A mollification framework for improperly posed problems
type: Item
instance of: scholarly article
title: A mollification framework for improperly posed problems
zbMATH Open document ID: 0896.65042
published in: Numerische Mathematik
publication date: 1998-02-01T00:00:00Z
review text: Let \(A\) represent a linear, translation-invariant operator between given Hilbert spaces for which the range is not dense. A general theory for the mollification of the improperly posed equation \(Au=f\) is developed utilizing the classical theory of semi-groups. The choice of the mollifier is fairly general and includes B-splines as well as the classical Gaussians. A translation-operator representation for the error associated with the application of mollification is derived and an error estimate for the mollification method as applied to the numerical solution of translation-invariant operator equation is given. Then the connection is analyzed between the degree of the mollification and the size of the discretization necessary to guarantee the boundedness of the amplification error. The results are applied to the construction of finite difference formulas for the inversion of translation-invariant operator equations and their use in connection with the numerical differentiation and half-differentiation of observational data.
Mathematics Subject Classification ID: 65J10
Mathematics Subject Classification ID: 45B05
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 65R30
Mathematics Subject Classification ID: 65J20
Mathematics Subject Classification ID: 65D25
zbMATH DE Number: 1111589
zbMATH Keywords: ill-posed problems
zbMATH Keywords: mollification
zbMATH Keywords: Hilbert spaces
zbMATH Keywords: translation-invariant operator equations
zbMATH Keywords: numerical differentiation
author: Markus Hegland
author: Robert S. Anderssen
reviewed by: Werner C. Rheinboldt
MaRDI profile type: Publication
full work available at URL: s002110050325
OpenAlex ID: W1971513836
DOI: 10.1007/S002110050325
description: scientific article; zbMATH DE number 1111589