Inverse Problems and Scattering Poles

在这笔资助下，Stefanov计划在两个领域进行研究：(1)逆问题和(2)共振（散射极点）。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。一些令人感兴趣的问题是各向异性介质中的反问题。在数学上，它们会导致从边界数据或散射数据中恢复黎曼度量的问题。一个例子是逆运动学问题，在这个问题中，人们必须从连接每两个边界点的测地线的长度中恢复有界域中的黎曼度量。这个问题在地球物理学中有自然的应用，并且与从边界上相关的双曲Dirichlet-to-Neumann （DN）映射中恢复度规的逆问题密切相关。pii计划在比已知的更大的度量类别中研究这个问题，并证明稳定性估计。相关的线性化问题——沿测地线积分的恢复常量也将被研究。PI也适用于（椭圆型）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，并且PI计划找到足够的条件，以保证人们可以直接从线性化问题的注入性中得出稳定性估计（因此是唯一性）。PI还将在共振（散射极）领域进行研究。PI计划研究一些可以构造非实准模的物理系统，并证明这些准模接近于不收敛于实轴的共振。两个这样的系统是传输问题和严格凸障碍物的散射问题。这个项目的动机来自于它可能的应用。逆问题在地质、医学、材料无损评价等领域都有广泛的应用。各向异性反演问题涉及介质的属性参数（电导率、波速等）不仅取决于位置，而且取决于方向。条件稳定性估计表明，在附加假设下，系统参数的微小变化会导致测量数据的微小变化，通常要求系数具有足够的平滑性。这证明了相应的逆问题是条件适定的。共振作为共振频率自然出现在无界域中的系统中。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的某些邻域的渐近分布，这为理解这些系统提供了计算和定性工具。

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.