A Mann iterative regularization method for elliptic Cauchy problems
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We investigate the Cauchy problem for linear elliptic operators with C^∞-coefficients at a regular set Ω⊂ R^2, which is a classical example of an ill-posed problem. The Cauchy data are given at the manifold Γ⊂∂Ω and our goal is to reconstruct the trace of the H^1(Ω) solution of an elliptic equation at ∂Ω / Γ. The method proposed here composes the segmenting Mann iteration with a fixed point equation associated with the elliptic Cauchy problem. Our algorithm generalizes the iterative method developed by Maz'ya et al., who proposed a method based on solving successive well-posed mixed boundary value problems. We analyze the regularizing and convergence properties both theoretically and numerically.
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