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Register a new userRecovery of a Smooth Metric via Wave Field and Coordinate Transformation Reconstruction
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In this paper, we study the inverse boundary value problem for the wave equation with a view towards an explicit reconstruction procedure. We consider both the anisotropic problem where the unknown is a general Riemannian metric smoothly varying in a domain, and the isotropic problem where the metric is conformal to the Euclidean metric. Our objective in both cases is to construct the metric, using either the Neumann-to-Dirichlet (N-to-D) map or Dirichlet-to-Neumann (D-to-N) map as the data. In the anisotropic case we construct the metric in the boundary normal (or semi-geodesic) coordinates via reconstruction of the wave field in the interior of the domain. In the isotropic case we can go further and construct the wave speed in the Euclidean coordinates via reconstruction of the coordinate transformation from the boundary normal coordinates to the Euclidean coordinates. Both cases utilize a variant of the Boundary Control method, and work by probing the interior using special boundary sources. We provide a computational experiment to demonstrate our procedure in the isotropic case with N-to-D data.
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We study the problem of unique recovery of a non-smooth one-form $mathcal A$ and a scalar function $q$ from the Dirichlet to Neumann map, $Lambda_{mathcal A,q}$, of a hyperbolic equation on a Riemannian manifold $(M,g)$. We prove uniqueness of the one-form $mathcal A$ up to the natural gauge, under weak regularity conditions on $mathcal A,q$ and under the assumption that $(M,g)$ is simple. Under an additional regularity assumption, we also derive uniqueness of the scalar function $q$. The proof is based on the geometric optic construction and inversion of the light ray transform extended as a Fourier Integral Operator to non-smooth parameters and functions.
Let $c$ be a piecewise smooth wave speed on $mathbb R^n$, unknown inside a domain $Omega$. We are given the solution operator for the scalar wave equation $(partial_t^2-c^2Delta)u=0$, but only outside $Omega$ and only for initial data supported outside $Omega$. Using our recently developed scattering control method, we prove that piecewise smooth wave speeds are uniquely determined by this map, and provide a reconstruction formula. In other words, the wave imaging problem is solvable in the piecewise smooth setting under mild conditions. We also illustrate a separate method, likewise constructive, for recovering the locations of interfaces in broken geodesic normal coordinates using scattering control.
We study the inverse problem of recovery a non-linearity $f(x,u)$, which is compactly supported in $x$, in the semilinear wave equation $u_{tt}-Delta u+ f(x,u)=0$. We probe the medium with either complex or real-valued harmonic waves of wavelength $sim h$ and amplitude $sim 1$. They propagate in a regime where the non-linearity affects the subprincipal but not the principal term, except for the zeroth harmonics. We measure the transmitted wave when it exits $text{supp}_x f$. We show that one can recover $f(x,u)$ when it is an odd function of $u$, and we can recover $alpha(x)$ when $f(x,u)=alpha(x)u^{2m}$. This is done in an explicit way as $hto0$.
We show that given two hyperbolic Dirichlet to Neumann maps associated to two Riemannian metrics of a Riemannian manifold with boundary which coincide near the boundary are close then the lens data of the two metrics is the same. As a consequence, we prove uniqueness of recovery a conformal factor (sound speed) locally under some conditions on the latter.
We consider wave equations on Lorentzian manifolds in case of low regularity. We first extend the classical solution theory to prove global unique solvability of the Cauchy problem for distributional data and right hand side on smooth globally hyperbolic space-times. Then we turn to the case where the metric is non-smooth and present a local as well as a global existence and uniqueness result for a large class of Lorentzian manifolds with a weakly singular, locally bounded metric in Colombeaus algebra of generalized functions.
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