Lipschitz stability for the electrostatic inverse boundary value problem with piecewise linear conductivities

Abstract

We consider the electrostatic inverse boundary value problem also known as electrical impedance tomography (EIT) for the case where the conductivity is a piecewise linear function on a domain ⊂ R^n and we show that a Lipschitz stability estimate for the conductivity in terms of the local Dirichlet-to-Neumann map holds true

Calderón’s Inverse Problem With a Finite Number of Measurements

“…where we called e l = e k l and emphasized the dependence on q i . By (7) and the assumption t k = c ′ (|k| s + 1), we obtain…”

“…These are the first stability estimates for the Gel'fand-Calderón and Calderón problems with a finite number of measurements. Lipschitz stability results have been previously known only when an infinite number of measurements are available [8,12,11,23,10,7,6] The exponentially growing constant is coherent with the exponential instability of the problem [29,19,26,38].…”

Lipschitz stability for the finite dimensional fractional Calderón problem with finite Cauchy data

“…Remark 4.8. If it is known that (u 1 , (−∆) s u 1 ) ∈ C 1 (p (1) ) and (u 2 , (−∆) s u 2 ), (v 2 , (−∆) s v 2 ) ∈ C 2 (p (2) ) for some functions p (1) , p (2) as in the definition of the finite Cauchy data, then it is possible to replace the term d(C 1 , C 2 ) in (19) by the quantity d(C 1 (p (1) ), C 2 (p (2) )).…”

Lipschitz stability for a piecewise linear Schrödinger potential from local Cauchy data

“…The use of complex frequencies was studied in [41,25]. Coming back to the object of the present paper, let us point out that various new aspects appear in comparison to prior results of stability under assumptions of piecewise constant or piecewise linear coefficients [8,5,12] which require novel arguments. I) Singular solutions.…”