Parallel subspace correction methods for semicoercive and nearly semicoercive convex optimization with applications to nonlinear PDEs
Young-Ju Lee, Jongho Park
TL;DR
This work develops a rigorous convergence theory for parallel subspace correction methods applied to semicoercive and nearly semicoercive convex optimization problems in Banach spaces, extending classical singular and nearly singular linear theory to nonlinear settings. Central ideas include a seminorm-based characterization of semicoercivity, stable space decompositions, and nonlinear Banach-space orthogonal decompositions that enable parameter-independent convergence in the nearly semicoercive regime. The authors prove descent properties and derive rates that depend on seminorm stability, local smoothness, and kernel structure, with epsilon-free estimates under suitable kernel decompositions. They demonstrate applicability to nonlinear PDEs with Neumann boundary conditions via two-level additive Schwarz methods for the s-Laplacian and a nonlinear mass term problem, showing scalability and robustness. The results lay groundwork for extending PSC methods to broader constrained and nonsmooth convex problems and to more complex nonlinear PDE systems.
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.