Parallel subspace correction methods for semicoercive and nearly semicoercive convex optimization with applications to nonlinear PDEs

TL;DR

This paper extends convergence analysis of parallel subspace correction methods from linear to nonlinear semicoercive convex optimization problems, with applications to nonlinear PDEs.

Contribution

It generalizes the theory of singular and nearly singular linear problems to a broader class of nonlinear convex optimization problems, providing new convergence estimates.

Findings

Convergence rate estimates in terms of seminorm stable decompositions.

Parameter-independent convergence for nearly semicoercive problems.

Application to nonlinear PDEs with Neumann boundary conditions.

Abstract

We present new convergence analyses for parallel subspace correction methods for unconstrained semicoercive and nearly semicoercive convex optimization problems, generalizing the theory of singular and nearly singular linear problems to a class of nonlinear problems. Our results demonstrate that the elegant theoretical framework developed for singular and nearly singular linear problems can be extended to unconstrained semicoercive and nearly semicoercive convex optimization problems. For semicoercive problems, we show that the convergence rate can be estimated in terms of a seminorm stable decomposition over the subspaces and the kernel of the problem, aligning with the theory for singular linear problems. For nearly semicoercive problems, we establish a parameter-independent convergence rate, assuming the kernel of the semicoercive part can be decomposed into a sum of local kernels, which aligns with the theory for nearly singular problems. To demonstrate the applicability of our results, we provide convergence analyses of two-level additive Schwarz methods for solving certain nonlinear partial differential equations with Neumann boundary conditions, within the proposed abstract framework.