Mathematical Sciences: Multidimensional Reconstruction Methods for Inverse Problems

这个反问题项目将发展从多维二阶偏微分方程组中恢复未知系数的数学方法。还将分析这些方法的稳定性和它们的计算代价。将对年龄结构人口的应用进行调查，并将从人口的某些函数中提取出生率和死亡率等信息。逆问题出现在描述包含系数或边界条件不确定性的物理系统的数学模型中。使用边界数据或内部测量形式的附加信息，寻求更准确的模型--模型的输出与测量数据尽可能接近的模型。逆问题的重要例子是使用CAT扫描的医学成像、声学超声和在搜索损坏或故障部分时对飞机机翼进行无损检测。

This project on inverse problems will develop mathematical recovery methods for unknown coefficients from second order partial differential equations in multi-dimensions. The stability of these methods and their computational costs will also be analyzed. Applications to age-structured populations will be investigated and information such as birth and death rates will be extracted from certain functionals of the population. Inverse problems arise in mathematical models describing physical systems containing uncertainties in coefficients or boundary conditions. Using additional information, in the form of boundary data or interior measurements, a more accurate model is sought -- one in which the output of the model conforms to the measured data as closely as possible. Important examples of inverse problems are medical imaging using CAT scans, acoustic sonograms, and non-destructive testing of aircraft wings in searching for damaged or failed sections.