Direct optimization of BPX preconditioners

TL;DR

This paper introduces a differentiable loss function for automatically constructing locally optimal BPX preconditioners, significantly reducing the condition number for various PDE models.

Contribution

It presents a novel differentiable optimization approach to tune BPX preconditioners without explicit eigenvalue estimation, improving their effectiveness.

Findings

Achieved 2 to 20 times smaller condition numbers for tested models

Constructed a parametric family of modified BPX preconditioners

Demonstrated effectiveness on multiple PDE types including Poisson and Helmholtz

Abstract

We consider an automatic construction of locally optimal preconditioners for positive definite linear systems. To achieve this goal, we introduce a differentiable loss function that does not explicitly include the estimation of minimal eigenvalue. Nevertheless, the resulting optimization problem is equivalent to a direct minimization of the condition number. To demonstrate our approach, we construct a parametric family of modified BPX preconditioners. Namely, we define a set of empirical basis functions for coarse finite element spaces and tune them to achieve better condition number. For considered model equations (that includes Poisson, Helmholtz, Convection-diffusion, Biharmonic, and others), we achieve from two to twenty times smaller condition numbers for symmetric positive definite linear systems.