Enclosure Methods for the Helmholtz-type Equations

This paper serves as a survey of enclosure-type methods used to determine the obstacles or inclusions embedded in the background medium from the near-field measurements of propagating waves. A type of complex geometric optics waves that exhibits exponential decay with distance from some critical level surfaces (hyperplanes, spheres or other types of level sets of phase functions) are sent to probe the medium. One can easily manipulate the speed of decay such that the waves can only detect the material feature that is close enough to the level surfaces. As a result of sending such waves with level surfaces moving along each direction, one should be able to pick out those that enclose the inclusion. The problem that Calderón proposed in 80’s [3] was whether one can determine the electrical conductivity by making voltage and current measurements at the boundary of the medium. Such electrical methods are also known as Electrical Impedance Tomography (EIT) and have broad applications in medical imaging, geophysics and so on. A breakthrough in solving the problem was due to Sylvester and Uhlmann. In [26], they constructed the complex geometric optics (CGO) solutions to the conductivity equation and proved the unique determination of C∞ isotropic conductivity from the boundary measurements in three and higher dimensional spaces. The result has been extended to conductivities with 3/2 derivatives in three dimensions and L∞ conductivities in two dimensions. The inverse problem in this paper concerns reconstructing an obstacle or a jump-type inclusion embedded in a known background medium, which is not included in the previous results when considering electrostatics. Several methods are proposed to solve the problem based on utilizing, generally speaking, two special types of solutions. The Green’s type solutions were considered first by Isakov [13], and several sampling methods [4, 14, 1, 2] and probing methods [10, 24] were developed. On the other hand, with the CGO solutions at disposal, the enclosure method was introduced by Ikehata [8, 9] with the idea as described in the first paragraph. Another method worth mentioning uses the oscillating-decaying type of solutions and was