Local Inverse Problems

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The principal investigator will study inverse problems arising in seismic imaging, cosmology, and medical imaging. The first of these is the mathematical problem of recovering the structure of the Earth from seismic measurements. The project takes into account that the Earth is best modeled as an elastic medium; that it consist of several cores, and that it is anisotropic (i.e., that the speed of seismic signals can depend on the direction). A second problem is motivated by applications to cosmology: to determine the stage of the early Universe, to the extent possible, from the cosmic microwave background radiation measurements. Finally, the principal investigator will study the mathematics of new medical imaging methods that use two different waves to form an image: one wave (electromagnetic, for example) is sent to the human body to excite the cells, which creates another type of wave (acoustic, for example) that one measures away from the body. In all the aforementioned examples, one is especially interested in solving local problems using local information: when measurements are done locally (say, on some part of the boundary of the relevant domain), one's desire is to recover the object locally, as well. Indeed, in many practical applications such measurements can only be done locally, and very often one is interested in the object under scrutiny only in a region near the point where the measurement is taken. The goal is to understand how much information is included in the measured data, to determine how sensitive such information is to noise and measurement errors, and to devise a means of using this data to reconstruct the object. More specifically, the project will pursue the following avenues of research: (1) recovery of a Riemannian metric, up to isometry, on a compact manifold with boundary near a strictly convex boundary point from localized lens/distance boundary data; (2) recovery of a Lorentzian metric up to a gauge transformation from boundary measurements; (3) recovery of both types of metrics from wave equation data on the boundary; (4) solving the inverse problem for the elastic geophysics model with piecewise smooth Lame parameters; (5) inversion of the geodesic and the light ray transforms; and (6) solving inverse source problems that arise in medical imaging. Theses include both linear and nonlinear problems, from well-posed to mildly ill-posed to ill-posed. The aim is to recover the leading term in the underlying hyperbolic equation, which determines the geometry. Unlike much previous work in the area, this project allows for the existence of conjugate points, albeit modulo some additional geometric assumptions. The problem of local recovery a Riemannian metric in boundary/lens rigidity, up to isometry, is very challenging because simple linearization does not work, and there is inherent nonuniqueness. The principal investigator expects to combine ideas from tensor tomography, Melrose's scattering calculus, and other new ideas to deal with the nonlinearity. His methods would prove global uniqueness and stability as well, under the condition that the manifold admits a strictly convex foliation. Existence of conjugate points is not excluded. The inverse problems in time-space, including the integral geometry ones, lack ellipticity because only space-like singularities are recoverable. This makes the problem ill-posed. The light ray transform is a restricted ray transform, requiring specialized microlocal tools, not even fully developed yet for nonflat metrics. Those problems are of interest in geometry (where they are called rigidity problems), in the part of the theory of partial differential equations known as inverse kinematic and inverse boundary-value problems, and in applications like geophysics.