Increasing Stability for the Conductivity and Attenuation Coefficients

Abstract

Abstract. In this work we consider stability of recovery of the conductivity and attenuation coefficients of the stationary Maxwell and Schrödinger equations from a complete set of (Cauchy) boundary data. By using complex geometrical optics solutions we derive some bounds which can be viewed as an evidence of increasing stability in these inverse problems when frequency is growing.

Inverse Scattering Source Problems

“…As I mentioned before, there are many scientist and researcher have been working on inverse scattering and more specifically on inverse source problems. To expand your knowledge and further mathematical development in this field of research, please see the result authors in [29][30][31][32][33][34][35][36][37][38][39][40][41], which were discussed different aspects of the problems.…”

“…Also authors in [23] proved a stability estimate using just Dirichlet data. Increasing stability for the Schrodinger potential from the complete set of the boundary data (the Dirichlet-to Neumann map) was demonstrated in [25,26]. They showed that the boundary condition for elastic waves they considered the following equation…”

Inverse source problems without (pseudo) convexity assumptions

“…Nevertheless, F. John in [16] showed that, in the continuation problem for the Helmholtz equation from the unit disk onto any larger disk, the stability estimate is still of a logarithmic type, uniformly with respect to the wave numbers. Increasing stability for the Schrödinger potential and conductivity coefficient has been demonstrated in [13], [14] respectively. In particular, in [13] we also traced the dependence of increasing stability on the attenuation.…”

“…In what follows we denote B(R) = {x : |x| < R}. To demonstrate uniqueness in the direct homogeneous scattering problem (3), (4), (5) as in (9), (10), (11) we have (14) Im…”

Runge approximation and stability improvement for a partial data Calderón problem for the acoustic Helmholtz equation

“…In this article we study improvement of stability effects in Runge approximation originating from the interplay of geometry and an increasing frequency parameter for the acoustic Helmholtz equation. These effects had first been observed in [15] and have subsequently been the object of intensive study, both in the context of unique continuation [24,23,21,51,50] and with regards to their effects on inverse problems [10,26,22,12,13,19,20,25,28,29,5,42,6,27]. Due to the notorious instability in many inverse problems, these improved stability estimates are of great significance, both from a theoretical and practical point of view [9].…”