Inverse Problems and Scattering Poles

在此资助下，Stefanov计划从事两个领域的工作：（1）逆问题和（2）共振（散射极）。在反问题方面，他计划研究一些偏微分方程的反问题和相关的几何反问题。一些感兴趣的问题是各向异性介质中的反问题。从数学上讲，它们会导致从边界数据或散射数据恢复黎曼度量的问题。一个例子是逆运动学问题，其中一个必须恢复一个黎曼度量在一个有界域的长度的测地线连接每两个边界点。这个问题在物理学中有着自然的应用，并且与从边界上的双曲Dirichlet-to-Neumann（DN）映射中恢复度量的反问题密切相关。PI计划研究这个问题的更大类的指标比已知的和证明稳定性估计。也将研究相关的线性化问题--从沿着测地线的积分中恢复张量. PI也将工作在（椭圆）各向异性反边值问题。还将研究开发一种抽象方法来获得条件稳定性估计的可能性，PI计划找到保证可以导出稳定性估计的条件（因此唯一性）直接从线性化问题的注入性。PI还将在共振领域进行研究PI计划研究一些物理系统，在这些系统中，人们可以构造非实准模，并证明这些准模接近于不收敛到真实的轴的共振。两个这样的系统是传输问题和散射的一个严格凸的障碍。这个项目的动机来自于它的可能的应用。反问题在地质、医学、材料无损评价等方面有着广泛的应用。各向异性反问题涉及到恢复具有性质（电导率、波速等）的介质参数。这不仅取决于位置，还取决于方向。条件稳定性估计表明，在额外的假设条件下，系统参数的微小变化会导致测量数据的微小变化，通常需要系数具有足够的平滑性.这表明相应的反问题是条件适定的。无界域中系统的共振频率。它们对应于测量数据中的峰值。对于某些系统，我们计划描述共振在实轴附近的渐近分布，这为理解这些系统提供了计算和定性的工具。

ABSTRACTDMS-0070823Under this grant, Stefanov plans to work in two areas: (1) Inverse Problems and(2) Resonances (scattering poles). In the area of Inverse Problems, he plans tostudy some inverse problems for partial differential equations and relatedinverse problems in geometry. Some of the problems of interest are inverseproblems in anisotropic media. Mathematically, they lead to problems ofrecovering a Riemannian metric from boundary data or scattering data. Oneexample is the inverse kinematic problem, where one has to recover a Riemannianmetric in a bounded domain from the lengths of geodesics connecting each twoboundary points. This problem has natural applications in geophysics and isclosely connected to the inverse problem of recovering a metric from theassociated hyperbolic Dirichlet-to-Neumann (DN) map on the boundary. The PIplans to study this problem for larger classes of metrics than already known andto prove also stability estimates. The related linearized problem - recovery oftensors from integrals along geodesics will also be studied. The PI will alsowork on the (elliptic) anisotropic inverse boundary value problem. Thepossibility of developing an abstract approach to obtaining conditionalstability estimates will be also studied and the PI plans to find sufficientconditions that would guarantee that one can derive a stability estimate (andtherefore uniqueness) directly from the injectivity of the linearized problem.The PI will also conduct research in the area of resonances (scattering poles).The PI plans to study some physical systems where one can construct non-realquasimodes and to show that those quasimodes are close to resonances notconverging to the real axis. Two such systems are the transmission problem andscattering by a strictly convex obstacle.The motivation for this project comes from its possible applications. InverseProblems have numerous applications to geology, medicine, non-destructiveevaluation of materials, etc. The anisotropic inverse problems are related torecovering parameters of media with properties (conductivity, wave speed, etc.)depending not only on the position but also on the direction. Conditionalstability estimates show that small changes of the parameters of the system leadto small changes in the measured data under additional assumptions usuallyrequiring sufficient smoothness of the coefficients. This demonstrates that thecorresponding inverse problem is conditionally well-posed. Resonances arisenaturally as resonance frequencies for systems in unbounded domains. Theycorrespond to peaks in the measured data. For some systems, we plan to describethe asymptotic distribution of the resonances in some neighborhood of the realaxis, which provides computational and qualitative tools for understanding thosesystems.