Spectral Problems in One and Two Dimensions

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This work investigates the numerical and analytic solution of inversespectral problems in one and two dimensions. In one dimension, theappearance of an eigenparameter in the boundary condition of aSturm-Liouville problem causes a loss of self-adjointness. Althoughuniqueness of the inverse problem has been established, there are noconstructive schemes available that lend themselves to numericalcomputation. This work (with William Rundell) develops and analyzes twoconstructive schemes involving to recover the potential in this type ofproblem. In two dimensions, the eigenvalues of particular membranes are usedto find an approximation to a function representing the nonhomogeneity inthe boundary value problem governing the elastic membrane. Projection of theboundary value problem and its coefficients onto appropriate vector spacesleads to a matrix inverse problem, which is solved using optimizationtechniques. This work will consider various domains and investigate therecovery of multiple coefficients. Theoretical questions regarding circulardomains will also be investigated. In particular, the recovery of a radialdensity using techniques from differential geometry and the properties ofthe 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 sizeof a brain tumor prior to a patient's treatment. An engineer does notdismantle an airplane to determine the level of corrosion in its wing.Instead external measurements of an object are made and used to determinethe internal properties of the object. This research focuses on the use ofvibrational information to determine physical parameters of an object. Ifthese parameters are known, the vibration is modeled mathematically by aboundary value problem. If the parameters are not known, but the vibrationis known, then the problem to be solved is an inverse boundary valueproblem - also known as an inverse spectral problem. This project developsseveral constructive algorithms for the solution of this type of problem. Itis important to realize that while mathematical inverse problems often havemultiple solutions, their physical counterpart may not. Choosing the"correct" solution is also addressed in this work.