Multigrid Methods in Lattice Field Computations

TL;DR

This paper reviews multigrid methods and explores their potential to significantly improve computational efficiency in lattice field calculations, including solving Dirac equations and updating configurations with near-constant complexity.

Contribution

It introduces the application of multigrid techniques to lattice field computations, demonstrating potential for near-constant time operations in complex simulations.

Findings

Potential for O(n) solutions of Dirac equations

Achieving O(1) operations in gauge fixing and updates

Demonstrated on simple model problems

Abstract

The multigrid methodology is reviewed. By integrating numerical processes at all scales of a problem, it seeks to perform various computational tasks at a cost that rises as slowly as possible as a function of $n$, the number of degrees of freedom in the problem. Current and potential benefits for lattice field computations are outlined. They include: $O(n)$ solution of Dirac equations; just $O(1)$ operations in updating the solution (upon any local change of data, including the gauge field); similar efficiency in gauge fixing and updating; $O(1)$ operations in updating the inverse matrix and in calculating the change in the logarithm of its determinant; $O(n)$ operations per producing each independent configuration in statistical simulations (eliminating CSD), and, more important, effectively just $O(1)$ operations per each independent measurement (eliminating the volume factor as well). These potential capabilities have been demonstrated on simple model problems. Extensions to real life are explored.