Application of denoising methods to regularizationof ill-posed problems

Linear systems of equations and linear least-squares problems with a matrix whose singular values “cluster” at the origin and with an error-contaminated data vector arise in many applications. Their numerical solution requires regularization, i.e., the replacement of the given problem by a nearby one, whose solution is less sensitive to the error in the data. The amount of regularization depends on a parameter. When an accurate estimate of the norm of the error in the data is known, this parameter can be determined by the discrepancy principle. This paper is concerned with the situation when the error is white Gaussian and no estimate of the norm of the error is available, and explores the possibility of applying a denoising method to both reduce this error and to estimate its norm. Applications to image deblurring are presented.