Carleman Weight Research Articles

Nowhere conformally homogeneous manifolds and limiting Carleman weights

In this note we prove that a generic Riemannian manifold of dimension $\geq 3$ does not admit any nontrivial local conformal diffeomorphisms. This is a conformal analogue of a result of Sunada concerning local isometries, and makes precise the principle that generic manifolds in high dimensions do not have conformal symmetries. Consequently, generic manifolds of dimension $\geq 3$ do not admit nontrivial conformal Killing vector fields near any point. As an application to the inverse problem of Calderón on manifolds, this implies that generic manifolds of dimension $\geq 3$ do not admit limiting Carleman weights near any point.

Limiting Carleman weights and anisotropic inverse problems

In this article we consider the anisotropic Calderon problem and related inverse problems. The approach is based on limiting Carleman weights, introduced in Kenig-Sjoestrand-Uhlmann (Ann. of Math. 2007) in the Euclidean case. We characterize those Riemannian manifolds which admit limiting Carleman weights, and give a complex geometrical optics construction for a class of such manifolds. This is used to prove uniqueness results for anisotropic inverse problems, via the attenuated geodesic X-ray transform. Earlier results in dimension $n \geq 3$ were restricted to real-analytic metrics.

Carleman estimates and inverse problems for Dirac operators

We consider limiting Carleman weights for Dirac operators and prove corresponding Carleman estimates. In particular, we show that limiting Carleman weights for the Laplacian also serve as limiting weights for Dirac operators. As an application we consider the inverse problem of recovering a Lipschitz continuous magnetic field and electric potential from boundary measurements for the Pauli Dirac operator.

Comparative studies of the globally convergent convexification algorithm with application to imaging of antipersonnel land mines

A comparative study of various aspects on the globally convergent convexification algorithm for coefficient inverse problems is described. Numerical results of the algorithm with application to detection of antipersonnel land mines are presented. The aspects studied include initial guess and layer size for the layer stripping approach, dimensions of the basis for spatial and pseudo-frequency approximations, the lower limit of the integrals with respect to pseudo-frequency κ, the stopping criteria of steepest descent method in the least squares minimization, the tails to compensate the truncated integration with respect to κ, the parameter λ associated with the Carleman weight function, and adding different noise levels to the input data for the coefficient inverse problems. With our new implementation, the convexification algorithm is very efficient and feasible to be applied in real time to detect antipersonnel land mines in the field.

Determining nonsmooth first order terms from partial boundary measurements

We extend results of Dos Santos Ferreira-Kenig-Sjöstrand-Uhlmann (Comm. Math. Phys., 2007) to less smooth coefficients, and we show that measurements on part of the boundary for the magnetic Schrödinger operator determine uniquely the magnetic field related to a Hölder continuous potential. We give a similar result for determining a convection term. The proofs involve Carleman estimates, a smoothing procedure, and an extension of the Nakamura-Uhlmann pseudodifferential conjugation method to logarithmic Carleman weights.

Four variations in global Carleman weights applied to inverse and controllability problems

This paper reviews four variants of global Carleman weights that are especially adapted to some singular controllability and inverse problems in partial differential equations. These variants arise when studying: i) one measurement stationary source inverse problems for the heat equation with discontinuous coefficients, ii) one measurement stationary potential inverse problems for the heat equation with discontinuous coefficients, iii) null controllability for fluid-structure problems in mobile domains and iv) recovering coefficients from locally supported boundary observations for the wave equation. In all the case we explain how to explicitly construct the Carleman weight.

Global Convexity in a Three-Dimensional Inverse Acoustic Problem

We consider an inverse scattering problem (ISP) for the acoustic equation $% u_{tt}=c_{}^2(x)\Delta u,u|_{t=0}=0,u_t|_{t=0}=\delta (x),x\in {\Bbb R}^3.$ The ISP consists of the determination of the speed of sound $c(x)$ inside a bounded domain $\Omega \subset {\Bbb R}^3$ given $c(x)$ outside $\Omega $ and measurements of the amplitude $u(x,t)$ of the sound at the boundary $% \partial \Omega ,\;u|_{_{\partial \Omega }}=\varphi (x,t).$ This problem is nonoverdetermined since only a single source location at $\left\{ 0\right\}$ is counted. Assuming regularity of the rays generated by $c(x)$ and using the Carleman's weight functions, we construct a cost functional $J_\lambda $. The main result is Theorem 3.1, which claims global strict convexity of $J_\lambda $ on "reasonable" compact sets of solutions. Therefore, global convergence on such a set of a number of standard minimization algorithms to the unique global minimum of $J_\lambda $ (i.e., solution of the ISP) is guaranteed. This in turn shows a possibility of constructions of numerical methods for this ISP which would not be affected by the problem of local minima.

Global convexity in a single-source 3-D inverse scattering problem

The authors consider nonoverdetermined 3-D inverse scattering problems based on the telegraph equation u tt = Δu + a(x)u t + b(x)u + δ(x, t), u| t<0 = 0, x ∈ R 3 . The goal is to recover the media properties represented by the coefficients a(x) or b(x) from, for example, backscattering data. Such a problem models imaging in certain biological tissues, in murky water, and in some geophysical and atmospheric phenomena. The main restriction is in the consideration of only finitely many Fourier harmonics of the solution u(x, t) (in the time variable t only). This seems to be acceptable for practical computations. The main mathematical tool is the construction of uniformly convex cost functionals on compact convex subsets of the solutions. This assures a global convergence of the minimization algorithm, which can be applied in the case of large media inhomogeneities. Since the technique is based on the so-called Carleman's weight functions, the approach is called Carleman's weight method.