An $$\ell ^2-\ell ^q$$ Regularization Method for Large Discrete Ill-Posed Problems

Abstract

Ill-posed problems arise in many areas of science and engineering. Their solutions, if they exist, are very sensitive to perturbations in the data. Regularization aims to reduce this sensitivity. Typically, regularization methods replace the original problem by a minimization problem with a fidelity term and a regularization term. Recently, the use of a p-norm to measure the fidelity term, and a q-norm to measure the regularization term, has received considerable attention. The relative importance of these terms is determined by a regularization parameter. This paper discussed how the latter parameter can be determined with the aid of the discrepancy principle. We primarily focus on the situation when $$p=2$$ and $$0<q\le 2$$ , where we note that when $$0<q<1$$ , the minimization problem may be non-convex.