Inverse Problems for Wave Phenomena

这个数学研究项目由planen Stefanov负责研究波传播现象中产生的逆问题。它们包括：(a)从测地线射线变换中恢复函数或张量场的积分几何逆问题；(b)合成孔径雷达（SAR）成像变换的微局部反演；(c)黎曼几何中的非线性边界和透镜刚性问题；(d)变速波动方程的反边界问题；(e)多波医学成像中波动方程的反问题。在(a)、(c)、(d)中，我们假设存在共轭点，目的是研究它们对可逆性及其稳定性的影响。该项目的统一主题是在复杂几何中进行反演。在所有这些情况下，一个直接的反演都失败了，因为微局部Bolker条件不再成立。在重建中，人们经常在共轭对或SAR的镜像点等处得到伪影。Stefanov将研究的基本问题是这些人工制品是否确实存在；如果可能的话，如何去除它们；以及（如果不可能的话）如何描述它们。长期目标是了解复杂介质中波动现象的线性和非线性反问题及其稳定性，其中焦散是不可避免的，并且可能失去稳定性。这些问题的研究工作基于并将进一步发展数学中几个领域的方法：黎曼几何和积分几何、反问题、微局部分析，包括解析微局部分析，以及经典泛函分析和偏微分方程理论。Plemen Stefanov的这个数学研究项目是由合成孔径雷达成像（SAR）、医学成像和地球物理学中出现的问题所激发的。在SAR中，Stefanov将研究实际问题，例如什么是产生图像重建最佳数据的最佳飞行路径；Stefanov还将为这个问题没有确切答案的情况开发一种近似重建算法。本项目研究的边界和透镜刚度问题是地震层析成像的数学基础：从地震波的传播时间中恢复地球（和太阳）的内部结构；它们也发生在超声成像的调查中。测地线射线变换问题是后者的一种特殊情况（线性化）。地震学中也出现了反波动方程问题，因为它试图利用更多关于波的信息——不仅仅是到达的时间，还有波的形状。多波问题是热声、光声成像等新兴医学成像方法的数学基础；它们属于新型的多波（混合）医学成像方法，将一种波的高对比度与另一种波的高分辨率相结合。

This mathematics research project by Plamen Stefanov concerns the study of inverse problems arising in wave propagation phenomena. They include the following: (a) inverse problems in integral geometry of recovery a function or a tensor field from the geodesic ray transform; (b) microlocal inversion of transforms appearing in Synthetic Aperture Radar (SAR) imaging; (c) non-linear boundary and lens rigidity questions in Riemannian geometry; (d) inverse boundary problems for wave equations with variable speed; and (e) inverse problems for the wave equation arising in multiwave medical imaging. In (a), (c), and (d), we assume that there are conjugate points and the goal is to study their effect on the invertibility and its stability. The unifying theme in the project is that the inversion is done in complex geometry. A straightforward inversion in all those cases fails due to the fact that the microlocal Bolker condition does not hold anymore. One often gets artifacts in the reconstruction placed at conjugate pairs or at mirror points in SAR, etc. The fundamental questions that Stefanov will study are whether those artifacts are indeed there; how to remove them, if possible; and, (if not possible) how to characterize them. The long term goal is to understand linear and non-linear inverse problems and their stability for wave phenomena in complex media, where caustics are inevitable and stability might be lost. The work on those problems is based upon, and will further develop, methods in several areas in mathematics: Riemannian and Integral geometry, Inverse Problems, microlocal analysis, including analytic microlocal analysis, as well as classical functional analysis and PDE theory. This mathematics research project by Plemen Stefanov is motivated by problems arising in Synthetic Aperture Radar imaging (SAR), medical imaging and geophysics. In SAR, Stefanov will study practical questions such as what are the best flight paths that produce the best data for image reconstruction; Stefanov will also develop an approximate reconstruction algorithm for situations where there is no exact answer to this question. The boundary and the lens rigidity problems studied in this project are the mathematical foundation of Seismic Tomography: to recover the inner structure of the Earth (and the Sun) from the travel times of seismic waves; they also occur in the investigation of ultrasound imaging. The geodesic ray transform problems are a special case (linearization) of the latter. The inverse wave equation problems arise in Seismology as well, as an attempt to use more information about the wave -- not just the time of arrival but also the shape of the wave. The multiwave problems are the mathematical foundation of emerging medical imaging methods like Thermoacoustic and Photoacoustic Tomography; they belong to the new class of multiwave (hybrid) medical imaging methods which combine the high contrast of one wave with the high resolution of another.