We study the inverse problem for the second order self-adjoint hyperbolic equation with the boundary data given on a part of the boundary. This paper is the continuation of the author's paper [E]. In [E] we presented the crucial local step of the proof. In this paper we prove the global step. Our method is a modification of the BC-method with some new ideas. In particular, the way of the determination of the metric is new.
We consider an inverse problem associated with $n$-dimensional asymptotically hyperbolic orbifolds $(n \geq 2)$ having a finite number of cusps and regular ends. By observing solutions of the Helmholtz equation at the cusp, we introduce a generalized $S$-matrix, and then show that it determines the manifolds with its Riemannian metric and the orbifold structure.
We give a survey of author's results on the inverse hyperbolic problems with time-dependent and time-independent coefficients. We consider the case of hyperbolic equations with Yang-Mills potentials and the case of domains with obstacles. In particular, we gave a stability estimate for the broken X-ray transform in the case of one convex obstacle in $\mathbb{R}^2$.
We prove that the scattering matrix at all energies restricted to an open subset of the boundary determines an asymptotically hyperbolic manifold modulo isometries that are equal to the identity on the open subset where the scattering matrix is known.
In this paper we prove stability estimates of logarithmic type for an inverse problem consisting in the determination of unknown portions of the boundary of a domain in $\mathbb{R}^n$, from a knowledge, in a finite time observation, of overdetermined boundary data for initial boundary value problem for anisotropic wave equation.
We consider an inverse problem of reconstructing two spatially varying coefficients in an acoustic equation of hyperbolic type using interior data of solutions with suitable choices of initial condition. Using a Carleman estimate, we prove Lipschitz stability estimates which ensures unique reconstruction of both coefficients. Our theoretical results are justified by numerical studies on the reconstruction of two unknown coefficients using noisy backscattered data.
We consider the inverse problem for the second order self-adjoint hyperbolic equation in a bounded domain in $\R^n$ with lower order terms depending analytically on the time variable. We prove that, assuming the BLR condition, the time-dependent Dirichlet-to-Neumann operator prescribed on a part of the boundary uniquely determines the coefficients of the hyperbolic equation up to a diffeomorphism and a gauge transformation. As a by-product we prove a similar result for the nonself-adjoint hyperbolic operator with time-independent coefficients.
In the present paper we consider an inverse source problem for time-fractional mixed parabolic-hyperbolic equation with the Caputo derivative. In case, when hyperbolic part of the considered mixed type equation is wave equation, the uniqueness of source and solution are strongly influenced by initial time and generally is ill-posed. However, when the hyperbolic part is time fractional, the problem is well posed if end time is large. Our method relies on the orthonormal system of eigenfunctions of the operator with respect to space variable. We proved the uniqueness and stability of certain wea…
In this paper, a nonlinear inverse boundary value problem for the second-order hyperbolic equation with nonlocal conditions is studied. To investigate the solvability of the original problem, we first consider an auxiliary inverse boundary value problem and prove its equivalence (in a certain sense) to the original problem. Then using the Fourier method and contraction map pings principle, the existence and uniqueness theorem for auxiliary problem is proved. Further, on the basis of the equivalency of these problems the existence and uniqueness theorem for the classical solution of the conside…
We study the inverse problems for the second order hyperbolic equations of general form with time-dependent coefficients assuming that the boundary data are given on a part of the boundary. The main result of this paper is the determination of the time-dependent Lorentzian metric by the boundary measurements. This is achieved by the adaptation of a variant of the Boundary Control method developed by the author in [E1], [E2].
We present a modification of the BC-method in the inverse hyperbolic problems. The main novelty is the study of the restrictions of the solutions to the characteristic surfaces instead of the fixed time hyperplanes. The main result is that the time-dependent Dirichlet-to-Neumann operator prescribed on a part of the boundary uniquely determines the coefficients of the self-adjoint hyperbolic operator up to a diffeomorphism and a gauge transformation. In this paper we prove the crucial local step. The global step of the proof will be presented in the forthcoming paper.
We prove stability for a formally determined inverse problem for a hyperbolic PDE where the coefficients depend on space and time variables. The hyperbolic operator has constant wave speed and we study the recovery of zeroth order and first order coefficients and the space dimension can be one or higher. We use a modification of the Bukhgeim-Klibanov method to obtain our results.
We study stability aspects for the determination of space and time-dependent lower order perturbations of the wave operator in three space dimensions with point sources. The problems under consideration here are formally determined and we establish Lipschitz stability results for these problems. The main tool in our analysis is a modified version of Bukgheim-Klibanov method based on Carleman estimates.
This paper consists of two parts. In the first part we describe the recent works on the inverse problems for the wave equation in $(n+1)$-dimensional space equipped with pseudo-Riemannian metric with Lorentz signature. We study the conditions of the existence of black (or white) holes for these wave equations. In the second part we prove energy type estimates on a finite time interval in the presence of black or white holes. We use these estimates to prove the nonuniqueness of the inverse problems.
We study the stability in the inverse problem of determining the time dependent zeroth-order coefficient $q(t,x)$ arising in the wave equation, from boundary observations. We derive, in dimension $n\geq 2$, a log-type stability estimate in the determination of $q$ from the Dirichlet-to-Neumann map, in a subset of our domain assuming that it is known outside this subset. Moreover, we prove that we can extend this result to the determination of $q$ in a larger region, and then in the whole domain provided that we have much more data.
This paper is devoted to the identification of the unknown smooth coefficient c entering the hyperbolic equation $c(x)\partial_{t}^{2}u - Δu = 0$ in a bounded smooth domain in $\R^{d}$ from partial (on part of the boundary) dynamic boundary measurements. In this paper we prove that the knowledge of the partial Cauchy data for this class of hyperbolic PDE on any open subset $Γ$ of the boundary determines explicitly the coefficient $c$ provided that $c$ is known outside a bounded domain. Then, through construction of appropriate test functions by a geometrical control method, we derive a formula…
In the present paper, we consider a non self adjoint hyperbolic operator with a vector field and an electric potential that depend not only on the space variable but also on the time variable. More precisely, we attempt to stably and simultaneously retrieve the real valued velocity field and the real valued potential from the knowledge of Neumann measurements performed on the whole boundary of the domain. We establish in dimension n greater than two, stability estimates for the problem under consideration. Thereafter, by enlarging the set of data we show that the unknown terms can be stably re…
A uniqueness result in the inverse problem for an inhomogeneous hyperbolic system on a real vector bundle over a smooth compact manifold, based on energy measurements for improperly known sources, is established.
We consider an inverse boundary value problem for the hyperbolic partial differential equation $ (-i\partial_{t} + A_{0}(t,x))^2 u(t,x) - \sum_{j=1}^n (-i\partial_{x_j} + A_{j}(t,x))^2 u(t,x) + V(t,x)u(t,x) = 0 $ with time dependent vector and scalar potentials ($\mathcal{A}= (A_{0},...,A_{m})$ and $V(t,x)$ respectively) on a bounded, smooth cylindric domain $(-\infty,\infty)\timesΩ$. Using a geometric optics construction we show that the boundary data allows us to recover integrals of the potentials along `light rays' and we then establish the uniqueness of these potentials modulo a gauge tra…
We consider a restricted Dirichlet-to-Neumann map associated to a wave type operator on a Riemannian manifold with boundary. The restriction corresponds to the case where the Dirichlet traces are supported on one subset of the boundary and the Neumann traces are restricted on another subset. We show that the restricted Dirichlet-to-Neumann map determines the lower order terms in the wave equation, up the natural gauge invariances, along a convex foliation of the manifold. We allow the lower order terms to be non-self-adjoint, and in particular, the corresponding physical system may have dissip…
We study inverse mean curvature flow with free boundary supported on geodesic spheres in hyperbolic space. Starting from any convex hypersurface inside a geodesic ball with a free boundary, the flow converges to a totally geodesic disk in finite time. Using the convergence result, we show a Willmore type inequality.
We construct a solution to inverse mean curvature flow on an asymptotically hyperbolic 3-manifold which does not have the convergence properties needed in order to prove a Penrose--type inequality. This contrasts sharply with the asymptotically flat case. The main idea consists in combining inverse mean curvature flow with work done by Shi--Tam regarding boundary behavior of compact manifolds. Assuming the Penrose inequality holds, we also derive a nontrivial inequality for functions on $S^2$.
This paper is concerned with robust preconditioning of wave equations constrained linear inverse problems from boundary observation data. The main result of this paper is a concept for regularization parameter robust preconditioning. Analogous concepts have been developed for control problems based on elliptic partial equations before.
We consider inverse curvature flows in hyperbolic space with starshaped initial hypersurface, driven by positive powers of a homogeneous curvature function. The solutions exist for all time and, after rescaling, converge to a sphere.
We introduce the shifted inverse curvature flow in hyperbolic space. This is a family of hypersurfaces in hyperbolic space expanding by $F^{-p}$ with positive power $p$ for a smooth, symmetric, strictly increasing and $1$-homogeneous curvature function $f$ of the shifted principal curvatures with some concavity properties. We study the maximal existence and asymptotical behavior of the flow for horo-convex hypersurfaces. In particular, for $0<p\leq 1$ we show that the limiting shape of the solution is always round as the maximal existence time is approached. This is in contrast to the asymptot…
In this article we are concerned with an inverse boundary value problem for a non-linear wave equation of divergence form with space dimension $n\geq 3$. In particular the so called the interior determination problem. This non-linear wave equation has a trivial solution, i.e. zero solution. By linearizing this equation at the trivial solution, we have the usual linear isotropic wave equation with the speed $\sqrt{γ(x)}$ at each point $x$ in a given spacial domain. For any small solution $u=u(t,x)$ of this non-linear equation, we have the linear isotropic wave equation perturbed by a divergence…
In this paper we give a more geometrical formulation of the main theorem in [E1] on the inverse problem for the second order hyperbolic equation of general form with coefficients independent of the time variable. We apply this theorem to the inverse problem for the equation of the propagation of light in a moving medium (the Gordon equation). Then we study the existence of black and white holes for the general hyperbolic and for the Gordon equation and we discuss the impact of this phenomenon on the inverse problems.
We describe the general setting for the optical Aharonov-Bohm effect based on the inverse problem of the identification of the coefficients of the governing hyperbolic equation by the boundary measurements. We interpret the inverse problem result as a possibility in principle to detect the optical Aharonov-Bohm effect by the boundary measurements.
We prove gradient estimates for hypersurfaces in the hyperbolic space $\mathbb{H}^{n+1},$ expanding by negative powers of a certain class of homogeneous curvature functions. We obtain optimal gradient estimates for hypersurfaces evolving by certain powers $p>1$ of $F^{-1}$ and smooth convergence of the properly rescaled hypersurfaces. In particular, the full convergence result holds for the inverse Gauss curvature flow of surfaces without any further pinching condition besides convexity of the initial hypersurface.
We consider a second-order hyperbolic equation on an open bounded domain $Ω$ in $\mathbb{R}^n$ for $n\geq2$, with $C^2$-boundary $Γ=\paΩ=\bar{Γ_0\cupΓ_1}$, $Γ_0\capΓ_1=\emptyset$, subject to non-homogeneous Neumann boundary conditions on the entire boundary $Γ$. We then study the inverse problem of determining the interior damping coefficient of the equation by means of an additional measurement of the Dirichlet boundary trace of the solution, in a suitable, explicit sub-portion $Γ_1$ of the boundary $Γ$, and over a computable time interval $T>0$. Under sharp conditions on the complementary pa…
We consider inverse curvature flows in $\Hh$ with star-shaped initial hypersurfaces and prove that the flows exist for all time, and that the leaves converge to infinity, become strongly convex exponentially fast and also more and more totally umbilic. After an appropriate rescaling the leaves converge in $C^\infty$ to a sphere.
The paper is devoted to the development of the theory of inverse problems for evolution equations with terms rapidly oscillating in time. A new approach to setting such problems is developed for the case in which additional constraints are imposed only on several first terms of the asymptotics of the solution rather that on the whole solution. This approach is realized in the case of a multidimensional hyperbolic equation with unknown absolute term.
We consider an inverse problem associated with some 2-dimensional non-compact surfaces with conical singularities, cusps and regular ends. Our motivating example is a Riemann surface $\mathcal M = Γ\backslash{\bf H}^2$ associated with a Fuchsian group of the 1st kind $Γ$ containing parabolic elements. $\mathcal M$ is then non-compact, and has a finite number of cusps and elliptic singular points, which is regarded as a hyperbolic orbifold. We introduce a class of Riemannian surfaces with conical singularities on its finite part, having cusps and regular ends at infinity, whose metric is asympt…
In analogy with radiation fields for asymptotically Euclidean manifolds, introduced by F.G. Friedlander, we define radiation fields for asymptotically hyperbolic manifolds. We use them to give a translation representation of the wave group and to obtain the scattering matrix for such manifolds. Furthermore, we prove a support theorem for the radiation fields which generalizes to this setting the well known support theorem of Helgason and Lax and Phillips for the horocyclic Radon transform. This is used to show that the manifold and the metric, up to invariants, are determined by the scattering…
Suppose $q_i(x)$, $i=1,2$ are smooth functions on $\R^3$ and $U_i(x,t)$ the solutions of the initial value problem {gather*} \pa_t^2 U_i- ΔU_i - q_i(x) U_i = δ(x,t), \qquad (x,t) \in \R^3 \times \R U_i(x,t) =0, \qquad \text{for} ~ t<0. {gather*} Pick $R,T$ so that $0 < R < T$ and let $C$ be the vertical cylinder $\{(x,t) \, : |x|=R, ~ R \leq t \leq T \}$. We show that if $(U_1, U_{1r}) = (U_2, U_{2r})$ on $C$ then $q_1 = q_2$ on the annular region $R \leq |x| \leq (R+T)/2$ provided there is a $γ>0$, independent of $r$, so that \[\int_{|x|=r} | Δ_S (q_1 - q_2)|^2 \, dS_x \leq γ\int_{|x|=r} |q_1…
In this article we are concerned with an inverse initial boundary value problem for a non-linear wave equation in space dimension $n\geq 2$. In particular we consider the so called interior determination problem. This non-linear wave equation has a trivial solution, i.e. zero solution. By linearizing this equation at the trivial solution, we have the usual linear wave equation with a time independent potential. For any small solution $u=u(t,x)$ of this non-linear equation, it is the perturbation of linear wave equation with time-independent potential perturbed by a divergence with respect to $…
Scattering is defined on compact manifolds with boundary which are equipped with an asymptotically hyperbolic metric, $g.$ A model form is established for such metrics close to the boundary. It is shown that the scattering matrix at energy $ζ$ exists and is a pseudo-differential operator of order $2ζ+1 - \dim X.$ The symbol of the scattering matrix is then used to show that except for a countable set of energies the scattering matrix at one energy determines the diffeomorphism class of the metric modulo terms vanishing to infinite order at the boundary. An analogous result is proved for potent…
We consider the multidimensional inverse problem of determining the conductivity coefficient of a hyperbolic equation in an infinite cylindrical domain, from a single boundary observation of the solution. We prove H{ö}lder stability with the aid of a Carleman estimate specifically designed for hyperbolic waveguides.
We study the well-posedness of the Bayesian inverse problem for scalar hyperbolic conservation laws where the statistical information about inputs such as the initial datum and (possibly discontinuous) flux function are inferred from noisy measurements. In particular, the Lipschitz continuity of the measurement to posterior map as well as the stability of the posterior to approximations, are established with respect to the Wasserstein distance. Numerical experiments are presented to illustrate the derived estimates.
