Stable Rank-Adaptive Dynamically Orthogonal Runge–Kutta Schemes
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2211.07852.pdf
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Author(s)
Charous, Aaron
Lermusiaux, Pierre F. J.
Date Issued
February 8, 2024
Journal
SIAM Journal on Scientific Computing
Publisher
Society for Industrial & Applied Mathematics (SIAM)
Citation
Charous, Aaron and Lermusiaux, Pierre F. J. 2024. "Stable Rank-Adaptive Dynamically Orthogonal Runge–Kutta Schemes." SIAM Journal on Scientific Computing, 46 (1).
Version
Author's final manuscript
Abstract
We develop two new sets of stable, rank-adaptive Dynamically Orthogonal Runge-Kutta (DORK) schemes that capture the high-order curvature of the nonlinear low-rank manifold. The DORK schemes asymptotically approximate the truncated singular value decomposition at a greatly reduced cost while preserving mode continuity using newly derived retractions. We show that arbitrarily high-order optimal perturbative retractions can be obtained, and we prove that these new retractions are stable. In addition, we demonstrate that repeatedly applying retractions yields a gradient-descent algorithm on the low-rank manifold that converges superlinearly when approximating a low-rank matrix. When approximating a higher-rank matrix, iterations converge linearly to the best low-rank approximation. We then develop a rank-adaptive retraction that is robust to overapproximation. Building off of these retractions, we derive two rank-adaptive integration schemes that dynamically update the subspace upon which the system dynamics are projected within each time step: the stable, optimal Dynamically Orthogonal Runge-Kutta (so-DORK) and gradient-descent Dynamically Orthogonal Runge-Kutta (gd-DORK) schemes. These integration schemes are numerically evaluated and compared on an ill-conditioned matrix differential equation, an advection-diffusion partial differential equation, and a nonlinear, stochastic reaction-diffusion partial differential equation. Results show a reduced error accumulation rate with the new stable, optimal and gradient-descent integrators. In addition, we find that rank adaptation allows for highly accurate solutions while preserving computational efficiency.
Subjects
Applied Mathematics
Computational Mathematics
MIT Department
Massachusetts Institute of Technology. Department of Mechanical Engineering
Massachusetts Institute of Technology. Center for Computational Science and Engineering
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Creative Commons Attribution-Noncommercial-ShareAlike
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DOI of Published Version
https://doi.org/10.1137/22m1534948
