Mathematical Sciences: Inverse Boundary Problems

PI：西尔维斯特DMS-9423849 西尔维斯特将继续他的偏微分和常微分方程的反问题的研究，因为它们涉及到从测量上或附近的外边界的非均匀介质的属性的确定。 应用包括医学成像、雷达成像和分层和非分层介质中的声学成像。 我们的目标是开发一个数学上严格和数值稳定的层剥离算法，首先在一个，然后在二维和三维。 偏微分方程是物理世界数学建模的基础。 数学分析的作用与其说是建立方程，不如说是提供关于解的定性和定量信息。 这可能包括关于独特性，平滑性和增长的问题的答案。 此外，分析经常发展出解的近似方法和对这些近似的准确性的估计。

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.