Regularization with Differential Operators: An Iterative Approach

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.

Keywords:

• Ill-posed problems
• Regularization
• Generalized inverses
• Iterative methods
• Preconditioned conjugate gradient method

AMS(MOS) subject classifications:

• 65J10