Linear regularizing algorithms for positive solutions of linear inverse problems

Abstract

Linear-regularization methods provide a simple technique for deter­mining stable approximate solutions of linear ill-posed problems such as Fredholm equations of the first kind, Cauchy problems for elliptic equations and backward solution of forward parabolic equations. In most of these problems the solution must be positive to satisfy physical plausibility. In this paper we consider ill-posed first-kind convolution equations and related problems such as numerical differentiation, Radon transform and Laplace-transform inversion. We investigate several linear regularization algorithms which provide positive approximate solutions for these problems at least in the absence of errors on the data. For noisy data the solution is not necessarily positive. Because the appearance of negative values can then only be an effect of the noise, the negative part of the solution should be negligible with a suitable choice of the regularization parameter. A price to pay for ensuring positivity is always, however, a reduction in resolution.