Browsing by Author "Blazek, Kirk D."

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    A mathematical framework for inverse wave problems in heterogeneous media
    (IOP Publishing, 2013) Blazek, Kirk D.; Stolk, Christiaan; Symes, William W.; The Rice Inversion Project
    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.
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    Well-posedness of Initial/Boundary Value Problems for Hyperbolic Integro-differential Systems with Nonsmooth Coefficients
    (2008-10) Blazek, Kirk D.; Stolk, Christiaan; Symes, William W.
    In the late 1960's, J.-L. Lions and collaborators showed that energy estimates could be used to establish existence, uniqueness, and continuous dependence on initial data for finite energy solutions of initial/boundary value problems for various linear partial differential evolution equations with nonsmooth coefficients. The second author has recently treated second order hyperbolic systems, for example linear elastodynamics, by similar methods, and extended these techniques to demonstrate continuous dependence and even differentiability (in a suitable sense) of the solution as function of the coefficients. In the present paper, we extend Lions' results in a different direction, to first order symmetric hyperbolic integrodifferential systems (such as linear viscoelasticity) with bounded and measurable coefficients. We show that the initial value problem is well-posed in an appropriate space of finite-energy weak solutions, and that solutions of this class are continuous as functions of the coefficients and data. This result is sharp, in the sense that solutions are not in general locally uniformly continuous in coefficients and data. Solutions are however (G�teaux-)differentiable as a function of the coefficients for fixed data, in case the data is smooth enough that the time derivative 1 of the solution is itself a finite-energy weak solution. The continuity result combines with the well- known domain of influence properties for hyperbolic systems with smooth coefficients to show that viscoelasticity with bounded, measureable coefficients predicts finite wave propagation speed.