RUI: Inverse Spectral Problems in One and Two Dimensions

本文研究了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，本征参数的出现会导致自伴性的丧失。尽管反问题的唯一性已经确定，但还没有适合于数值计算的构造性方案。这项工作(与威廉·伦德尔)开发和分析了两个涉及恢复这类问题的潜力的建设性方案。在二维情况下，利用特定薄膜的特征值来近似表示弹性薄膜边值问题中的非均匀函数。将边值问题及其系数投影到适当的向量空间，得到一个矩阵逆问题，该问题可用最优化技术求解。这项工作将考虑不同的领域，并研究多个系数的恢复。此外，还将探讨有关圆形区域的理论问题。特别是，将研究利用微分几何技术恢复辐射密度以及振动圆膜的径向光谱特性。在许多情况下，直接测量物体的特性是不现实的。在病人接受治疗之前，医生不会通过手术来确定脑瘤的大小。工程师不是为了确定飞机机翼的腐蚀程度而拆解飞机，而是对物体进行外部测量，并用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。在这些参数已知的情况下，用充分值问题对振动进行数学建模。如果参数未知，但振动已知，则要解决的问题是逆边值问题，也称为逆谱问题。这个项目开发了几个用于解决这类问题的构造性算法。重要的是要认识到，虽然数学反问题通常有多个解，但它们的物理对应问题可能不会。选择“正确”的解决方案也在这项工作中讨论。

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.