Dynamically Orthogonal Runge-Kutta Schemes with Perturbative Retractions for the Dynamical Low-Rank Approximation

Abstract

. Whether due to the sheer size of a computational domain, the fine resolution required, 5 or the multiples scales and stochasticity of the dynamics, the dimensionality of a system must often 6 be reduced so that problems of interest become computationally tractable. In this paper, we develop 7 retractions for time-integration schemes that efficiently and accurately evolve the dynamics of a 8 system’s low-rank approximation. Through differential geometry, we analyze the error incurred 9 at each time step due to the high-order curvature of the manifold of fixed-rank matrices. We first 10 obtain a novel, explicit, computationally inexpensive set of algorithms that we refer to as perturbative 11 retractions and show that the set converges to an ideal retraction that projects optimally and exactly 12 to the manifold of fixed-rank matrices by reducing what we define as the projection-retraction error. 13 Furthermore, each perturbative retraction itself exhibits high-order convergence to the best low- 14 rank approximation of the full-rank solution. Using perturbative retractions, we then develop a new 15 class of integration techniques that we refer to as dynamically orthogonal Runge-Kutta (DORK) 16 schemes. DORK schemes integrate along the nonlinear manifold, updating the subspace upon which 17 we project the system’s dynamics as it is integrated. Through numerical test cases, we demonstrate 18 our schemes for matrix addition, real-time data compression, and deterministic and stochastic partial 19 differential equations. We find that DORK schemes are highly accurate by incorporating knowledge 20 of the dynamic, nonlinear manifold’s high-order curvature and computationally efficient by limiting 21 the growing rank needed to represent the evolving dynamics. 22 schemes that account for the 774 evolution of the reduced-order integration space during the time-step. We show that 775 DORK schemes are (i) highly accurate by incorporating knowledge of the dynamic, 776 nonlinear manifold’s high-order curvature from stage to stage and (ii) computationally 777 efficient by limiting the growing rank needed to represent the evolving dynamics L . In 778 our real-time data compression example, we show that these retractions may be used 779 when the dynamics are data-driven (rather than model-driven), and the perturbative 780 retractions with high-order corrections drift from the best low-rank approximation at 781 a much slower rate than retractions with low-order corrections. Note that adjusting 782 how the dynamics are calculated to correct the drift in the first place is important 783 to obtain the most accurate scheme. Lastly, we show that the retractions may be 784 used in deterministic and stochastic differential equations: the dynamical low-rank 785 approximation and the retractions are agnostic to the nature of the mathematical 786 spaces we choose to compress, whether deterministic or stochastic. In both cases, the 787 low-rank approximation converges quickly not just to the best approximation, but to 788 the full-rank solution. In the stochastic case, this offers an efficient methodology for 789 uncertainty quantification in nonlinear problems with non-Gaussian statistics (e.g., 790 [42, 44, 41, 40]). 791