Applications of microlocal analysis to some hyperbolic inverse problems
Applications of microlocal analysis to some hyperbolic inverse problems
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微局域分析在一些双曲反问题中的应用
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发表时间：
2015
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通讯作者：
Andrew J. Homan
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作者：
Andrew J. Homan
霍曼，安德鲁J博士，普渡大学，2015年5月。微局部分析在某些双曲反问题中的应用。主要教授：普拉门·斯特凡诺夫。本论文主要研究了三个反问题：热声层析成像中的超声波恢复、合成孔径雷达中的奇异点消除以及广义Radon变换的内射性和稳定性。每一个问题都是用微局部方法处理的。在阻尼波方程下的热声层析成像的上下文中，我显示的唯一性和稳定性的问题与完整的数据，提供了一个重建算法的小衰减与完整的数据，并获得稳定性估计可见的奇异性与部分数据。关于合成孔径雷达的这一章构造了几个无限维的地面反射率函数的微局部族，这些函数在使用合成孔径雷达成像时出现微局部规则。最后，基于与周汉明的合作，我们给出了一类解析广义Radon变换的解析微局部正则性，并以此证明了定义在解析黎曼流形上的一类广义Radon变换的内射性和稳定性.
Homan, Andrew J. PhD, Purdue University, May 2015. Applications of Microlocal Analysis to Some Hyperbolic Inverse Problems. Major Professor: Plamen Stefanov. This thesis compiles my work on three inverse problems: ultrasound recovery in thermoacoustic tomography, cancellation of singularities in synthetic aperture radar, and the injectivity and stability of some generalized Radon transforms. Each problem is approached using microlocal methods. In the context of thermoacoustic tomography under the damped wave equation, I show uniqueness and stability of the problem with complete data, provide a reconstruction algorithm for small attenuation with complete data, and obtain stability estimates for visible singularities with partial data. The chapter on synthetic aperture radar constructs microlocally several infinite-dimensional families of ground reflectivity functions which appear microlocally regular when imaged using synthetic aperture radar. Finally, based on a joint work with Hanming Zhou, we show the analytic microlocal regularity of a class of analytic generalized Radon transforms, using this to show injectivity and stability for a generic class of generalized Radon transforms defined on analytic Riemannian manifolds.