Inverse Boundary Problems

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This project will address the mathematical theory of several fundamental inverse problems arising in many areas of science and technology, including medical imaging, geophysics, and nondestructive testing. Four major topics of research are proposed. The first is travel-time tomography in anisotropic media. In mathematical terms this involves the determination of a Riemannian metric (anisotropic sound speed) in the interior of a domain from the lengths of geodesics joining points of the boundary (travel times) and from other kinematic information. The second topic is electric impedance tomography (EIT), also called Calderon's problem. In this inverse method one attempts to determine the conductivity of a medium by making voltage and current measurements at its boundary. The third topic focuses on coupled-physics or hybrid inverse problems. In this type of inverse problem one attempts to determine the internal properties of a medium by combining two types of waves through a physical principle; namely, one wave that has high resolution (e.g., ultrasound) and another that provides high contrast. Examples are photoacoustic tomography (PAT), thermoacoustic tomography (TAT), and transient elastography (TE). The fourth topic is on invisibility and cloaking: how to make objects invisible to different types of waves.In inverse boundary problems one attempts to determine the internal properties of a medium by making measurements at the boundary of the medium. In other words, can one "see" what is inside by making measurements on the outside? An example is a CT scan, a commonly used medical imaging technique. One measures the response of the body to X-rays and makes an image of what is inside from this information. The project will investigate new proposed medical imaging techniques, such as photoacoustic tomography. This combines the high resolution of one imaging method with the high contrast capabilities of another. One important medical imaging application is breast cancer detection. Ultrasound provides a high (sub-millimeter) resolution, but suffers from low contrast. On the other hand, many tumors absorb much more energy from electromagnetic waves than do healthy cells. Photoacoustic tomography consists of sending relatively harmless optical radiation into tissues. This causes heating (with increases of the temperature in the millikelvin range), which results in the generation of propagating ultrasound waves (the photo-acoustic effect). Another area that the project will explore is to image the Earth's interior by measuring the time that it takes for seismic waves to traverse it. In this way one attempts a journey through the center of the Earth with the help of information provided by earthquakes. A final major topic of research in the project is the study of cloaking and invisibility. Can one make objects invisible to light, sound, and other types of waves?