Kirsch, Andreas; Rieder, Andreas Inverse problems for abstract evolution equations. II: Higher order differentiability for viscoelasticity. (English) Zbl 1427.35361 SIAM J. Appl. Math. 79, No. 6, 2639-2662 (2019). MSC: 35R30 Inverse problems for PDEs 35F10 Initial value problems for linear first-order PDEs 86A22 Inverse problems in geophysics Keywords:full waveform seismic inversion; viscoelastic wave equation; adjoint state method; nonlinear inverse and ill-posed problem; higher order Fréchet derivative PDF BibTeX XML Cite \textit{A. Kirsch} and \textit{A. Rieder}, SIAM J. Appl. Math. 79, No. 6, 2639--2662 (2019; Zbl 1427.35361) Full Text: DOI References: [1] J. O. Blanch, J. O. A. Robertsson, and W. W. Symes, Modeling of a constant Q: Methodology and algorithm for an efficient and optimally inexpensive viscoelastic technique, Geophysics, 60 (1995), pp. 176-184, https://doi.org/10.1190/1.1443744. [2] T. Bohlen, Viskoelastische FD-Modellierung seismischer Wellen zur Interpretation gemessener Seismogramme, Ph.D. thesis, Christian-Albrechts-Universität zu Kiel, Kiel, Germany, 1998, https://bit.ly/2LM0SWr. [3] T. Bohlen, Parallel 3-D viscoelastic finite difference seismic modelling, Comput. Geosci., 28 (2002), pp. 887-899, https://doi.org/10.1016/S0098-3004(02)00006-7. [4] J. Dieudonné, Foundations of Modern Analysis, Pure Appl. Math. 10-I, Academic Press, New York, NY, 1969. [5] G. Fabien-Ouellet, E. Gloaguen, and B. Giroux, Time domain viscoelastic full waveform inversion, Geophysic. J. Int., 209 (2017), pp. 1718-1734, https://doi.org/10.1093/gji/ggx110. [6] A. Fichtner, Full Seismic Waveform Modelling and Inversion, Adv. Geophy. Environ. Mech. Math., Springer-Verlag, Berlin, 2011, https://doi.org/10.1007/978-3-642-15807-0. [7] A. Fichtner and J. Trampert, Hessian kernels of seismic data functionals based upon adjoint techniques, Geophys. J. Int., 185 (2011), pp. 775-798, https://doi.org/10.1111/j.1365-246X.2011.04966.x. [8] F. Hettlich and W. Rundell, A second degree method for nonlinear inverse problems, SIAM J. Numer. Anal., 37 (2000), pp. 587-620, https://doi.org/10.1137/S0036142998341246. · Zbl 0946.35115 [9] B. Hofmann, On Ill-Posedness and Local Ill-Posedness of Operator Equations in Hilbert Spaces, Tech. Report 97-8, Fakultät für Mathematik, Technische Universität Chemnitz-Zwickau, D-09107 Chemnitz, Germany, 1997, http://www.qucosa.de/fileadmin/data/qucosa/documents/4197/data/a008.pdf. [10] A. Kirsch and A. Rieder, Inverse problems for abstract evolution equations with applications in electrodynamics and elasticity, Inverse Problems, 32 (2016), 085001, 24, https://doi.org/10.1088/0266-5611/32/8/085001. · Zbl 1416.65164 [11] P. Monk, Finite Element Methods for Maxwell’s Equations, Numer. Math. Sci. Comput., Oxford University Press, New York, 2003, https://doi.org/10.1093/acprof:oso/9780198508885.001.0001. · Zbl 1024.78009 [12] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci. 44, Springer-Verlag, New York, 1983, https://doi.org/10.1007/978-1-4612-5561-1. · Zbl 0516.47023 [13] J. Virieux and S. Operto, An overview of full-waveform inversion in exploration geophysics, Geophysics, 74 (2009), pp. 1-26, https://doi.org/10.1190/1.3238367. [14] P. Yang, R. Brossier, L. Métivier, and J. Virieux, A review on the systematic formulation of \(3\) D multiparameter full waveform inversion in viscoelastic medium, Geophys. J. Int., 207 (2016), pp. 129-149, https:/doi.org/10.1093/gji/ggw262. [15] E. Zeidler, Nonlinear Functional Analysis and Its Applications. I: Fixed-Point Theorems, Springer-Verlag, New York, 1986. · Zbl 0583.47050 [16] U. Zeltmann, The Viscoelastic Seismic Model: Existence, Uniqueness and Differentiability with Respect to Parameters, Ph.D. thesis, Karlsruhe Institute of Technology, Karlsruhe, Germany, 2018, http://dx.doi.org/10.5445/IR/1000093989. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.