In fields such as oil and gas prospecting, mapping the interior of a planet, or medical imaging, one determines properties of the interior of an object such as the location of oil/gas deposits in the interior of the earth, characterize the interior composition of a planet, or determine if an interior lump in the body is cancerous or not. Since drilling or cutting is often expensive or unfeasible in these situations, these objects are probed by non-invasive methods such as sound waves generated on the boundary of the object. The expectation is that the interior composition of the object will influence the incoming waves and the response wave, also measured only on the boundary of the object, provides a mathematical window into the interior of the object. The principal investigator (PI) will study the mathematics behind this imaging technique. Further, the PI will train graduate students and postdocs in this type of mathematics through mini-courses, seminars, personal conversations and workshops. Some of these students and postdocs will use these skills to solve practical problems for companies exploring for oil, building imaging devices, or involved in remote sensing.
Problems like those described above, but with over-determined data, where the unknown function depends on fewer variables than the data, have received a lot of attention. The PI focuses on the less studied formally determined problems where the unknown function depends on the same number of variables as the data. Such problems, in two or more space dimensions, are harder but very useful in situations where data acquisition is difficult, and their investigation is one of the important challenges in the field. This project will study the following problems in three space dimensions: the fixed angle scattering problem, the backscattering problem, the point source problem, and the incoming spherical wave problem. Recently, using an adaptation of the Bukhgeim-Klibanov method, the PI and his collaborators proved stability for the fixed angle scattering problem for coefficients which are even in one of the variables and proved uniqueness for the problem of recovering a coefficient given data from the point source problem as well as the incoming spherical wave problem. This project aims at further adapting the Bukhgeim-Klibanov method to use Robbiano-Tataru type Carleman estimates and unique continuation arguments to tackle the proposed problems.
This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
