Tangential cone condition for the full waveform forward operator in the viscoelastic regime: the nonlocal case. (English) Zbl 1536.35371
Summary: We discuss mapping properties of the parameter-to-state map of full waveform inversion and generalize the results of M. Eller and A. Rieder [Inverse Probl. 37, No. 8, Article ID 085011, 17 p. (2021; Zbl 1486.65149)] from the acoustic to the viscoelastic wave equation. In particular, we establish injectivity of the Fréchet derivative of the parameter-to-state map for a semidiscrete seismic inverse problem in the viscoelastic regime. Here the finite-dimensional parameter space is restricted to functions having global support in the propagation medium (the nonlocal case) and that are locally linearly independent. As a consequence, we deduce local conditional well-posedness of this nonlinear inverse problem. Furthermore, we show that the tangential cone condition holds, which is an essential prerequisite in the convergence analysis of a variety of inversion algorithms for nonlinear ill-posed problems.
MSC:
| 35R30 | Inverse problems for PDEs |
| 35L45 | Initial value problems for first-order hyperbolic systems |
| 35R09 | Integro-partial differential equations |
| 65J20 | Numerical solutions of ill-posed problems in abstract spaces; regularization |
Keywords:
nonlinear ill-posed problem; tangential cone condition; Lipschitz stability; full waveform seismic inversion; viscoelastic wave equationCitations:
Zbl 1486.65149References:
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