Some Linear and Nonlinear Geometric Inverse Problems

Abstract

Inverse problems is an area at the interface of several disciplines and has become a prominent research topic due to its potential applications. A wide range of these problems can be formulated under various geometric settings, and we call them {\it geometric inverse problems}. In this thesis, we study several linear and nonlinear geometric inverse problems. See Chapter 1 for a more detailed introduction. Chapter 2 is devoted to an integral geometry problem regarding the invertibility of local weighted X-ray transforms of functions along general smooth curves. We extend the results on the local invertibility of geodesic ray transform proved by Uhlmann and Vasy \cite{UV15} to X-ray transforms along general curves. In particular, our method shows that the geodesic nature of the curves does not play an essential role in this problem. In Chapter 3, as a joint work with Yernat Assylbekov, we consider the boundary rigidity problem with respect to Hamiltonian systems involving both magnetic fields and potentials. We establish various rigidity results of such systems on compact manifolds with boundary. Unlike the cases of geodesic or magnetic systems, knowing boundary data of one energy level is insufficient for unique determination of our systems, we provide some counterexamples. Given a bounded domain in $\mathbb{R}^n$ with a conformally Euclidean metric, in Chapter 4, we develop an explicit reconstruction procedure for the inverse problem of recovering a semigeodesic neighborhood of the boundary of the domain and the conformal factor in this neighborhood from some internal data. The key ingredient is the relation between the reconstruction procedure and a Cauchy problem of the conformal Killing equation. This is a joint work with Leonid Pestov and Gunther Uhlmann.