A mathematical framework for inverse wave problems in heterogeneous media

TL;DR

This paper develops a mathematical framework for inverse wave problems in heterogeneous media, focusing on hyperbolic systems with measurable coefficients, enabling better understanding and solution of inverse problems in complex materials.

Contribution

It introduces a theoretical foundation for inverse wave problems with measurable coefficients, ensuring conditions for optimization methods like Newton's method are satisfied.

Findings

Finite speed of wave propagation in heterogeneous media.

Conditions for the applicability of Newton's method in inverse problems.

Framework accommodates spatial heterogeneity in material properties.

Abstract

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.