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

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−2dx2 ) 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 |F ec −2 − c −2 � (ξ)| ≤ Ce2R|ξ| Λe − Λ ∗ , ξ ∈ R n , for all ec ∈ U ∩ V , where Λ is the Dirichlet-to-Neumann map corre- e sponding to the wave speed ec and k·k∗ is a norm capturing certain regularity properties of the Dirichlet-to-Neumann maps

Time reversal method with stabilizing boundary conditions for Photoacoustic tomography

We study an inverse initial source problem that models photoacoustic tomo- graphy measurements with array detectors, and introduce a method that can be viewed as a modi fi cation of the so called back and forth nudging method. We show that the method converges at an exponential rate under a natural visibility condition, with data given only on a part of the boundary of the domain of wave propagation. In this paper we consider the case of noiseless measurements

Hyperbolic inverse problem with data on disjoint sets

We consider a restricted Dirichlet-to-Neumann map associated to a wave type operator on a Riemannian manifold with boundary. The restriction corresponds to the case where the Dirichlet traces are supported on one subset of the boundary and the Neumann traces are restricted on another subset. We show that the restricted Dirichlet-to-Neumann map determines the geometry and the lower order terms in the wave equation, up the natural gauge invariances, along a convex foliation of the manifold. The main novelty is the recovery of the lower order terms when the supports of the Dirichlet traces are disjoint from the set on which the Neumann traces are restricted. We allow the lower order terms to be non-self-adjoint, and in particular, the corresponding physical system may have dissipation of energy

A finite element data assimilation method for the wave equation

We design a primal-dual stabilized finite element method for the numerical approximation of a data assimilation problem subject to the acoustic wave equation. For the forward problem, piecewise affine, continuous, finite element functions are used for the approximation in space and backward differentiation is used in time. Stabilizing terms are added on the discrete level. The design of these terms is driven by numerical stability and the stability of the continuous problem, with the objective of minimizing the computational error. Error estimates are then derived that are optimal with respect to the approximation properties of the numerical scheme and the stability properties of the continuous problem. The effects of discretizing the (smooth) domain boundary and other perturbations in data are included in the analysis

Simulations on the Accuracy of Laser-Flash Data Analysis Methods

The Laser-Flash thermal diffusivity measurement method can be considered one of the most succesful applications of photothermal techniques. This due to the phenomenological and experimental simplicity and ease of reaching better than 1% accuracy over a wide temperature range. The method is based on observing the temperature rise of the sample back face resulting from the absorption of a laser pulse at the other face. There are various approaches for the data reduction and, especially for high temperature measurements where heat loss effects need to be accounted for, they are based on approximations. This is because the inverse function relating thermal properties and heat exchange conditions with the temperature rise temporal shape is not available in closed form. Therefore, detailed error propagation calculations analyses that would take into account all the steps of the data analysis procedures have not in general been performed for data. In this work, simulations of the noise sensitivity and accuracy of selected data reduction schemes were studied using synthetic data. The work was done in connection with the design of a high temperature laser-flash instrument for the measurement of ceramic composites for fusion reactor applications