Approximate regularized solutions to improperly posed linear integral and operator equations

Abstract

Many problems in mathematical analysis lead to what is usually called in the literature improperly posed problems. The theory and numerical methods of investigation and approximation of these problems — which involve additional difficulties that are not encountered in properly posed problems — have been the subject of intensive research during the past two decades, and continue to present many challenging questions. Improperly posed problems have also been the main theme of numerous conferences and addresses.In the present paper we will report on some recent results for obtaining approximate regularized solutions (and pseudo solutions) of linear operator equations of the first and second kinds. Applications to integral equations will be given.The underlying philosophy of many approaches to regularization resides in the sense we should understand an approximate solution of an improperly posed problem and in effecting numerically these approximations. We provide computable approximate regularized solutions, as well as convergence rates which are optimal in the context of operator equations considered. We also highlight some aspects of the role of generalized inverses and reproducing kernel spaces in regularization and computational methods for operator and integral equations.KeywordsIntegral EquationLinear OperatorOperator EquationGeneralize InverseContinuous DependenceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.