Inverse Boundary Problems

University Of Washington, Seattle WA

Investigators

Abstract

The proposer plans to continue his work on inverse boundary problems. For the case of Electrical Impedance Tomography (EIT) he plans to consider the problem in which one makes electrical measurements only on part of the boundary. In EIT he also plans to consider the case of conductivities having jump type singularities as well as the identifiability problem for anisotropic conductors. Another inverse boundary problem that the proposer plans to consider is whether the geodesic distance (travel times) between points on the boundary of a domain or Riemannian manifold with boundary, satisfying certain conditions, determines uniquely the metric in the interior up to an isometry which is the identity on the boundary. The third inverse problem is the problem of Reflection Seismology. In particular the proposer plans to consider the problem of seismic migration when caustics are present and the case in which the Earth is modeled as an anisotropic elastic medium instead of an acoustic medium. Inverse boundary problems are a class of problems in which one seeks to determine the internal properties of a medium by performing measurements along the boundary of the medium. These inverse problems arise in many important physical situations, ranging from geophysics to medical imaging to the non-destructive evaluation of materials. The proposer plans to consider the case of Electrical Impedance Tomography (EIT). The measurements in this case are voltage and current measurements from which one attempts to infer the internal conductivity of the medium. EIT has been proposed as a diagnostic tool in medicine since organs and tissues have quite different conductivities. Another important inverse boundary value problem arising in geophysics that the proposer plans to consider is Reflection Seismology. In this case one attempts to determine the substructure of the Earth by measuring the response at the surface produced by impulses located also on the surface. One of the main applications of this inverse problem is in locating hydrocarbons.Other potential applications are to improving ultrasound and sonar.