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SUMMARY:Local analysis of the inverse problem associated with the Helmholt
 z equation -- Lipschitz stability and iterative reconstruction - de Hoop\,
  M (Purdue University)
DTSTART:20111213T100000Z
DTEND:20111213T103000Z
UID:TALK34963@talks.cam.ac.uk
CONTACT:Mustapha Amrani
DESCRIPTION:We consider the Helmholtz equation on a bounded domain\, and t
 he Dirichlet-to-Neumann map as the data. Following the work of Alessandrin
 i and Vessalla\, we establish conditions under which the inverse problem d
 efined by the Dirichlet-to-Neumann map is Lipschitz stable. Recent advance
 s in developing structured massively parallel multifrontal direct solvers 
 of the Helmholtz equation have motivated the further study of iterative ap
 proaches to solving this inverse problem. We incorporate structure through
  conormal singularities in the coefficients and consider partial boundary 
 data. Essentially\, the coefficients are finite linear combinations of pie
 cewise constant functions. We then establish convergence (radius and rate)
  of the Landweber iteration in appropriately chosen Banach spaces\, avoidi
 ng the fact the coefficients originally can be $L^{infty}$\, to obtain a r
 econstruction. Here\, Lipschitz (or possibly H"{o}lder) stability replaces
  the so-called source condition. We accommodate the exponential growth of 
 the Lipschitz constant using approximations by finite linear combinations 
 of piecewise constant functions and the frequency dependencies to obtain a
  convergent projected steepest descent method containing elements of a non
 linear conjugate gradient method. We point out some correspondences with d
 iscretization\, compression\, and multigrid techniques.\n \nJoint work wit
 h E. Beretta\, L. Qiu and O. Scherzer.\n\n
LOCATION:Seminar Room 1\, Newton Institute
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