A Lipschitz stable reconstruction formula for the inverse problem for the wave equation

Abstract

We consider the problem to reconstruct a wave speed c ∈ C ∞ (M ) in a domain M ⊂ R n from acoustic boundary measurements modelled by the hyperbolic Dirichlet-to-Neumann map Λ. We introduce a reconstruction formula for c that is based on the Boundary Control method and incorporates features also from the complex geometric optics solutions approach. Moreover, we show that the reconstruction formula is locally Lipschitz stable for a low frequency component of c −2 under the assumption that the Riemannian manifold (M, c −2 dx 2 ) has a strictly convex function with no critical points. That is, we show that for all bounded C 2 neighborhoods U of c, there is a C 1 neighborhood V of c and constants C, R > 0 such thatfor all c ∈ U ∩ V , where Λ is the Dirichlet-to-Neumann map corresponding to the wave speed c and · * is a norm capturing certain regularity properties of the Dirichlet-to-Neumann maps.1991 Mathematics Subject Classification. Primary: 35R30.

Unique continuation for the Helmholtz equation using stabilized finite element methods

“…concludes the proof. (14) and (u h , z h ) ∈ V h × W h be the solution to the perturbed problem (23). Then there are C > 0 and α ∈ (0, 1) such that for all k, h > 0 with kh 1…”

“…Let ω ⊂ B ⊂ Ω be defined as in Corollary 2. Let u ∈ H 2 (Ω) be the solution to the unperturbed problem (14) and (u h , z h ) ∈ V h × W h be the solution to the perturbed problem (23). Then there are C > 0 and α ∈ (0, 1) such that for all k, h > 0 with k 2 h 1…”

Stable determination of coefficients in the dynamical Schrödinger equation in a magnetic field

“…We refer to this paper for a longer bibliography in the case of the wave equation. In [30] Liu and Oksanen consider the problem to reconstruct a wave speed c from acoustic boundary measurements modelled by the hyperbolic Dirichlet to Neumann map. They introduced a reconstruction formula for c that is based on the Boundary Control method and incorporates features also from the complex geometric optics solutions approach.…”

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“…However in practice some non trapping assumptions on the domain Ω are required for the hypothesis of the Corollary 1 to hold c.f. [20,35,33]. These non trapping assumptions may not be required if using the boundary control method and the full source to solution map [4,3].…”