{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:31:38Z","timestamp":1787337098007,"version":"build-2736575974"},"reference-count":46,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2006,1]]},"abstract":"<jats:p>Scanning acoustic microscopy based on focused ultrasound waves is a promising new tool in medical imaging. In this work we apply an adaptive hybrid FEM\/FDM (finite element methods\/finite difference methods) method to an inverse scattering problem for the time-dependent acoustic wave equation, where one seeks to reconstruct an unknown sound velocity c(x) from a single measurement of wave-reflection data on a small part of the boundary, e.g., to detect pathological defects in bone. Typically, this corresponds to identifying an unknown object (scatterer) in a surrounding homogeneous medium.<\/jats:p>\n                  <jats:p>The inverse problem is formulated as an optimal control problem, where we use an adjoint method to solve the equations of optimality expressing stationarity of an associated augmented Lagrangian by a quasi-Newton method. To treat the problem of multiple minima of the objective function, the optimization procedure is first performed on a coarse grid to smooth the high frequency error, generating a starting point for optimization steps on successively refined meshes. Local refinement based on the results of previous steps will improve computational efficiency of the method.<\/jats:p>\n                  <jats:p>As the main result then, an a posteriori error estimate is proved for the error in the Lagrangian, and a corresponding adaptive method is formulated, where the finite element mesh is refined from residual feedback. The performance of the adaptive hybrid method and the usefulness of the a posteriori error estimator for problems with limited boundary data are illustrated in three dimensional numerical examples.<\/jats:p>","DOI":"10.1137\/050631252","type":"journal-article","created":{"date-parts":[[2008,2,6]],"date-time":"2008-02-06T16:45:19Z","timestamp":1202316319000},"page":"382-402","source":"Crossref","is-referenced-by-count":35,"title":["An Adaptive Hybrid FEM\/FDM Method for an Inverse Scattering Problem in Scanning Acoustic Microscopy"],"prefix":"10.1137","volume":"28","author":[{"given":"Larisa","family":"Beilina","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Christian","family":"Clason","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,25]]},"reference":[{"key":"R1","doi-asserted-by":"crossref","unstructured":"V. Akcelik, G. 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