PFLARE library

Created by: Steven Dargaville

This library contains methods which can be used to solve linear systems in parallel with PETSc, with interfaces in C/Fortran/Python.

It aims to provide fast & scalable iterative methods for asymmetric linear systems, in parallel and on both CPUs and GPUs.

Some examples of asymmetric linear systems that PFLARE can scalably solve include:

• Advection equations
• Streaming operators from Boltzmann applications
• Space-time discretisations
• Heavily anisotropic Poisson/diffusion equations

without requiring Gauss-Seidel methods. This includes time dependent or independent equations, with structured or unstructured grids, with lower triangular structure or without.

Methods available in PFLARE

PFLARE adds new methods to PETSc, including:

1. Polynomial approximate inverses, e.g., GMRES and Neumann polynomials
2. Reduction multigrids, e.g., AIRG, nAIR and lAIR
3. CF splittings, e.g., PMISR DDC
4. Methods to extract diagonally dominant submatrices

Quick start

You can get started with PFLARE in one of four ways:

• PFLARE is available directly through the PETSc configure (from the PETSc 3.25 release) with: --download-pflare, see docs/installation.md
• To run the Jupyter notebooks in notebooks/ in your browser without requiring a local install, click the Binder badge above
• To download a Docker image with PFLARE installed, run docker run -it stevendargaville/pflare, then make check inside the container
• To build from source, see docs/installation.md

Documentation

For details about PFLARE, please see:

Path	Contents
docs/new_methods.md	Details on the new methods added by PFLARE
docs/installation.md	How to install PFLARE
docs/use_pflare.md	How to use PFLARE
docs/gpus.md	Using GPUs with PFLARE
docs/reuse.md	Re-using components of PFLARE
docs/options.md	List of the options available in PFLARE
docs/faq.md	Frequently asked questions and help!

and the Jupyter notebooks:

Path	Contents
notebooks/01_getting_started.ipynb	Introduce PFLARE
notebooks/02_pcpflareinv.ipynb	Examine some of the approximate inverses found in PCPFLAREINV
notebooks/03_cf_splitting.ipynb	Visualise the C/F splitting and explore the PMISR-DDC algorithm
notebooks/04_pcair.ipynb	Introduce PCAIR and the AIRG method
notebooks/05_parallel.ipynb	Discuss PCAIR, parallelism and GPUs
notebooks/06_reuse.ipynb	Discuss PCAIR and reuse

More examples

For more ways to use the library please see the Fortran/C examples and the Makefile in tests/, along with the Python examples in python/.

References & citing

Please see the references below for more details. If you use PFLARE in your work, please consider citing [1-3].

1. S. Dargaville, R. P. Smedley-Stevenson, P. N. Smith, C. C. Pain, AIR multigrid with GMRES polynomials (AIRG) and additive preconditioners for Boltzmann transport, Journal of Computational Physics 518 (2024) 113342
2. S. Dargaville, R. P. Smedley-Stevenson, P. N. Smith, C. C. Pain, Coarsening and parallelism with reduction multigrids for hyperbolic Boltzmann transport, The International Journal of High Performance Computing Applications 39(3) (2025) 364-384
3. S. Dargaville, R. P. Smedley-Stevenson, P. N. Smith, C. C. Pain, Solving advection equations with reduction multigrids on GPUs (2025) http://arxiv.org/abs/2508.17517
4. T. A. Manteuffel, S. Münzenmaier, J. Ruge, B. Southworth, Nonsymmetric Reduction-Based Algebraic Multigrid, SIAM Journal on Scientific Computing 41 (2019) S242–S268
5. T. A. Manteuffel, J. Ruge, B. S. Southworth, Nonsymmetric algebraic multigrid based on local approximate ideal restriction (lAIR), SIAM Journal on Scientific Computing 40 (2018) A4105–A4130
6. A. Ali, J. J. Brannick, K. Kahl, O. A. Krzysik, J. B. Schroder, B. S. Southworth, Constrained local approximate ideal restriction for advection-diffusion problems, SIAM Journal on Scientific Computing (2024) S96–S122
7. T. Zaman, N. Nytko, A. Taghibakhshi, S. MacLachlan, L. Olson, M. West, Generalizing reduction-based algebraic multigrid, Numerical Linear Algebra with Applications 31(3) (2024) e2543
8. Loe, J.A., Morgan, R.B, Toward efficient polynomial preconditioning for GMRES. Numerical Linear Algebra with Applications 29 (2022) e2427