Generative modeling of Sparse Approximate Inverse Preconditioners

TL;DR

This paper introduces a deep learning approach using autoencoders to generate sparse approximate inverse preconditioners for matrix systems from elliptic PDE discretizations, leveraging the matrices' inherent properties.

Contribution

It presents a novel deep learning paradigm for producing high-performance SPAI preconditioners tailored to matrices from elliptic differential operators.

Findings

Effective generation of SPAI preconditioners for various PDE discretizations.

Promising results demonstrating the approach's potential.

Leverages properties of matrices from differential operators.

Abstract

We present a new deep learning paradigm for the generation of sparse approximate inverse (SPAI) preconditioners for matrix systems arising from the mesh-based discretization of elliptic differential operators. Our approach is based upon the observation that matrices generated in this manner are not arbitrary, but inherit properties from differential operators that they discretize. Consequently, we seek to represent a learnable distribution of high-performance preconditioners from a low-dimensional subspace through a carefully-designed autoencoder, which is able to generate SPAI preconditioners for these systems. The concept has been implemented on a variety of finite element discretizations of second- and fourth-order elliptic partial differential equations with highly promising results.