The principle of indirect elimination

TL;DR

This paper introduces the principle of indirect elimination, a method to identify and remove bad-converging modes in algorithms solving discretized differential equations, enhancing convergence especially in physics applications.

Contribution

It presents a novel, general approach to improve convergence of standard algorithms by using their own solutions to identify problematic modes.

Findings

Effective removal of bad-converging modes in Dirac equation simulations

Applicable to algorithms like Conjugate Gradient, relaxation, and multigrid

Enhances convergence in physics computations

Abstract

The principle of indirect elimination states that an algorithm for solving discretized differential equations can be used to identify its own bad-converging modes. When the number of bad-converging modes of the algorithm is not too large, the modes thus identified can be used to strongly improve the convergence. The method presented here is applicable to any standard algorithm like Conjugate Gradient, relaxation or multigrid. An example from theoretical physics, the Dirac equation in the presence of almost-zero modes arising from instantons, is studied. Using the principle, bad-converging modes are removed efficiently. Applied locally, the principle is one of the main ingredients of the Iteratively Smooting Unigrid algorithm.