RUI: Inverse Spectral Problems in One and Two Dimensions

本文研究了一维和二维反谱问题的数值解和解析解。在一维情况下，aSturm-Liouville问题边界条件中特征参数的出现导致了自伴随性的丧失。虽然反问题的唯一性已经确立，但目前还没有适合于数值计算的建设性方案。这项工作（与威廉伦德尔）开发并分析了涉及恢复这类问题的潜力的两种建设性方案。在二维空间中，利用特定膜的特征值，在控制弹性膜的边值问题中找到表示非均匀性的函数的近似值。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，利用优化技术求解。这项工作将考虑不同的领域，并研究多系数的恢复。关于循环域的理论问题也将被研究。特别是，利用微分几何技术恢复径向密度和振动圆膜的径向谱特性将被研究。在许多情况下，直接测量对象的属性是不实际的。在病人接受治疗之前，医生不会通过手术来确定脑肿瘤的大小。工程师不会为了确定机翼的腐蚀程度而拆卸飞机。相反，物体的外部测量是用来确定物体的内部属性的。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，用边值问题对振动进行数学建模。如果参数不知道，但振动是已知的，那么要解决的问题是一个反边值问题-也称为反谱问题。这个项目开发了几个建设性的算法来解决这类问题。重要的是要认识到，虽然数学逆问题通常有多个解决方案，但它们的物理对应物可能没有。选择“正确”的解决方案也在这项工作中得到解决。

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.