Hierarchical adaptive low-rank format with applications to discretized PDEs

TL;DR

This paper introduces a new adaptive hierarchical low-rank matrix format that efficiently approximates discretized PDEs, especially handling localized singularities and dynamic features in time-dependent problems.

Contribution

It presents a flexible, adaptive matrix format that combines hierarchical partitioning with low-rank approximation for PDE discretizations, enabling efficient solutions of complex linear systems.

Findings

Effective approximation of functions with localized singularities.

Efficient solution of linear systems from PDE discretizations.

Successful application to nonlinear and time-dependent PDEs.

Abstract

A novel compressed matrix format is proposed that combines an adaptive hierarchical partitioning of the matrix with low-rank approximation. One typical application is the approximation of discretized functions on rectangular domains; the flexibility of the format makes it possible to deal with functions that feature singularities in small, localized regions. To deal with time evolution and relocation of singularities, the partitioning can be dynamically adjusted based on features of the underlying data. Our format can be leveraged to efficiently solve linear systems with Kronecker product structure, as they arise from discretized partial differential equations (PDEs). For this purpose, these linear systems are rephrased as linear matrix equations and a recursive solver is derived from low-rank updates of such equations. We demonstrate the effectiveness of our framework for stationary and time-dependent, linear and nonlinear PDEs, including the Burgers' and Allen-Cahn equations.