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 represent 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.
Highlights
Inverse problems for waves in heterogeneous materials occur in seismology, ultrasonic nondestructive evaluation and biomedical imaging, some electromagnetic imaging technologies, and elsewhere
Inverse Problems 29 (2013) 065001 wave fields on space-time hypersurfaces, and nonlinear least-squares methods developed for their numerical solution, that is, for finding the coefficients in the systems of partial differential equations chosen to model the wave physics
Numerical solutions to discretized inverse problems respect the underlying continuum physics only insofar as they converge under refinement of the discretization, and such convergence generally requires that the continuum problems have well-behaved solutions
Summary
Inverse problems for waves in heterogeneous materials occur in seismology, ultrasonic nondestructive evaluation and biomedical imaging, some electromagnetic imaging technologies, and elsewhere. It is very useful: as will be explained, it justifies certain infeasible-model methods for inverse problems in wave propagation Another useful by-product of the theory is the finite speed of propagation property for hyperbolic systems with bounded, measurable coefficients, a result which so far as we can tell is new. Having explained our main results and some important examples to which they apply, we devote the sixth section to a discussion of the related literature and various implications for inverse problems in wave propagation, including the importance of the general (operator coefficient) case of the abstract theory in formulating certain inversion algorithms, representation of energy sources, and several other matters not addressed in the remainder of this paper.
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