Full operator preconditioning and the accuracy of solving linear systems

TL;DR

This paper introduces full operator preconditioning (FOP), a technique that transforms operator equations before discretization to significantly enhance the accuracy of solutions to ill-conditioned linear systems.

Contribution

The paper proposes FOP, a novel operator-level transformation method that improves solution accuracy and condition numbers, with applications to polynomial approximation, spectral methods, and finite-element discretizations.

Findings

FOP significantly improves solution accuracy for ill-conditioned systems.

Traditional preconditioning improves condition numbers but not accuracy; FOP addresses this.

FOP-based preconditioners reduce errors in solving differential equations.

Abstract

Unless special conditions apply, the attempt to solve ill-conditioned systems of linear equations with standard numerical methods leads to uncontrollably high numerical error. Often, such systems arise from the discretization of operator equations with a large number of discrete variables. In this paper we show that the accuracy can be improved significantly if the equation is transformed before discretization, a process we call full operator preconditioning (FOP). It bears many similarities with traditional preconditioning for iterative methods but, crucially, transformations are applied at the operator level. We show that while condition-number improvements from traditional preconditioning generally do not improve the accuracy of the solution, FOP can. A number of topics in numerical analysis can be interpreted as implicitly employing FOP; we highlight (i) Chebyshev interpolation in polynomial approximation, and (ii) Olver-Townsend's spectral method, both of which produce solutions of dramatically improved accuracy over a naive problem formulation. In addition, we propose a FOP preconditioner based on integration for the solution of fourth-order differential equations with the finite-element method, showing the resulting linear system is well-conditioned regardless of the discretization size, and demonstrate its error-reduction capabilities on several examples. This work shows that FOP can improve accuracy beyond the standard limit for both direct and iterative methods.