A Framework for the Solution of Tree-Coupled Saddle-Point Systems

TL;DR

This paper introduces a parallelizable direct method and structure-exploiting preconditioners for solving saddle-point systems with a tree-based block structure, enhancing efficiency and convergence in iterative methods.

Contribution

It presents a novel framework combining direct and preconditioned iterative methods tailored for tree-structured saddle-point systems, with theoretical analysis and numerical validation.

Findings

Preconditioners improve convergence of MINRES and GMRES.

Eigenvalue analysis shows favorable spectral properties.

Numerical experiments confirm theoretical predictions.

Abstract

We consider the solution of saddle-point systems with a tree-based block structure, introducing a parallelizable direct method for their solution. As our key contribution, we then propose several structure-exploiting preconditioners to be used during applications of the MINRES and GMRES algorithms and analyze their properties. We adapt several concepts originating in the field of multigrid methods, obtaining a variety of problem-adapted multi-level methods. We analyze the complexity of all algorithms, and derive a number of results on eigenvalues of the preconditioned system and convergence of iterative methods. We validate our theoretical findings through a range of numerical experiments.