Regularization of ill-posed problems

Summary

A problem is ill-posed in the sense of Hadamard when it is not a well-posed problem. This means that a solution does not exist, is not unique, or does not depend continuously on the data. Regularization replaces an ill-posed problem by a family of nearby well-posed problems, indexed by a parameter. The parameter is chosen from the noise level so that the regularized solutions converge to the true solution as the noise tends to zero. It is the basic tool for inverse problems such as image deblurring, tomography and parameter identification.