Regularization with Differential Operators: An Iterative Approach

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Abstract
It is the purpose of this paper to introduce iterative counterparts to the regularization method of Tikhonov-Phillips with general regularization (or smoothing) operators, To this effect, an explicit formula for the regularized approximations x α is determined which has the form where x 0 represents a certain well-posed component of x T = KL − and r α(λ) = (λ + α)−1. L − is a generalized inverse of L introduced earlier by Eldén; it depends on the underlying operator K. The iterative algorithms will be defined by replacing the rational functions r α 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 explicity. This is shown for a model problem where L is a discretization of the derivative operator.