REGULARIZATION WITH DIFFERENTIAL-OPERATORS - AN ITERATIVE APPROACH

It is the purpose of this paper to introduce iterative counterparts to the regularization method of Tikhonov-Phillips with general regularization (or smoothing) operators, \\b - Kx\\2 + alpha\\Lx\\2 --> min To this effect, an explicit formula for the regularized approximations x(alpha) is determined which has the form x(alpha) = x0 + L-r(alpha)(T*T)T*b, where x0 represents a certain well-posed component of x, T = KL- and r(alpha)(lambda) = (lambda + alpha)-1. L- is a generalized inverse of L introduced earlier by Elden; it depends on the underlying operator K. The iterative algorithms will be defined by replacing the rational functions r(alpha) by adequate polynomials. The methods to be considered-including the preconditioned conjugate gradient method-are highly efficient because they admit a recursive computation of the regularized approximations and because the generalized inverse L- need not be computed explicitly. This is shown for a model problem where L is a discretization of the derivative operator.