RUI: Inverse Spectral Problems in One and Two Dimensions

本文研究一维和二维逆谱问题的数值解和解析解。在一维Sturm-Liouville问题的边界条件中，特征参数的出现导致自伴性的丧失。虽然反问题的唯一性已经建立，但没有建设性的计划，使自己的数值计算。这项工作（与威廉伦德尔）开发和分析两个建设性的计划，涉及到恢复这种类型的问题的潜力。在二维情况下，利用特定薄膜的本征值来近似描述弹性薄膜边值问题中的非均匀性。将边值问题及其系数投影到适当的向量空间上会导致一个矩阵逆问题，该问题使用优化技术来解决。本工作将考虑不同的领域和研究的恢复多个系数。有关circulardomains的理论问题也将被调查。特别地，将研究利用微分几何技术恢复径向密度和振动圆膜的径向谱的性质。在许多情况下，直接测量物体的性质是不实际的。 医生不会在病人治疗前进行手术来确定脑瘤的大小。 工程师拆卸飞机并不是为了确定机翼的腐蚀程度，而是通过对物体的外部测量来确定物体的内部特性。 本研究的重点是利用振动信息来确定物体的物理参数。如果这些参数是已知的，振动的数学模型由abounce值问题。 如果参数未知，但振动是已知的，那么要解决的问题是一个逆边值问题-也称为逆谱问题。 本项目开发了几种构造性算法来解决这类问题。 重要的是要认识到，虽然数学逆问题往往有多个解决方案，他们的物理对应可能没有。 选择“正确”的解决方案也在这项工作中解决。

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.