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On application of generalized discrepancy principle to iterative methods for nonlinear ill-posed problems. (English) Zbl 1068.47081

The equation

F(x)=f,(1)

where a nonlinear operator F acts on a pair of Hilbert spaces (f:H 1 H 2 ) is considered in the paper. The element f is approximately known, f δ -f H 2 <δ. The operator F is FrĂ©chet differentiable and satisfies the conditions F ' (x)1 and F ' (x)-F '' (y)x-y for any x,yH 1 . The original problem is ill-posed, particularly the solution of (1) with the exact data may be nonunique. The following iteratively regularized scheme is used to minimize a functional Φ(x)=F(x)-f δ H 2 2 :x n+1 =ξ-θ(F ' * (x n )F ' (x n ),α n )F ' * (x n ){F(x n )-f δ -F ' (x n )(x n -ξ)}. Here ξ is an element of H 1 and a source type condition is fulfilled; θ(λ,α) is a function of a spectral parameter λ and α>0. A novel generalized discrepancy principle

F(x N )-f δ 2 τδF(x n )-f δ 2 ,(2)

where 0nN,τ1 is suggested in the paper. It is proved (under a source type condition) that if N=N(δ) is chosen by (2), then lim δ0 x N(δ) -x ¯0, where x ¯ is a solution of (1). Convergence rates for various generating functions θ=θ(λ,α) are obtained.

MSC:
47J06Nonlinear ill-posed problems
65F22Ill-posedness, regularization (numerical linear algebra)