非均匀介质中波动反演问题的一个数学框架
A mathematical framework for inverse wave problems in heterogeneous media
摘要：
本文基于系数有界可测的双曲型线性积分微分方程组，为波传播反问题的一些常用公式提供了理论基础。这些时间相关的偏微分方程的系数表示参数的空间变化的材料的力学性能。岩石、人造材料和其他波传播环境通常在各种尺度下表现出力学性质的空间异质性，代表这些性质的系数函数必须模拟这种异质性。我们展示了如何选择域（类的非光滑系数函数）和数据定义（弱解的痕迹），使逆波问题的优化配方满足一些先决条件的应用牛顿法及其亲属。这些结果来自一类抽象的一阶演化系统的性质，其中各种物理波系统作为具体实例出现。有限速度的传播线性波有界的，可测量的力学参数场是这个理论的副产品之一。
英文摘要：
This paper provides a theoretical foundation for some common formulations of inverse problems in wave propagation, based on hyperbolic systems of linear integro-differential equations with bounded and measurable coefficients. The coefficients of these time-dependent partial differential equations respresent parametrically the spatially varying mechanical properties of materials. Rocks, manufactured materials, and other wave propagation environments often exhibit spatial heterogeneity in mechanical properties at a wide variety of scales, and coefficient functions representing these properties must mimic this heterogeneity. We show how to choose domains (classes of nonsmooth coefficient functions) and data definitions (traces of weak solutions) so that optimization formulations of inverse wave problems satisfy some of the prerequisites for application of Newton's method and its relatives. These results follow from the properties of a class of abstract first-order evolution systems, of which various physical wave systems appear as concrete instances. Finite speed of propagation for linear waves with bounded, measurable mechanical parameter fields is one of the by-products of this theory.
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