Global convergence of damped semismooth Newton methods for ℓ1 Tikhonov regularization. (30th January 2015)

• Record Type: Journal Article
• Authors: Hans, Esther; Raasch, Thorsten
• Is Part Of: Inverse problems. Volume 31:Number 2(2015:Feb.)
• Publication Date: 2015-01-30
• Subjects: ℓ1-Tikhonov regularization -- semismooth Newton methods -- global convergence -- inverse problems -- sparsity constraints; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/0266-5611/31/2/025005
• Languages: English
• ISSNs: 0266-5611

Abstract

Abstract: We are concerned with Tikhonov regularization of linear ill-posed problems with ℓ1 coefficient penalties. Griesse and Lorenz (2008 Inverse Problems 24 035007 ) proposed a semismooth Newton method for the efficient minimization of the corresponding Tikhonov functionals. In the class of high-precision solvers for such problems, semismooth Newton methods are particularly competitive due to their superlinear convergence properties and their ability to solve piecewise affine equations exactly within finitely many iterations. However, the convergence of semismooth Newton schemes is only local in general. In this work, we discuss the efficient globalization of B(ouligand)-semismooth Newton methods for ℓ1 Tikhonov regularization by means of damping strategies and suitable descent with respect to an associated merit functional. Numerical examples are provided which show that our method compares well with existing iterative, globally convergent approaches.