Truncated Nonsmooth Newton Multigrid Methods for Block-Separable Minimization Problems

TL;DR

The paper introduces a globally convergent Truncated Nonsmooth Newton Multigrid (TNNMG) method for block-separable convex minimization problems, with flexible algorithmic options and theoretical convergence guarantees.

Contribution

It provides a proof of global convergence for TNNMG under weak conditions and discusses customizable algorithmic strategies for various problem types.

Findings

Proves global convergence of TNNMG under weak assumptions

Discusses flexible algorithmic choices for different problems

Applicable to nonlinear and nonsmooth PDE discretizations

Abstract

The Truncated Nonsmooth Newton Multigrid (TNNMG) method is a robust and efficient solution method for a wide range of block-separable convex minimization problems, typically stemming from discretizations of nonlinear and nonsmooth partial differential equations. This paper proves global convergence of the method under weak conditions both on the objective functional, and on the local inexact subproblem solvers that are part of the method. It also discusses a range of algorithmic choices that allows to customize the algorithm for many specific problems. Numerical examples are deliberately omitted, because many such examples have already been published elsewhere.