scholarly journals A sharp stability estimate for tensor tomography in non-positive curvature

2020 ◽  
Author(s):  
Gabriel P. Paternain ◽  
Mikko Salo
Keyword(s):  
Positive Curvature ◽  
Initial Boundary ◽  
Stability Estimate ◽  
Tensor Fields ◽  
Ray Transform ◽  
Tensor Tomography ◽  
Beam Geometry ◽  
Sharp Stability ◽  
Simply Connected ◽  
Natural Parametrization

AbstractWe consider the geodesic X-ray transform acting on solenoidal tensor fields on a compact simply connected manifold with strictly convex boundary and non-positive curvature. We establish a stability estimate of the form $$L^2\mapsto H^{1/2}_{T}$$ L 2 ↦ H T 1 / 2 , where the $$H^{1/2}_{T}$$ H T 1 / 2 -space is defined using the natural parametrization of geodesics as initial boundary points and incoming directions (fan-beam geometry); only tangential derivatives at the boundary are used. The proof is based on the Pestov identity with boundary term localized in frequency.

scholarly journals Sharp stability estimate for the geodesic ray transform

2020 ◽  
Vol 36 (2) ◽  
pp. 025013 ◽  
Author(s):  
Yernat M Assylbekov ◽  
Plamen Stefanov
Keyword(s):  
Stability Estimate ◽  
Ray Transform ◽  
Sharp Stability ◽  
Geodesic Ray Transform

scholarly journals Stability of the Non-abelian X-ray Transform in Dimension $$\ge 3$$

2021 ◽  
Author(s):  
Jan Bohr
Keyword(s):  
Stability Estimate ◽  
Scattering Data ◽  
Regularity Conditions ◽  
Uniform Bounds ◽  
Ray Transform ◽  
Shallow Layer ◽  
Matrix Potential ◽  
Novel Method ◽  
Statistical Consistency ◽  
Appl Math

AbstractNon-abelian X-ray tomography seeks to recover a matrix potential $$\Phi :M\rightarrow {\mathbb {C}}^{m\times m}$$ Φ : M → C m × m in a domain M from measurements of its so-called scattering data $$C_\Phi $$ C Φ at $$\partial M$$ ∂ M . For $$\dim M\ge 3$$ dim M ≥ 3 (and under appropriate convexity and regularity conditions), injectivity of the forward map $$\Phi \mapsto C_\Phi $$ Φ ↦ C Φ was established in (Paternain et al. in Am J Math 141(6):1707–1750, 2019). The present article extends this result by proving a Hölder-type stability estimate. As an application, a statistical consistency result for $$\dim M =2$$ dim M = 2 (Monard et al. in Commun Pure Appl Math, 2019) is generalised to higher dimensions. The injectivity proof in (Paternain et al. in Am J Math 141(6):1707–1750, 2019) relies on a novel method by Uhlmann and Vasy (Invent Math 205(1):83–120, 2016), which first establishes injectivity in a shallow layer below $$\partial M$$ ∂ M and then globalises this by a layer stripping argument. The main technical contribution of this paper is a more quantitative version of these arguments, in particular, proving uniform bounds on layer depth and stability constants.

scholarly journals AN INFINITE NUMBER OF CLOSED FLRW UNIVERSES FOR ANY VALUE OF THE SPATIAL CURVATURE

2012 ◽  
Vol 18 ◽  
pp. 77-80
Author(s):  
HELIO V. FAGUNDES
Keyword(s):  
Finite Volume ◽  
Infinite Number ◽  
Negative Curvature ◽  
Large Scale ◽  
Positive Curvature ◽  
Cosmological Models ◽  
Infinite Volume ◽  
Spatial Curvature ◽  
Multiply Connected ◽  
Simply Connected

The Friedman-Lemaître-Robertson-Walker cosmological models are based on the assumptions of large-scale homogeneity and isotropy of the distribution of matter and energy. They are usually taken to have spatial sections that are simply connected; they have finite volume in the positive curvature case, and infinite volume in the null and negative curvature ones. I want to call the attention to the existence of an infinite number of models, which are based on these same metrics, but have compact, finite volume, multiply connected spatial sections. Some observational implications are briefly mentioned.

scholarly journals Recovery of transversal metric tensor in the Schrödinger equation from the Dirichlet-to-Neumann map

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Mourad Bellassoued ◽  
Zouhour Rezig
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Compact Riemannian Manifold ◽  
Stability Estimate ◽  
Metric Tensor ◽  
X Ray ◽  
Ray Transform ◽  
Boundary Measurements ◽  
Dirichlet To Neumann Map ◽  
Direct Use

<p style='text-indent:20px;'>In this paper, we deal with the inverse problem of determining simple metrics on a compact Riemannian manifold from boundary measurements. We take this information in the dynamical Dirichlet-to-Neumann map associated to the Schrödinger equation. We prove in dimension <inline-formula><tex-math id="M1">\begin{document}$ n\geq 2 $\end{document}</tex-math></inline-formula> that the knowledge of the Dirichlet-to-Neumann map for the Schrödinger equation uniquely determines the simple metric (up to an admissible set). We also prove a Hölder-type stability estimate by the construction of geometrical optics solutions of the Schrödinger equation and the direct use of the invertibility of the geodesical X-ray transform.</p>

scholarly journals Smoothness of solutions of parabolic equations in regions with edges

1981 ◽  
Vol 84 ◽  
pp. 159-168 ◽  
Author(s):  
A. Azzam ◽  
E. Kreyszig
Keyword(s):  
Parabolic Equation ◽  
Boundary Value Problem ◽  
Bounded Domain ◽  
Parabolic Equations ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Smoothness Of Solutions ◽  
Initial Boundary Value ◽  
Simply Connected

We consider the mixed initial-boundary value problem for the parabolic equationin a region Ω × (0, T], where x = (x1, x2) and Ω ⊂ R2 is a simply-connected bounded domain having corners.

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