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A MANN ITERATIVE REGULARIZATION METHOD FOR ELLIPTIC CAUCHY PROBLEMS

2001; Taylor & Francis; Volume: 22; Issue: 7-8 Linguagem: Inglês

10.1081/nfa-100108313

1532-2467

Heinz W. Engl, Antonio Leitão,

Sparse and Compressive Sensing Techniques

Abstract 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. †Supported by CNPq under grant 200049-94.1 and FWF project F–1308 within Spezial-forschungsbereich 13. Acknowledgments Notes †Supported by CNPq under grant 200049-94.1 and FWF project F–1308 within Spezial-forschungsbereich 13. 1For a formal definition of well posed problems, see e.g. Citation[Ba] or Citation[EHN]. 2The Cauchy data (f, g) is called consistent if the corresponding problem EquationCP has a solution. Otherwise (f, g) is called inconsistent Cauchy data. 3For details on the definition of the Sobolev spaces see Citation[Ad] or Citation[DaLi]. 4The corresponding definitions can be found in Citation[BrPe]. 5See Citation[Je] for a complete proof. 6It is obvious that this Picard iteration may fail to converge in this general framework. 7Due to the geometrical interpretation, we adopt the notation used in Citation[Gr] and Citation[EnSc]. In Citation[Do], matrices satisfying iv) are called normal matrices. 8For simplicity we denote by ∥·∥ the norm of the Hilbert space H, but meant is the norm ∥·∥* introduced in Definition 4. 9See URL http://www.netlib.org for details.