904 research outputs found
Uniqueness for a wave propagation inverse problem in a half space
No full textLassas, Matti; Cheney, M.; Uhlmann, Gunther. (1997). Uniqueness for a wave propagation inverse problem in a half space. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/3152
Boundary determination of a Riemannian metric by the localized boundary distance function
No full text
Some Linear and Nonlinear Geometric Inverse Problems
No full textThesis (Ph.D.)--University of Washington, 2015Inverse 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 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
Geodesic X-Ray Transform on Asymptotically Hyperbolic Manifolds
No full textThesis (Ph.D.)--University of Washington, 2020This dissertation contains work of the author and joint work with C. Robin Graham concerning the geodesic X-ray transform in the setting of asymptotically hyperbolic manifolds. It is divided into three self contained chapters, each addressing a different question. The topic of the first chapter is the local injectivity of the X-ray transform, extending a result proved by Uhlmann and Vasy on compact manifolds with boundary. Assuming knowledge of the X-ray transform for geodesics contained in a small neighborhood of a boundary point we show local injectivity for asymptotically hyperbolic metrics even modulo O(\rho^5) in dimension 3 and higher. In the second chapter we construct examples of asymptotically hyperbolic metrics demonstrating that in the asymptotically hyperbolic setting absence of conjugate points does not suffice to exclude boundary conjugate points. The construction uses techniques developed by Gulliver and clarifies the definition of a simple asymptotically hyperbolic manifold, formulated by Graham, Guillarmou, Stefanov and Uhlmann. In the third chapter we show a stability estimate for the X-ray transform on simple asymptotically hyperbolic manifolds, extending to this setting the work of Stefanov and Uhlmann on simple compact manifolds with boundary
Inverse Boundary-Value Problems on an Infinite Slab
No full textThesis (Ph.D.)--University of Washington, 2015In this work, we study the stability aspect of two inverse boundary-value problems (IBVPs) on an infinite slab with partial data. The uniqueness aspects of these IBVPs were considered and studied by Li and Uhlmann for the case of the Schrödinger equation as well as by Krupchyk, Lassas and Uhlmann for the case of the magnetic Schrödinger equation. Here we quantify the method of uniqueness proposed by Li and Uhlmann and prove a log-log stability estimate for the IBVPs associated to the Schrödinger equation. The boundary measurements considered in these problems are modelled by partial knowledge of the Dirichlet-to-Neumann map; more precisely, we establish log-log stability estimates for each of the following two IBVPs: in the first inverse problem, the corresponding Dirichlet and Neumann data are known on different boundary hyperplanes of the slab; in the second inverse problem, they are known on the same boundary hyperplane of the slab
Nonlinear PDEs: regularity, rigidity, and an inverse problem
No full textThesis (Ph.D.)--University of Washington, 2021Based on joint work with Arunima Bhattacharya, we obtain a sharp regularity result for Lagrangian mean curvature type equations with possibly H\"older continuous Lagrangian phases. Along the way, the constant rank theorem of Bian and Guan is generalized, and a different, lower regularity way to prove strict convexity is developed. Next, based on joint work with Yu Yuan, we show that smooth semiconvex solutions of the sigma-2 equation are quadratic polynomials if they are entire. Finally, a Calder\'on type inverse problem for quasilinear elliptic equations is discussed, where the author improves a recent result of Mu\~noz and Uhlmann using boundary jet linearization
- …
