Dynamic Inverse Wave Problems - Part I: Regularity for the Direct\n Problem

2018/08/02 by Thies Gerken, Gerken, Thies, Simon Grützner +1
Earth and Planetary Sciences · Engineering · Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Hydraulic Fracturing and Reservoir Analysis #Numerical methods in inverse problems #Seismic Imaging and Inversion Techniques

paper · pdf · doi:10.48550/arxiv.1808.00804

openalex publication_date 2018/08/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For parameter identification problems the Fr 'echet-derivative of the\nparameter-to-state map is of particular interest. In many applications, e.g. in\nseismic tomography, the unknown quantity is modeled as a coefficient in a\nlinear differential equation, therefore computing the derivative of this map\ninvolves solving the same equation, but with a different right-hand side. It\nthen remains to show that this right-hand side is regular enough to ensure the\nexistence of a solution. For second-order hyperbolic PDEs with time-dependent\nparameters the needed results are not as readily available as in the stationary\ncase, especially when working in a variational framework. This complicates for\nexample the reconstruction of a time-dependent density in the wave equation. To\novercome this problem we extend the existing regularity results to the\ntime-dependent case.\n

Related

• Recover all Coefficients in Second-Order Hyperbolic Equations from Finite Sets of Boundary Measurements
• Stability for a formally determined inverse problem for a hyperbolic PDE with space and time dependent coefficients
• A Lipschitz stable reconstruction formula for the inverse problem for the wave equation
• Inverse Boundary Value Problem for Non-linear Hyperbolic Partial Differential Equations
• On an inverse dynamic problem for the wave equation with a potential on a real line