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Parallel projection methods and the resolution of ill-posed problems. (English) Zbl 0819.65142

To solve an overdetermined system \(G(x) = 0\), \(G : \Omega \subset \mathbb{R}^ n \to \mathbb{R}^ m\), \(m \geq n\), \(\Omega\) closed and convex, the authors devide \(G\) into \(s\) blocks, \(G = (G_ 1, \dots, G_ s)^ T\), and apply the iteration method \(x^{k+1} = P (\Phi (x^ k))\), where \(\varphi (x) = \sum^ s_{i=1} \lambda_ i J^ +_ i (x)G_ i (x)\), \(\lambda_ i \geq 0\), \(\sum \lambda_ i = 1\), \(J^ +_ i\) is the Moore-Penrose pseudoinverse to the Jacobian matrix \(J_ i\) of \(G_ i\), and \(P\) is the metric projection to \(\Omega\).
A local convergence theorem of the method is formulated and the proof is outlined. Extensive numerical examples are presented. The authors solve a nonlinear integral equation of the first kind arising in inverse gravimetry.
Reviewer: G.Vainikko (Espoo)

MSC:

65R20 Numerical methods for integral equations
65R30 Numerical methods for ill-posed problems for integral equations
65H10 Numerical computation of solutions to systems of equations
65Y05 Parallel numerical computation
45G10 Other nonlinear integral equations
86A22 Inverse problems in geophysics
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References:

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