arXiv subjects
Publications and source records attributed to Gunther Uhlmann.
In this paper we show that for metrics with conormal singularities that correspond to class C^{1,α} with α>0, the reflected wave is more regular than the incident wave in a Sobolev sense. This is helpful in the analysis of the multiple scattering series since higher order terms can be effectively `peeled off'.
We show that for a simple surface with boundary the attenuated ray transform in the presence of a unitary connection and a skew-Hermitian Higgs field is injective modulo the natural obstruction for functions and vector fields. We also show that the connection and the Higgs field are uniquely determined by the scattering relation modulo a gauge transformation. The proofs involve a Pestov type energy identity for connections together with holomorphic gauge transformations which arrange the curvature of the connection to have definite sign.
This letter announces and summarizes results obtained in arXiv:1111.5051 and considers several natural extensions. The aforementioned paper proposes a procedure to reconstruct coefficients in a second-order, scalar, elliptic equation from knowledge of a sufficiently large number of its solutions. We present this derivation and extend it to show which parameters may or may not be reconstructed for several hybrid (also called coupled physics) imaging modalities including photo-acoustic tomography, thermo-acoustic tomography, transient elastography, and magnetic resonance elastography. Stability estimates are also proposed.
This paper concerns the reconstruction of possibly complex-valued coefficients in a second-order scalar elliptic equation posed on a bounded domain from knowledge of several solutions of that equation. We show that for a sufficiently large number of solutions and for an open set of corresponding boundary conditions, all coefficients can be uniquely and stably reconstructed up to a well characterized gauge transformation. We also show that in some specific situations, a minimum number of such available solutions equal to $I_n=\frac12n(n+3)$ is sufficient to uniquely and globally reconstruct the unknown coefficients. This theory finds applications in several coupled-physics medical imaging modalities including photo-acoustic tomography, transient elastography, and magnetic resonance elastography.
In this work we study the inverse boundary value problem of determining the refractive index in the acoustic equation. It is known that this inverse problem is ill-posed. Nonetheless, we show that the ill-posedness decreases when we increase the frequency and the stability estimate changes from logarithmic type for low frequencies to a Lipschitz estimate for large frequencies.
The advent of transformation optics and metamaterials has made possible devices producing extreme effects on wave propagation. Here we give theoretical designs for devices, Schrödinger hats, acting as invisible concentrators of waves. These exist for any wave phenomenon modeled by either the Helmholtz or Schrödinger equations, e.g., polarized waves in EM, pressure waves in acoustics and matter waves in QM, and occupy one part of a parameter space continuum of wave-manipulating structures which also contains standard transformation optics based cloaks, resonant cloaks and cloaked sensors. For EM and acoustic Schrödinger hats, the resulting centralized wave is a localized excitation. In QM, the result is a new charged quasiparticle, a \emph{quasmon}, which causes conditional probabilistic illusions. We discuss possible solid state implementations.
We prove identification of coefficients up to gauge by Cauchy data at the boundary for elliptic systems on oriented compact surfaces with boundary or domains of $\mathbb{C}$. In the geometric setting, we fix a Riemann surface with boundary, and consider both a Dirac-type operator plus potential acting on sections of a Clifford bundle and a connection Laplacian plus potential (i.e. Schrödinger Laplacian with external Yang-Mills field) acting on sections of a Hermitian bundle. In either case we show that the Cauchy data determines both the connection and the potential up to a natural gauge transformation: conjugation by an endomorphism of the bundle which is the identity at the boundary. For domains of $\mathbb{C}$, we recover zeroth order terms up to gauge from Cauchy data at the boundary in first order elliptic systems.
In this paper we study inverse boundary value problems with partial data for the magnetic Schrödinger operator. In the case of an infinite slab in $R^n$, $n\ge 3$, we establish that the magnetic field and the electric potential can be determined uniquely, when the Dirichlet and Neumann data are given either on the different boundary hyperplanes of the slab or on the same hyperplane. This is a generalization of the results of [41], obtained for the Schrödinger operator without magnetic potentials. In the case of a bounded domain in $R^n$, $n\ge 3$, extending the results of [2], we show the unique determination of the magnetic field and electric potential from the Dirichlet and Neumann data, given on two arbitrary open subsets of the boundary, provided that the magnetic and electric potentials are known in a neighborhood of the boundary. Generalizing the results of [31], we also obtain uniqueness results for the magnetic Schrödinger operator, when the Dirichlet and Neumann data are known on the same part of the boundary, assuming that the inaccessible part of the boundary is a part of a hyperplane.
We study the problem of recovery the source $a(t,x)F(x)$ in the wave equation in anisotropic medium with $a$ known so that $a(0,x)\not=0$ with a single measurement. We use Carleman estimates combined with geometric arguments and give sharp conditions for uniqueness. We also study the non-linear problem of recovery the sound speed in the equation $u_{tt} -c^2(x)Δu =0$ with one measurement. We give sharp conditions for stability, as well. An application to thermoacoustic tomography is also presented.
We consider an operator $Δ^2 + A(x)\cdot D+q(x)$ with the Navier boundary conditions on a bounded domain in $R^n$, $n\ge 3$. We show that a first order perturbation $A(x)\cdot D+q$ can be determined uniquely by measuring the Dirichlet--to--Neumann map on possibly very small subsets of the boundary of the domain. Notice that the corresponding result does not hold in general for a first order perturbation of the Laplacian.
We show that a first order perturbation $A(x)\cdot D+q(x)$ of the polyharmonic operator $(-Δ)^m$, $m\ge 2$, can be determined uniquely from the set of the Cauchy data for the perturbed polyharmonic operator on a bounded domain in $R^n$, $n\ge 3$. Notice that the corresponding result does not hold in general when $m=1$.
We present a new algorithm for reconstructing an unknown source in Thermoacoustic and Photoacoustic Tomography based on the recent advances in understanding the theoretical nature of the problem. We work with variable sound speeds that might be also discontinuous across some surface. The latter problem arises in brain imaging. The new algorithm is based on an explicit formula in the form of a Neumann series. We present numerical examples with non-trapping, trapping and piecewise smooth speeds, as well as examples with data on a part of the boundary. These numerical examples demonstrate the robust performance of the new algorithm.
It is generally believed that transformation optics based cloaking, besides rendering the cloaked region invisible to detection by scattering of incident waves, also shields the region from those same waves. We demonstrate a coupling between the cloaked and uncloaked regions, exposing a difference between cloaking for rays and waves. Interior resonances allow this coupling to be amplified, and careful choice of parameters leads to effective cloaks with degraded shielding. As one application, we describe how to use transformation optics to hide sensors in the cloaked region and yet enable the sensors to efficiently measure waves incident on the exterior of the cloak, an effect similar to the plasmon based approach of Alu' and Engheta.
We prove that a potential $q$ can be reconstructed from the Dirichlet-to-Neumann map for the Schrodinger operator $-Δ_g + q$ in a fixed admissible 3-dimensional Riemannian manifold $(M,g)$. We also show that an admissible metric $g$ in a fixed conformal class can be constructed from the Dirichlet-to-Neumann map for $Δ_g$. This is a constructive version of earlier uniqueness results by Dos Santos Ferreira et al. on admissible manifolds, and extends the reconstruction procedure of Nachman in Euclidean space. The main points are the derivation of a boundary integral equation characterizing the boundary values of complex geometrical optics solutions, and the development of associated layer potentials adapted to a cylindrical geometry.
We study the mathematical model of thermoacoustic and photoacoustic tomography when the sound speed has a jump across a smooth surface. This models the change of the sound speed in the skull when trying to image the human brain. We derive an explicit inversion formula in the form of a convergent Neumann series under the assumptions that all singularities from the support of the source reach the boundary.
We establish a decoupling result for the $P$ and $S$ waves of linear, isotropic elasticity, in the setting of twice-differentiable Lamé parameters. Precisely, we show that the $P\leftrightarrow S$ components of the wave propagation operator are regularizing of order one on $L^2$ data, by establishing the diagonalization of the elastic system modulo a $L^2$-bounded operator. Effecting the diagonalization in the setting of twice-differentiable coefficients depends upon the symbol of the conjugation operator having a particular structure.
We study the asymptotic behavior of an incompressible fluid around a bounded obstacle. The problem is modeled by the stationary Navier-Stokes equations in an exterior domain in $\R^n$ with $n\ge 2$. We will show that, under some assumptions, any nontrivial velocity field obeys a minimal decaying rate $\exp(-Ct^2\log t)$ at infinity. Our proof is based on appropriate Carleman estimates.
Consider a real-analytic orientable connected complete Riemannian manifold $M$ with boundary of dimension $n\ge 2$ and let $k$ be an integer $1\le k\le n$. In the case when $M$ is compact of dimension $n\ge 3$, we show that the manifold and the metric on it can be reconstructed, up to an isometry, from the set of the Cauchy data for harmonic $k$-forms, given on an open subset of the boundary. This extends a result of [13] when $k=0$. In the two-dimensional case, the same conclusion is obtained when considering the set of the Cauchy data for harmonic $1$-forms. Under additional assumptions on the curvature of the manifold, we carry out the same program when $M$ is complete non-compact. In the case $n\ge 3$, this generalizes the results of [12] when $k=0$. In the two-dimensional case, we are able to reconstruct the manifold from the set of the Cauchy data for harmonic $1$-forms.