The proposer will study problems in scattering theory and several non-linear inverse problems that can be formulated as boundary and lens rigidity questions and related tensor tomography inverse problems. Boundary and lens rigidity for compact Riemannian manifolds with boundary are inverse problems where one wants to recover the manifold from the distance (travel times)between each two boundary points, or the scattering relation on the boundary. A linearization of these problems is the integral geometry problem of recovering of a tensor field from integrals along maximal geodesics. The proposer will study this type of questions for various systems: Riemannian manifolds, asymptotically hyperbolic manifolds, Lorentzian manifolds, and non-metric Hamiltonians. The case of caustics will be studied carefully. The motivation to study those problems comes both from pure mathematics: rigidity questions in geometry, inverse scattering, inverse boundary value problems for hyperbolic equations, math theory of relativity, integral geometry of tensors; as well from applications to geophysics, medical imaging, oil exploration, non-destructive testing, cosmology and conformal field theory, etc. The proposer will study stable ways to recover the parameters of the system (the metric, the Hamiltonian, etc.). One of goals of this project in the scattering theory part is to study the asymptotic distribution of resonances and it relation to the classical mechanical behavior of the system.
The Inverse Problems that will be studied in this project serve as mathematical models in many practical situations: in medicine for imaging the internal structure of a human body (CT, ultrasound, thermoacoustic tomography); in non-destructive material testing; in geophysics for obtaining information about the inner structure of the earth from travel times of seismic waves, or from travel times of reflected acoustic waves; in oil exploration; in theory of relativity, theoretical physics, etc. Riemannian metrics model anisotropic media, where the speed of wave propagation may depend not only on the position but also on the direction and this project's emphasis is on the study of anisotropic media. Anisotropy naturally occurs in Earth, in the human body, etc. The proposed research will develop further the mathematical tools of travel time tomography, and will analyze the stability in various situations. The theory of resonances, as a part of scattering theory, is of fundamental importance for quantum mechanics, quantum chemistry. Resonances can be observed as certain peak frequencies and one of the objectives of this research is to relate them to the properties of the system.
The PI studied fundamental questions arising in the mathematical theory of Inverse Problems and Tomography motivated by applications in medical imaging, and seismology. We use methods coming from Microlocal Analysis, Geometry and the PDE theory. We are mainly concerned with stability, besides uniqueness. One set of problems consists of the non-linear traveltime tomography problem arising in geophysics and the related integral geometry problems of integrating functions or tensor fields along geodesics. In particular, we study geometries with conjugate points (caustics). We analyze stability or the lack if it, support theorems, uniqueness and recovery of singularities. We also study several problems arising in medical imaging, like Thermo- and Photo-acoustic Tomography (TAT and PAT), SPECT, and coupled-physics imaging, closely related to TAT and PAT. We present a thorough analysis of the TAT/PAT problems for variable speeds both with full data and with partial data. We present if and only if conditions for uniqueness and such conditions for stability, as well as a direct reconstruction for full data. We also study the case of a discontinuous sound speed modeling brain imaging. We show that recovery of both the speed and the source is unstable on the linearization level. We show that the SPECT problem of recovering both the attenuation and the source is ill posed in general and has no unique solution; but under some assumptions it can be well posed. We find that the stability of the problem is linked to a certain Hamiltonian system. We also present numerical examples illustrating the theory. We study an integral geometry problem related to SAR and we show that in general, artifcats due to left/right ambiguity exist even for curved flight paths; they cannot be removed; and are of equal strength. If the support of the function is known a priori to be compact, then recovery is possible but it is not equivalent to the R^*R technique. We also published a work that looks into what makes abstract non-linear inverse problems stable or not. We show that stability (and injectivity) of the linearization, even with loss of finitely many derivatives, leads to Holder stability of the non-linear problem; and give and review the techniques for proving stability of the linearization, using microlocal methods.
