Operator-theoretic and regularization approaches to ill-posed problems

Abstract

A general framework of regularization and approximation methods for ill-posed problems is developed. Three levels in the resolution processes are distinguished and emphasized in this expository-research paper: philosophy of resolution, regularization–approximation schema, and regularization algorithms. Dilemmas and methodologies of resolution of ill-posed problems and their numerical implementations are examined with particular reference to the problem of finding numerically minimum weighted-norm least squares solutions of first kind integral equations (and more generally of linear operator equations with non-closed range). An emphasis is placed on the role of constraints, function space methods, the role of generalized inverses, and reproducing kernels in the regularization and stable computational resolution of these problems. The thrust of the contribution is devoted to the interdisciplinary character of operator-theoretic and regularization methods for ill-posed problems, in particular in mathematical geoscience.