| Keywords Estimation of distributed parameter systems, Medical applications, Operator methods
Abstract
In the two-dimensional setup, harmonic conjugation and the Cauchy-Riemann equations provide classical links between harmonic and analytic functions. They allow one to express direct or inverse Cauchy type boundary problems for Laplace operator as reconstruction issues for analytic functions of the complex variable, from its values on part of the boundary. However, such links still exist in dimension n where analytic functions can be defined as gradients of harmonic functions. In dimensions 2 and 3, we formulate and solve some of those inverse problems as best constrained approximation issues in Hardy classes of the domain. The approximation criterion involves the part of the boundary where data is available while a (regularization) constraint acts on the complementary part, where no measurements are available. In 2D and 3D circular or spherical situations, we describe efficient and stable resolution schemes, using Toeplitz or Hankel operators and appropriate bases. This approach provides an original and constructive way for solving inverse Cauchy type problems, numerically efficient and robust with respect to numerical or experimental errors on measurements. Those appear in multiple industrial and real-life applications. We will illustrate our algorithms by numerical computations concerning: - a 2D problem concerning the identification of a Robin exchange coefficient on part of the boundary, related to corrosion detection, - a 3D issue of boundary data propagation, for the inverse EEG (electroencephalography) problem in medical engineering, consisting in finding out sources in the brain from electrical measurements taken on the scalp.
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