# Sparse Hierarchical Preconditioners Using Piecewise Smooth Approximations of Eigenvectors

## TL;DR

This paper introduces a hierarchical preconditioning method that improves the efficiency of iterative solvers for PDE discretizations by focusing on accurately approximating smooth eigenvectors, leading to near-optimal solution times.

## Contribution

It presents a novel hierarchical approximate factorization approach that constructs sparse preconditioners with low complexity, tailored to smooth eigenvectors in elliptic PDE problems.

## Key findings

- Achieves $O(n)$ or $O(n \, \log n)$ construction complexity.
- Demonstrates optimal $O(n)$ solution times on large elliptic problems.
- Preconditioners are exact on near-kernel modes for elasticity equations.

## Abstract

When solving linear systems arising from PDE discretizations, iterative methods (such as Conjugate Gradient, GMRES, or MINRES) are often the only practical choice. To converge in a small number of iterations, however, they have to be coupled with an efficient preconditioner. The efficiency of the preconditioner depends largely on its accuracy on the eigenvectors corresponding to small eigenvalues, and unfortunately, black-box methods typically cannot guarantee sufficient accuracy on these eigenvectors. Thus, constructing the preconditioner becomes a problem-dependent task. However, for a large class of problems, including many elliptic equations, the eigenvectors corresponding to small eigenvalues are smooth functions of the PDE grid. In this paper, we describe a hierarchical approximate factorization approach which focuses on improving accuracy on the smooth eigenvectors. The improved accuracy is achieved by preserving the action of the factorized matrix on piecewise polynomial functions of the grid. Based on the factorization, we propose a family of sparse preconditioners with $O(n)$ or $O(n \log{n})$ construction complexities. Our methods exhibit the optimal $O(n)$ solution times in benchmarks run on large elliptic problems of different types, arising for example in flow or mechanical simulations. In the case of the linear elasticity equation the preconditioners are exact on the near-kernel rigid body modes.