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Perturbative Retractions with High-Order Convergence to the Best Low-Rank Approximation

Charous, A., and P.F.J. Lermusiaux, 2022. Perturbative Retractions with High-Order Convergence to the Best Low-Rank Approximation. SIAM Journal on Scientific Computing, sub-judice.

Whether due to the sheer size of a computational domain, the fine resolution required, or the multiples scales and stochasticity of the dynamics, the dimensionality of a system must often be reduced so that problems of interest become computationally tractable. In this paper, we develop retractions for time-integration schemes that efficiently and accurately evolve the dynamics of a system’s low-rank approximation. Through differential geometry, we analyze the error incurred at each time step due to the high-order curvature of the manifold of fixed-rank matrices. We obtain a novel, explicit, computationally inexpensive set of algorithms that we refer to as perturbative retractions and show that the set converges to an ideal retraction that projects optimally and exactly to the manifold of fixed-rank matrices by reducing what we define as the projection-retraction error. Furthermore, each perturbative retraction itself exhibits high-order convergence to the best low-rank approximation of the full-rank solution. We show that their high-order corrections significantly reduce the numerical error accumulated over time when compared to a naive retraction with a first-order correction. Through numerical test cases, we demonstrate the retractions’ efficiency for matrix addition, real-time data compression, and deterministic and stochastic partial differential equations.