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- A hybrid method for sound-hard obstacle reconstructionPublication . Kress, Rainer; Serranho, PedroWe are interested in solving the inverse problem of acoustic wave scattering to reconstruct the position and the shape of sound-hard obstacles from a given incident field and the corresponding far field pattern of the scattered field. The method we suggest is an extension of the hybrid method for the reconstruction of sound-soft cracks as presented in [R. Kress, P. Serranho, A hybrid method for two-dimensional crack reconstruction, Inverse Problems 21 (2005) 773–784] to the case of sound-hard obstacles. The designation of the method is justified by the fact that it can be interpreted as a hybrid between a regularized Newton method applied to a nonlinear operator equation with the operator that maps the unknown boundary onto the solution of the direct scattering problem and a decomposition method in the spirit of the potential method as described in [A. Kirsch, R. Kress, On an integral equation of the first kind in inverse acoustic scattering, in: Cannon, Hornung (Eds.), Inverse Problems, ISNM, vol. 77, 1986, pp. 93–102. Since the method does not require a forward solver for each Newton step its computational costs are reduced. By some numerical examples we illustrate the feasibility of the method.
- A hybrid method for two-dimensional crack reconstructionPublication . Kress, Rainer; Serranho, PedroWe present a new method for solving the time-harmonic inverse scattering problem for sound-soft or perfectly conducting cracks in two dimensions. Our approach extends a method that was recently suggested by one of us for inverse obstacle scattering. It can be viewed as a hybrid between a regularized Newton iterationmethod applied to a nonlinear operator equation involving the operator that, for a fixed incident wave, maps the crack onto the far-field pattern of the scattered wave and a decomposition method due to Kirsch and Kress. As an important feature, in contrast to the traditional Newton iterations for solving inverse scattering problems, our method does not require a forward solver for each iteration step. The theoretical background of the method is based on the minimization of a cost function containing an additional penalty term to deal with reconstructing the full crack. Numerical examples illustrate the feasibility of the method and its stability with respect to noisy data. We expect that the method can also be extended to sound-hard cracks.
- Iterative and range test methods for an inverse source problem for acoustic wavesPublication . Alves, Carlos J. S.; Kress, Rainer; Serranho, PedroWe propose two methods for solving an inverse source problem for time-harmonic acoustic waves. Based on the reciprocity gap principle a nonlinear equation is presented for the locations and intensities of the point sources that can be solved via Newton iterations. To provide an initial guess for this iteration we suggest a range test algorithm for approximating the source locations. We give a mathematical foundation for the range test and exhibit its feasibility in connection with the iteration method by some numerical examples.
- A hybrid method for inverse scattering for shape and impedancePublication . Serranho, PedroWe present a hybrid method to numerically solve the inverse scattering problem for shape and impedance, given the far-field pattern for one incident direction. This method combines ideas of both iterative and decomposition methods, inheriting the advantages of each of them, such as getting good reconstructions and not needing a forward solver at each step. An optimization problem is presented as the theoretical background of the method and numerical results show its feasibility.
- A hybrid method for inverse scattering for sound-soft obstacles in R3Publication . Serranho, PedroWe present a hybrid method to numerically solve the inverse acoustic soundsoft obstacle scattering problem in R3, given the far-field pattern for one incident direction. This method combines ideas of both iterative and decomposition methods, inheriting advantages of each of them, such as getting good reconstructions and not needing a forward solver at each step. A related Newton method is presented to show convergence of the method and numerical results show its feasibility.
- Huygens’ principle and iterative methods in inverse obstacle scatteringPublication . Ivanyshyn, Olha; Kress, Rainer; Serranho, PedroThe inverse problem we consider in this paper is to determine the shape of an obstacle from the knowledge of the far field pattern for scattering of time-harmonic plane waves. In the case of scattering from a sound-soft obstacle, we will interpret Huygens’ principle as a system of two integral equations, named data and field equation, for the unknown boundary of the scatterer and the induced surface flux, i.e., the unknown normal derivative of the total field on the boundary. Reflecting the ill-posedness of the inverse obstacle scattering problem these integral equations are ill-posed. They are linear with respect to the unknown flux and nonlinear with respect to the unknown boundary and offer, in principle, three immediate possibilities for their iterative solution via linearization and regularization. In addition to presenting new results on injectivity and dense range for the linearized operators, the main purpose of this paper is to establish and illuminate relations between these three solution methods based on Huygens’ principle in inverse obstacle scattering. Furthermore, we will exhibit connections and differences to the traditional regularized Newton type iterations as applied to the boundary to far field map, including alternatives for the implementation of these Newton iterations.