Illustration of iterative linear solver behavior on simple 1D and 2D problems

TL;DR

This paper visually analyzes how classic iterative linear solvers behave on simple 1D and 2D problems, providing intuition on their convergence patterns in geometry processing tasks.

Contribution

It offers a visual and comparative study of classic solvers' behavior on structured linear systems from geometry processing.

Findings

Jacobi, Gauss-Seidel, SSOR, and CG exhibit distinct convergence behaviors.

Visualizations reveal solver dynamics on 1D and 2D grids.

Insights aid in understanding solver efficiency and stability.

Abstract

In geometry processing, numerical optimization methods often involve solving sparse linear systems of equations. These linear systems have a structure that strongly resembles to adjacency graphs of the underlying mesh. We observe how classic linear solvers behave on this specific type of problems. For the sake of simplicity, we minimise either the squared gradient or the squared Laplacian, evaluated by finite differences on a regular 1D or 2D grid. We observed the evolution of the solution for both energies, in 1D and 2D, and with different solvers: Jacobi, Gauss-Seidel, SSOR (Symmetric successive over-relaxation) and CG (conjugate gradient [She94]). Plotting results at different iterations allows to have an intuition of the behavior of these classic solvers.