Regularization strategy for an inverse problem for a 1 + 1 dimensional wave equation. (14th April 2016)

• Record Type: Journal Article
• Authors: Korpela, Jussi; Lassas, Matti; Oksanen, Lauri
• Is Part Of: Inverse problems. Volume 32:Number 6(2016:Jun.)
• Publication Date: 2016-04-14
• Subjects: inverse problems -- regularization theory -- wave equations; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/0266-5611/32/6/065001
• Languages: English
• ISSNs: 0266-5611

Abstract

Abstract: An inverse boundary value problem for a 1 + 1 dimensional wave equation with a wave speed c ( x ) is considered. We give a regularization strategy for inverting the map  : c ↦ Λ, where Λ is the hyperbolic Neumann-to-Dirichlet map corresponding to the wave speed c . That is, we consider the case when we are given a perturbation of the Neumann-to-Dirichlet map Λ ˜ = Λ + , where  corresponds to the measurement errors, and reconstruct an approximative wave speed c ˜ . We emphasize that Λ ˜ may not be in the range of the map  . We show that the reconstructed wave speed c ˜ satisfies ∥ c ˜ − c ∥ ≤ C ∥  ∥ 1 / 54 . Our regularization strategy is based on a new formula to compute c from Λ.