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A unified approach to regularization methods for linear ill-posed problems

The aim of this paper is to study regularization methods for linear ill-posed problems. Linear methods are Tikhonov-Phillips methods, iterative methods, truncated singular value decomposition, Backus-Gilbert-type methods and approximate inverse, for example. The first three are generally studied as filter methods where a special filter for the singular value decomposition can be computed. In the other methods mentioned the regularization is achieved by either smoothing the data or the solution. More general is the approximate inverse introduced by Louis (1996 Inverse Problems 12 175-90). Here we show that all these methods can be viewed either as smoothing the pseudo-inverse or equivalently as first smoothing the data and then applying the pseudo-inverse. The smoothing of the data or of the pseudo-inverse has to be at least of the order of the smoothing of the operator in the problem to be solved. Conditions for the order-optimality of the methods are given.