ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

15. Numerical Analysis and Scientific Computing

Tautenhan, Ulrich, Hochschule für Techn., Wirtsh. u. Sozialwesen Zittan/Görliz, Germany 1991 MS Classification: 65J20 Hämarik, Uno, Tartu University, Estonia; Raus, Toomas, Tartu University; Tautenhan, Ulrich, Hochschule für Techn., Wirtsh. u. Sozialwesen Zittan/Görliz, Germany On the a-posteriori patameter choice in regularization methods Consider the equation Au = f with linear bounded operator A from one Hilbert space to another. The range R(A) of A may be non-closed. Instead of f ∈ R(A) available is f_{δ}, ∥f_{δ} − f∥ ≤ δ. In the iterated Tikhonov method we take u_{α,0} = u_{0}, find iteratively u_{α,n} = (αI + A^{∗}A)^{−1}(αu_{α,n−1}A^{∗}f_{δ}) for n = 1, …, m and take u_{α} := u_{α,m}. Let u_{∗} be the nearest to u_{0} solution of Au = f. Let b = const > 1. As known, the pseudo-optimal error estimate holds by choice of α = α_{1} by Raus-Gferer rule from the equation (Au_{α} − f_{δ}, Au_{α,m+1} − f_{δ})^{1/2} = bδ. We propose to choose α = α_{2} from equation (Au_{α} − f_{δ}, Au_{α,m+1} − f_{δ}) = b∥Au_{α,m+1} − f_{δ}∥δ. One may prove that ∥u_{α_{2}} − u_{∗}∥ ≤ ∥u_{α_{1}} − u_{∗}∥.