PrecRTR

This is a collection of MATLAB files associated with the paper Implicit low-rank Riemannian schemes for the time integration of stiff partial differential equations.

We propose two implicit numerical schemes for the low-rank time integration of stiff nonlinear partial differential equations. Our approach uses the preconditioned Riemannian trust-region (RTR) method of Absil, Baker, and Gallivan, 2007. We demonstrate the efficiency of our method for solving the Allen-Cahn and the Fisher-KPP equations on the manifold of fixed-rank matrices. Furthermore, our approach allows us to avoid the restriction on the time step typical of methods that use the fixed-point iteration to solve the inner nonlinear equations. Finally, we demonstrate the efficiency of the preconditioner on the same variational problems presented in Sutti and Vandereycken, 2021.

The RTR method is executed using solvers from the Manopt package (version 7.0) with the Riemannian embedded submanifold geometry from Vandereycken, 2013.

The paper's calculations outlined in Appendix C ("Derivation of the Preconditioner") are implemented in the script getPrecon.

In the Manopt code, we inject our preconditioner in the trustregions, line 14-15, we have the trust-regions tCG solver, line 76 in the tCG solver, "P is the preconditioner" v = precon(x,u).

If there are any problems or bugs, feel free to contact me.

If you use PrecRTR for your research, please kindly cite the following paper:

Sutti, M., Vandereycken, B., Implicit low-rank Riemannian schemes for the time integration of stiff partial differential equations. J Sci Comput 101, 3 (2024). https://doi.org/10.1007/s10915-024-02629-8.

I) Version History

• Ver 1, 12 Sep 2023: initial release.

II) Contents

The main folder LFMS contains the following subfolders:

• mdatafiles: contains the *.mat datafiles generated by the main scripts.
• model: contains all the scripts related to each of the four models considered in the paper: the Allen-Cahn equation (ACE), the Fisher-KPP equation (FKPP), the ``Lyapunov'' variational problem (LYAP) and the nonlinear PDE variational problem (`NPDE`).
• plots: the folder where the plots are saved.
• postprocessing: contains the drivers and the utilities for plotting and visualizing data.
• reference_solutions: contains the reference solutions generated by the scripts Driver_*.m.
• utilities: contains all the utility functions shared by the four models.

III) Installation and Usage

No installation is required. Use the Driver_*.m files to run the calculations or generate the figures in the paper. The results of the simulations will be saved as matfiles in the mdatafiles: folder. Use the Driver_*.m files in the postprocessing folder to plot and visualize data. The figures will then be saved in the plots folder.

IV) License

The code written in this repository is GPL licensed.

V) Dependencies

• Manopt (for numerical optimization on manifolds)

• export_fig (for exporting publication-quality figures)