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Local analysis of the inverse problem associated with the Helmholtz equation -- Lipschitz stability and iterative reconstruction

de Hoop, M (Purdue University)
Tuesday 13 December 2011, 10:00-10:30

Seminar Room 1, Newton Institute


We consider the Helmholtz equation on a bounded domain, and the Dirichlet-to-Neumann map as the data. Following the work of Alessandrini and Vessalla, we establish conditions under which the inverse problem defined by the Dirichlet-to-Neumann map is Lipschitz stable. Recent advances in developing structured massively parallel multifrontal direct solvers of the Helmholtz equation have motivated the further study of iterative approaches 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 piecewise constant functions. We then establish convergence (radius and rate) of the Landweber iteration in appropriately chosen Banach spaces, avoiding the fact the coefficients originally can be $L^{\infty}$, to obtain a reconstruction. Here, Lipschitz (or possibly Hoelder) 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 nonlinear conjugate gradient method. We point out some correspondences with discretization, compression, and multigrid techniques. Joint work with E. Beretta, L. Qiu and O. Scherzer.


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