2015/09/09 by Alexandre Munnier, Munnier, Alexandre, Karim Ramdani +1
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Composite Material Mechanics #FOS: Mathematics #Numerical methods in inverse problems #math.AP
paper · pdf · doi:10.48550/arxiv.1509.02693
arxiv created 2015/09/09 · openalex publication_date 2015/09/09 · arxiv updated 2015/09/10 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
In this paper, we address a classical case of the Calderón (or conductivity) inverse problem in dimension two. We aim to recover the location and the shape of a single cavity ω (with boundary γ) contained in a domain Ω (with boundary Γ) from the knowledge of the Dirichlet-to-Neumann (DtN) map Λγ: f \longmapsto ∂n uf|Γ, where uf is harmonic in Ω∖ω, uf|Γ=f and uf|γ=cf, cf being the constant such that ∫γ∂n uf \rm ds=0. We obtain an explicit formula for the complex coefficients am arising in the expression of the Riemann map z\longmapsto a1 z + a0 + ∑m\leqslant -1 am zm that conformally maps the exterior of the unit disk onto the exterior of ω. This formula is derived by using two ingredients: a new factorization result of the DtN map and the so-called generalized Pólia-Szegö tensors (GPST) of the cavity. As a byproduct of our analysis, we also prove the analytic dependence of the coefficients am with respect to the DtN. Numerical results are provided to illustrate the efficiency and simplicity of the method.