2020/03/30 by Faouzi Triki, Triki, Faouzi, Yin Tao +1
Earth and Planetary Sciences · Environmental Science · Mathematics · #35R30 #65N21 #Analysis of PDEs (math.AP) #FOS: Mathematics #Groundwater flow and contamination studies #Numerical Analysis (math.NA) #Numerical methods in inverse problems #Seismic Imaging and Inversion Techniques
paper · doi:10.48550/arxiv.2003.13638
openalex publication_date 2020/03/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper concerns the reconstruction of a scalar coefficient of a second-order elliptic equation in divergence form posed on a bounded domain from internal data. This theory finds applications in multi-wave imaging, greedy methods to approximate parameter-dependent elliptic problems, and image treatment with partial differential equations. We first show that the inverse problem for smooth coefficients can be rewritten as a linear transport equation. Assuming that the coefficient is known near the boundary, we study the well-posedness of associated transport equation as well as its numerical resolution using discontinuous Galerkin method. We propose a regularized transport equation that allow us to derive rigorous convergence rates of the numerical method in terms of the order of the polynomial approximation as well as the regularization parameter. We finally provide numerical examples for the inversion assuming a lower regularity of the coefficient, and using synthetic data.