No abstract

We investigate the method of asymptotical regularization for solving nonlinear illposed problems F ( x ) = y, where. instead of y. noisy data y6 E Y with IIy -y6 11 < 6 axe given and F : D ( F ) C X -f Y is a nonlinear operator between Hilben spaces X and Y . Assuming cenin conditions concerning the nonlinear operator F and the smoothness of the unknown solution x we give convergence properties of the method and derive stability estimates, which show lhat the muracy of the asymptotical regularization method is order optimal provided that [he regularization parameter hi LC been chosen by a generalized discrepmcy principle.

In this paper we study the method of Lavrentiev regularization to reconstruct solutions x † of nonlinear ill-posed problems F(x) = y where instead of y noisy data y δ ∈ X with y − y δ δ are given and F : D(F) ⊂ X → X is a monotone nonlinear operator. In this regularization method regularized solutions x δ α are obtained by solving the singularly perturbed nonlinear operator equation F(x) + α(x − x) = y δ with some initial guess x. Assuming certain conditions concerning the nonlinear operator F and the smoothness of the element x − x † we derive stability estimates which show that the accuracy of the regularized solutions is order optimal provided that the regularization parameter α has been chosen properly.

For solving linear ill-posed problems regularization methods are required when the available data include some noise. In the present paper regularized approximations are obtained by a general regularization scheme in Hilbert scales which include well-known regularization methods such as the method of Tikhonov regularization and its higher-order forms, spectral methods, asymptotical regularization and iterative regularization methods. For both the cases of high-and low-order regularization, we study a priori and a posteriori rules for choosing the regularization parameter and provide order optimal error bounds that characterize the accuracy of the regularized approximations. These error bounds have been obtained under general smoothing conditions. The results extend earlier results and cover the case of finitely and infinitely smoothing operators. The theory is illustrated by a special ill-posed deconvolution problem arising in geoscience.

In this paper we study regularization methods to reconstruct the solution x* of the linear ill-posed problem Ax y, A X --+ Y, from noisy data y 6 Y. The regularization methods are of the general form Y + g(B-SA*A)B-SA*(y AY) where B denotes an unbounded self-adjoint strictly positive definite operator in the Hilbert space X. Assuming IIAxll [Ixll-a and IIx* :llp < E for some Y 6 X, a > 0 and p > 0 (llxllr IIBr/2x[I is the norm in a Hilbert scale (Xr)rs) we derive error estimates which show that the accuracy of the above regularization methods is order optimal provided that the function g ()0, the regularization parameter and the parameter s are chosen properly.