On a Family of Gradient-Type Projection Methods for Nonlinear Ill-Posed Problems. (18th August 2018)

• Record Type: Journal Article
• Authors: Leitão, Antonio; Svaiter, Benar F.
• Is Part Of: Numerical functional analysis and optimization. Volume 39:Number 11(2018)
• Page Start: 1153
• Page End: 1180
• Publication Date: 2018-08-18
• Subjects: Ill-posed problems -- nonlinear equations -- projection methods -- tangential cone condition; 65J20 -- 47J06; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2018.1466331
• Languages: English
• ISSNs: 0163-0563

Abstract

Abstract: We propose and analyze a family of successive projection methods whose step direction is the same as the Landweber method for solving nonlinear ill-posed problems that satisfy the Tangential Cone Condition (TCC). This family encompasses the Landweber method, the minimal error method, and the steepest descent method; thus, providing an unified framework for the analysis of these methods. Moreover, we define new methods in this family, which are convergent for the constant of the TCC in a range twice as large as the one required for the Landweber and other gradient type methods. The TCC is widely used in the analysis of iterative methods for solving nonlinear ill-posed problems. The key idea in this work is to use the TCC in order to construct special convex sets possessing a separation property, and to successively project onto these sets. Numerical experiments are presented for a nonlinear two-dimensional elliptic parameter identification problem, validating the efficiency of our method.