Seminario Análisis y Aplicaciones

Lugar: Aula Gris I, ICMAT

Fecha y hora: 10 de Mayo, 11:30-12:30
Ponente: Cristobal Meroño (UPM)
Título: Solving ill-posed inverse problems via the Born approximation
Abstract:
In this talk we discuss recent work on inverse problems for elliptic
partial differential equations revolving around the concept of Born
approximation, a tool originally introduced in the context of Scattering
Theory that has been extensively used in the physics and computational literature. We will focus on the inverse problem of recovering the potential of a Schrödinger operator from the knowledge of its Dirichlet-to-Neumann map. This is known as the Calderón problem, the mathematical model of Electrical Impedance Tomography.  This inverse problem is severely ill-posed, which makes the task of designing efficient algorithms to solve it particularly difficult. We will rigorously prove, in the simplified setting of radial potentials in the euclidean unit ball, the existence of the Born approximation, a function that encodes the whole DtN map and enjoys several interesting qualitative and quantitative approximation properties. We use this function to factorize the inverse problem into a linear (ill-posed but explicit) and a nonlinear (well-posed, Hölder continuous) part. This factorization gives a (partial) characterization of the set of DtN maps and we will show how this can be used to ultimately design efficient algorithms to solve the inverse problem. Our analysis is based on results on inverse spectral theory for Schrödinger operators on the half-line, in particular on the concept of A-amplitude introduced by Barry Simon in 1999. This talk is based on joint works with Juan Antonio Barceló, Carlos Castro, Fabricio Macià (UPM), Thierry Daudé (Besançon), and François Nicoleau (Nantes).