A LIPSCHITZ STABLE RECONSTRUCTION FORMULA FOR THE INVERSE PROBLEM FOR THE WAVE EQUATION

作者：Liu Shitao^*; Oksanen Lauri

来源：Transactions of the American Mathematical Society, 2016, 368(1): 319-335.

DOI：10.1090/tran/6332

• Inverse problems
• wave equation
• Lipschitz stability

We consider the problem to reconstruct a wave speed c E (M) in a domain M subset of R-n from acoustic boundary measurements modelled by the hyperbolic Dirichlet-to-Neumann map A. 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 that broken vertical bar F(c(-2) -c(-2))1 < C <= 2R(broken vertical bar xi broken vertical bar)
parallel to A - A parallel to, xi is an element of R-n
for all EE Un V, where -A- is the Dirichlet-to-Neumann map corresponding to the wave speed E and broken vertical bar.broken vertical bar, is a norm capturing certain regularity properties of the Dirichlet-to-Neumann maps.

• 出版日期2016-1