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Semiconvergence and relaxation parameters for projected SIRT algorithms. (English) Zbl 1254.65044

The problem is to solve \(Ax=b\) with \(b\) a vector of noisy observations. The SIRT (simultaneous iterative reconstruction technique) algorithms minimize \(\|Ax-b\|_M^2\) using iterates \(x_{k+1}=P_C(x_k+\lambda_k SZ^TM(b-Ax_k))\) where \(S\) and \(M\) are matrices defining the method, \(M\) is a weighting matrix and \(\lambda_k\) a sequence of relaxation parameters and \(P_C\) the unit matrix. In [T. Elfving, T. Nikazad and P. C. Hansen, ETNA, Electron. Trans. Numer. Anal. 37, 321–336 (2010; Zbl 1205.65148)] the choice of the relaxation parameter in SIRT algorithms was investigated via the analysis of the semiconvergence. In some applications, the solution \(x\) is constrained to some convex set \(C\). Then the P-SIRT uses \(P_C\) as a projection onto the set \(C\), which is the subject of this paper. The choice of the relaxation parameters is analysed in view of the semiconvergence. However also the step-length can be used to suppress noise in connection with semiconvergence.

MSC:

65F10 Iterative numerical methods for linear systems
65R32 Numerical methods for inverse problems for integral equations

Citations:

Zbl 1205.65148

Software:

AIR tools; SNARK93
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