Non-smooth Solutions to Least Squares Problems

Jodi Mead, Boise State University

Abstract

In an attempt to overcome the ill-posedness or illconditioning of inverse problems, regularization methods are implemented by introducing assumptions on the solution. Common regularization methods include total variation, L-curve, Generalized Cross Validation (GCV), and the discrepancy principle. It is generally accepted that all of these approaches except total variation unnecessarily smooth solutions, mainly because the regularization operator is in an L2 norm. Alternatively, statistical approaches to ill-posed problems typically involve specifying a priori information about the parameters in the form of Bayesian inference. These approaches can be more accurate than typical regularization methods because the regularization term is weighted with a matrix rather than a constant. The drawback is that the matrix weight requires information that is typically not available or is expensive to calculate. ¨ª The ¥ö2 method developed by the author and colleagues can be viewed as a regularization method that uses statistical information to find matrices to weight the regularization term. We will demonstrate that unique and simple L2 solutions found by this method do not unecessarily smooth solutions when the regularization term is accurately weighted with a diagonal matrix.