Mathematical Sciences: Inverse Boundary Problems

PI：西尔维斯特DMS-9423849西尔维斯特将继续他对偏微分方程式和常微分方程式中的反问题的研究，因为这些问题涉及到从外边界上或附近的测量确定非均匀介质的性质。应用包括层状和非层状介质中的医学成像、雷达成像和声学成像。目标是开发一种数学上严格且数值稳定的层剥离算法，首先是一层剥离算法，然后是二维和三维剥层算法。偏微分方程式是建立物理世界数学模型的基础。数学分析的作用与其说是创建方程，不如说是提供有关解的定性和定量信息。这可能包括回答有关唯一性、平稳性和成长性的问题。此外，分析经常开发出近似解的方法和对这些近似的精度的估计。

PI: Sylvester DMS-9423849 Sylvester will continue his research on inverse problems in partial differential and ordinary differential equations, as they relate to the determination of the properties of inhomogeneous media from measurements made on or near the outer boundary. Applications include medical imaging, radar imaging, and acoustical imaging in layered and non-layered media. The goal is to develop a mathematically rigorous and numerically stable layer stripping algorithm, first in one, and then in two and three dimensions. Partial differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations.