RECORD
Recovery of time dependent coefficients from boundary data for hyperbolic equations
Analysis of PDEs · Submitted 14 Jan 2019
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim The Cauchy data is observed on time-like parts of the space-time boundary and uniqueness is proved up to the natural gauge for the problem.
The compiled record
The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
Open the compiled record
The abstract, sentence by sentence
We study uniqueness of the recovery of a time-dependent magnetic vector-valued potential and an electric scalar-valued potential on a Riemannian manifold from the knowledge of the Dirichlet to Neumann map of a hyperbolic equation. The Cauchy data is observed on time-like parts of the space-time boundary and uniqueness is proved up to the natural gauge for the problem. The proof is based on Gaussian beams and inversion of the light ray transform on Lorentzian manifolds under the assumptions that the Lorentzian manifold is a product of a Riemannian manifold with a time interval and that the geodesic ray transform is invertible on the Riemannian manifold.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We study uniqueness of the recovery of a time-dependent magnetic vector-valued potential and an electric scalar-valued potential on a Riemannian manifold from the knowledge of the Dirichlet to Neumann map of a hyperbolic equation. | contribution 0.96 | 0.85 | 0.22 | asserted 0.66 |
| s1 | The Cauchy data is observed on time-like parts of the space-time boundary and uniqueness is proved up to the natural gauge for the problem. | result 0.92 | 0.89 | 0.58 | asserted 1.00 |
| s2 | The proof is based on Gaussian beams and inversion of the light ray transform on Lorentzian manifolds under the assumptions that the Lorentzian manifold is a product of a Riemannian manifold with a time interval and that the geodesic ray transform is invertible on the Riemannian manifold. | method 1.00 | 0.72 | 0.09 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v1
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
The paper
Original abstract
We study uniqueness of the recovery of a time-dependent magnetic vector-valued potential and an electric scalar-valued potential on a Riemannian manifold from the knowledge of the Dirichlet to Neumann map of a hyperbolic equation. The Cauchy data is observed on time-like parts of the space-time boundary and uniqueness is proved up to the natural gauge for the problem. The proof is based on Gaussian beams and inversion of the light ray transform on Lorentzian manifolds under the assumptions that the Lorentzian manifold is a product of a Riemannian manifold with a time interval and that the geodesic ray transform is invertible on the Riemannian manifold.
Human review
How the literature uses it
12 arXiv papers cite this in 25 sentences
Papers citing it · 4 sentences without a confident label
Uses its method or tool 5 papers
-
As in [ref] , we have the following estimate by the construction of $u_ρ$ (cf. [ref]) [equation] We construct Gaussian beam solutions of the forms [equation] concentrating near the same null-geodesic $ϑ(t)=(t,γ(t))$ as in [ref] .
Nonlinear ultrasound imaging modeled by a Westervelt equation · Construction of Gaussian beam solutions · 0.96
-
As in [ref] , we now have the Jacobi weighted ray transform of the first kind (cf. [ref] ) of $𝒞c_P^-9/2ρ^-3/2=:f$ along the geodesic $γ$ in $(Ω,g_P)$ passing through $x_0=γ(t_0)$, [equation] for any $Y∈𝕐_γ$.
On an inverse boundary value problem for a nonlinear elastic wave equation · Determination of $𝒞$ · 0.95
-
Adapting the approach in [ref] and [ref] , we now prove the following lemma.
Inverse problems for the spectral fractional Laplacian with inhomogeneous Dirichlet boundary data · Proof of Theorem [ref] · 0.97
Compares with it 1 paper
-
The only known results are in the case $G=U(1)$, see [ref] , and these results impose convexity assumptions not satisfied by the geometric setting of Figure [ref].
Inverse problem for the Yang-Mills equations · Discussion and comparison with previous literature · 0.83
Cites it as background 9 papers
-
Light ray transform results in the Lorentzian setting exist as well: [ref] for functions, [ref] for one-forms, and [ref] for higher order tensor fields.
The Lorentzian scattering rigidity problem and rigidity of stationary metrics · Introduction · 0.86
-
This transform has been studied in [ref] .
The Light Ray transform on Lorentzian manifolds · A time dependent potential · 0.89
-
We note here that Gaussian beams have been used to study various inverse problems [ref] .
On an inverse boundary value problem for a nonlinear elastic wave equation · Introduction · 0.90
What it cites · 2 sentences without a confident label
Compares with 1 paper
-
Let us point out that, even when $𝒜 = 0$, combining Theorem [ref] and [ref] gives a result on unique determination of smooth $q$ that falls outside the scope of the closest previous results [ref] .
The geodesic X-ray transform with matrix weights · On inversion of the geodesic ray transform · 0.92
Cites as background 1 paper
-
We refer the reader to [ref] for an introduction to the method and to [ref] for an example of a state of the art result in this direction.
Application of the boundary control method to partial data Borg-Levinson inverse spectral problem · Previous literature · 0.89
Discussion
Know something about this paper? Leave a comment, or highlight any passage above to comment on it.