# The Role of Energy Minimization in Algebraic Multigrid Interpolation

## TL;DR

This paper reviews algebraic multigrid convergence theory, proposes a weighted energy minimization approach for interpolation, and explores how energy minimization influences AMG performance through theoretical and numerical analysis.

## Contribution

It introduces a general weighted energy minimization functional for AMG interpolation and discusses its practical implications and limitations.

## Key findings

- Weighted energy minimization offers new insights into AMG interpolation.
- Numerical results highlight the role of energy minimization and constraint vectors.
- The proposed preconditioner aids in solving Sylvester- or Lyapunov-type equations.

## Abstract

Algebraic multigrid (AMG) methods are powerful solvers with linear or near-linear computational complexity for certain classes of linear systems, Ax=b. Broadening the scope of problems that AMG can effectively solve requires the development of improved interpolation operators. Such development is often based on AMG convergence theory. However, convergence theory in AMG tends to have a disconnect with AMG in practice due to the practical constraints of (i) maintaining matrix sparsity in transfer and coarse-grid operators, and (ii) retaining linear complexity in the setup and solve phase. This paper presents a review of fundamental results in AMG convergence theory, followed by a discussion on how these results can be used to motivate interpolation operators in practice. A general weighted energy minimization functional is then proposed to form interpolation operators, and a novel `diagonal' preconditioner for Sylvester- or Lyapunov-type equations developed simultaneously. Although results based on the weighted energy minimization typically underperform compared to a fully constrained energy minimization, numerical results provide new insight into the role of energy minimization and constraint vectors in AMG interpolation.