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

Christoph Hansknecht, Bernhard Heinzelreiter, John W. Pearson, Andreas Potschka

TL;DR

This work considers the solution of saddle‐point systems with a tree‐based block structure, introducing a parallelizable direct method for their solution and proposes several structure‐exploiting preconditioners to be used during applications of the GMRES algorithm and analyze their properties.

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.

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

TL;DR

This work considers the solution of saddle‐point systems with a tree‐based block structure, introducing a parallelizable direct method for their solution and proposes several structure‐exploiting preconditioners to be used during applications of the GMRES algorithm and analyze their properties.

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.

Paper Structure

This paper contains 5 sections, 1 theorem, 8 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation and Definitions
  3. Assumptions
  4. Direct Method
  5. Structure of Algorithm

Key Result

Lemma 1.1

Under Assumption ass:general_assumptions, system eq:2by2_coupled_system is invertible.

Figures (1)

  • Figure 1: Example of a tree-coupled system, based on a tree with $N = 3$ vertices $\mathbb{V} = \{1, 2, 3\}$ and $M = 2$ arcs $\mathbb{A} = \{a_{1} = (3, 1), a_{2} = (3,2)\}$. Vertices $1$ and $2$ are leaves each having a height of zero, a depth of one, and $3$ as their parent. Vertex $R = 3$ is an inner vertex with a height of one (equal to the height of $\mathbb{D}$), a depth of zero, and $1$ and $2$ as its children. The inner subgraph $\mathbb{D}^{\circ}$ consists of vertex $3$ without containing any arcs. The arcs entering $1$ and $2$ are $a_1$ and $a_2$ respectively, i.e., it holds that $k_{1} = 1$ and $k_{2} = 2$. Both arcs have $3$ as their tail while $\mathop{\mathrm{head}}\nolimits(a_1) = 1$ and $\mathop{\mathrm{head}}\nolimits(a_2) = 2$. The subtree rooted at $3$ is equal to the graph itself, whereas $\mathbb{D}_{\leq 1}$ and $\mathbb{D}_{\leq 2}$ consist of only the vertices $1$ and $2$ respectively without any arcs.

Theorems & Definitions (2)

  • Lemma 1.1
  • proof