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

Abstract

<p style='text-indent:20px;'>In this article, we discuss quantitative Runge approximation properties for the acoustic Helmholtz equation and prove stability improvement results in the high frequency limit for an associated partial data inverse problem modelled on [<xref ref-type="bibr" rid="b3">3</xref>,<xref ref-type="bibr" rid="b35">35</xref>]. The results rely on quantitative unique continuation estimates in suitable function spaces with explicit frequency dependence. We contrast the frequency dependence of interior Runge approximation results from non-convex and convex sets.</p>

Stability estimates for an inverse problem for Schrödinger operators at high frequencies from arbitrary partial boundary measurements

“…As mentioned in [20], generically (a2) does not pose major restrictions, as it is always possible to find arbitrarily large values of k such that the condition (a2) is fulfilled. For more details for this, we refer the reader to [20].…”

“…Up to now, there is no result for both Γ D and Γ N taken on arbitrary open subsets of the boundary except for the case when Γ D = Γ N (see e.g. [3,20,23,40]), to the best of our knowledge.…”

“…Such a result was verified in [33] by careful analysis in different frequency regions with full boundary measurements. Rigorous studies of increasing stability for different inverse potential problems can be found in [20,26,31,[34][35][36]41] and references therein.…”

“…Thus, there are still many open questions related to partial data case, especially for arbitrary boundary data. Recently, some results are presented with arbitrary boundary data, such as [20,41]. In [41], stability estimates were showed by the knowledge of the partial Robin-to-Dirichlet map at the fixed frequency k > 0.…”

“…The complex geometric optics (CGO) solutions and Carleman estimate are used to prove the main results in [41]. In [20], the approach of Runge approximation was used to prove stability improvement results in the high frequency limit for the acoustic Helmholtz equation by using partial DtN map with Γ D = Γ N = Γ, where Γ is an arbitrary nonempty open subset of ∂Ω.…”