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Conjugate gradient type methods for ill-posed problems. (English) Zbl 0830.65043
Pitman Research Notes in Mathematics Series. 327. Harlow: Longman Scientific & Technical. 134 p. (1995).
Let \(X\) and \(Y\) be Hilbert spaces, \(T\in {\mathcal L}(X, Y)\) a linear bounded operator with an unbounded inverse or not having an inverse at all. During the last 40 years, several regularization methods have been elaborated to solve the ill-posed problem \(Tx= y\), e.g. Tikhonov regularization and some “linear” iteration methods. Conjugate gradient type iteration methods proved to be more effective in practice but very complicated in theoretical investigations.
In this book, the author presents a systematic treatment of conjugate gradient type methods, applied to the normalized equation \(T^* Tx= T^* y\) or to the equation \(Tx= y\) directly if \(T\) is self-adjoint (semidefinite or not). The author deeply examines and answers the following questions:
– under what conditions converges/diverges the iteration as the iteration index goes to infinity?
– given the noise level \(\delta\) in the data, how can the stopping index be chosen such that the approximations are order-optimal?
– are there heuristic stopping rules for the case that no information about the noise level \(\delta\) is known?
Reviewer: G.Vainikko (Espoo)

65J10 Numerical solutions to equations with linear operators (do not use 65Fxx)
65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
47A50 Equations and inequalities involving linear operators, with vector unknowns