scholarly journals On an inverse boundary problem arising in brain imaging

2019 ◽  
Vol 267 (4) ◽  
pp. 2471-2502 ◽  
Author(s):  
Youjun Deng ◽  
Hongyu Liu ◽  
Gunther Uhlmann
Keyword(s):  
Brain Imaging ◽  
Boundary Problem ◽  
Inverse Boundary ◽  
Inverse Boundary Problem

An inverse boundary problem for fourth-order Schrödinger equations with partial data

2019 ◽  
Vol 69 (1) ◽  
pp. 125-138
Author(s):  
Zhiwen Duan ◽  
Shuxia Han
Keyword(s):  
Boundary Problem ◽  
Fourth Order ◽  
Schrodinger Equation ◽  
Geometrical Optics ◽  
Cauchy Data ◽  
Carleman Estimates ◽  
Schrödinger Equations ◽  
Inverse Boundary ◽  
Partial Data ◽  
Inverse Boundary Problem

Abstract In this paper, we show that in dimension n ≥ 3, the knowledge of the Cauchy data for the fourth-order Schrödinger equation measured on possibly very small subsets of the boundary determines uniquely the potential. The proof is based on the Carleman estimates and the construction of complex geometrical optics solutions.

The inverse boundary problem for the Rayleigh system

1995 ◽  
Vol 36 (12) ◽  
pp. 6688-6708 ◽  
Author(s):  
Richard Beals ◽  
G. M. Henkin ◽  
N. N. Novikova
Keyword(s):  
Boundary Problem ◽  
Inverse Boundary ◽  
Inverse Boundary Problem ◽  
Rayleigh System

scholarly journals Boundary regularity for the Ricci equation, geometric convergence, and Gel?fand?s inverse boundary problem

2004 ◽  
Vol 158 (2) ◽  
pp. 261-321 ◽  
Author(s):  
Michael Anderson ◽  
Atsushi Katsuda ◽  
Yaroslav Kurylev ◽  
Matti Lassas ◽  
Michael Taylor
Keyword(s):  
Boundary Problem ◽  
Boundary Regularity ◽  
Inverse Boundary ◽  
Geometric Convergence ◽  
Inverse Boundary Problem

Local uniqueness for the inverse boundary problem for the two-dimensional diffusion equation

2000 ◽  
Vol 11 (5) ◽  
pp. 473-489 ◽  
Author(s):  
N. I. GRINBERG
Keyword(s):  
Diffusion Equation ◽  
Boundary Problem ◽  
Light Propagation ◽  
Two Dimensional ◽  
Scattering Media ◽  
Absorption Coefficients ◽  
Inverse Boundary ◽  
Linear Transport Equation ◽  
Inverse Boundary Problem ◽  
Dirichlet To Neumann Map

We study an inverse boundary problem for the diffusion equation in ℝ2. Our motivation is that this equation is an approximation of the linear transport equation and describes light propagation in highly scattering media. The diffusion equation in the frequency domain is the nonself-adjoint elliptic equation div(D grad u) - (cμa + iω0) u = 0; ω0 ≠ 0, where D and μa are the diffusion and absorption coefficients. The inverse problem is the reconstruction of D and μa inside a bounded domain using only measurements at the boundary. In the two-dimensional case we prove that the Dirichlet-to-Neumann map, corresponding to any one positive frequency ω0, determines uniquely both the diffusion and the absorption coefficients, provided they are sufficiently slowly-varying. In the null-background case we estimate analytically how large these coefficients can be to guarantee uniqueness of the reconstruction.

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