Mathematical Sciences: Inverse Problems for Hyperbolic Partial Differential Equations

9404283 Symes本项目将侧重于地球物理学中出现的反问题，特别是根据在物体表面或接近物体表面进行的测量来确定或估计介质的折射率或波速。这类问题的场方程组是双曲型的，或与双曲组密切相关。将研究几个关键的数学问题，每个问题都具有重大的实际意义：1)在焦散存在的情况下波动方程线性化反问题的整体估计；2)奇点的非光滑传播与线性化之间的关系；3)目标函数的设计，以给出反问题的拟凸优化公式；以及4)衰减和色散的调节；例如，粘弹性反问题。纯粹的和应用的地球物理学的遥感技术使用测量地球表面附近的声场或电磁场来推断地球的内部结构。这些场由代表地球一部分的区域中的一些适当的场方程的解来模拟。这些方程的系数代表了分布的物理参数，如电导率、硬度、密度等。因此，遥感技术提出了反问题：根据边界附近的场的测量来估计区域内部的场方程的系数。这项研究涉及到那些实验设计和参数范围的模型，这些设计和参数范围使探测场由传播波组成。

9404283 Symes This project will focus on inverse problems arising in geophysics, in particular, the determination or estimation of the index of refraction, or wave velocity, of a medium from measurements made at or near the surface of the body. The field equations for such problems are hyperbolic, or are closely related to hyperbolic systems. Several key mathematical questions will be investigated, each with substantial practical implications: 1) global estimates for a linearized inverse problem for the wave equation in the presence of caustics; 2) the relation between nonsmooth propagation of singularities and linearization; 3) design of objective functions to give quasiconvex optimization formulations of inverse problems; and 4) accommodation of attenuation and dispersion; for example, inverse problems in viscoelasticity. The remote sensing techniques of pure and applied geophysics use measurements of an acoustic or electromagnetic field near the earth's surface to infer the internal structure of the earth. These fields are modeled by the solutions of some appropriate field equations in a domain representing a portion of the earth. The coefficients of these equations represent distributed physical parameters such as conductivity, stiffness, density, etc. Thus remote sensing techniques pose inverse problems: to estimate coefficients of field equations in the interior of a domain from measurements of fields near the boundary. This research concerns models of those experimental designs and parameter ranges which cause the probe fields to consist of propagating waves.