Universität zu Lübeck

******************** PLEASE NOTE THIS SEMINAR WILL TAKE PLACE ON TUESDAY ********************

Well-known iterative schemes for the solution of ill-posed linear equations are the Landweber iteration, the cg-iteration and semi-iterative algorithms like the $\nu$-methods. After introducing these methods, we show that for ill-posed problems a slight modification of the underlying three-term recurrence relation of the $\nu$-methods provides accelerated Landweber algorithms with better performance properties than the $\nu$-methods. The new semi-iterative methods are based on the family of co-dilated ultraspherical polynomials. Compared to the standard $\nu$-methods, the residual polynomials of the modified methods have a faster decay at the origin. This results in an earlier termination of the iteration if the spectrum of the involved operator is clustered around the origin. The convergence order of the modified methods turns out to be the same as for the original $\nu$-methods. The new algorithms are tested numerically and a simple adaptive scheme is developed in which an optimal dilation parameter is determined. At the end, the new semi-iterative methods are used to solve a parameter identification problem obtained from a model in elastography.