Title: Convergence Analyses of some Nonlinear Multi-Level Algorithms for Non-Quadratic Convex Optimization Problems via Space Decomposition and Subspace Correction

Abstract: Nonlinear multi-level methods, such as the full approximation storage (FAS) multigrid scheme, are widely used solvers for nonlinear problems. In this presentation, a new framework to analyze FAS-type methods for convex optimization problems is developed. FAS can be recast as an inexact version of a nonlinear multigrid method based on space decomposition and subspace correction, namely the successive subspace optimization (SSO) method of Jinchao XU and coauthors. The theory is quite general and is an abstraction of both SSO and the preconditioned steepest descent (PSD) method. In our algorithm, we show that the local problem in each subspace can be simplified to be linear and one gradient decent iteration is enough to ensure linear convergence of the FAS scheme.  This work is joint with Long Chen and Xiaozhe Hu.