Iterative thresholding low-rank time integration

2025/07/21 by Bachmayr, Markus, Dolbeault, Matthieu, Sachsenmaier, Polina
#65L70 #65M12 #FOS: Mathematics #Numerical Analysis (math.NA) #Primary 65F55 #Secondary 65Y20

paper · doi:10.48550/arxiv.2507.15848

Abstract

We develop time integration methods in low-rank representation that can adaptively adjust approximation ranks to achieve a prescribed accuracy, while ensuring that these ranks remain proportional to the corresponding best approximation ranks. Our approach relies on an iterative scheme combined with soft thresholding of the iterates. A model case of a time-dependent Schrödinger equation with low-rank matrix approximation is analyzed in detail, and the required modifications for second-order parabolic problems are described. Numerical tests illustrate the results for both cases.

Citations

• Robust high-order low-rank BUG integrators based on explicit Runge--Kutta methods
• Randomized low-rank Runge-Kutta methods
• A space-time variational formulation for the many-body electronic Schrödinger evolution equation
• Robust Implicit Adaptive Low Rank Time-Stepping Methods for Matrix Differential Equations
• Projected exponential methods for stiff dynamical low-rank approximation problems
• Second-order exponential splittings in the presence of unbounded and time-dependent operators: construction and convergence
• Implicit low-rank Riemannian schemes for the time integration of stiff partial differential equations
• A space-time adaptive low-rank method for high-dimensional parabolic partial differential equations
• Implicit integration of nonlinear evolution equations on tensor manifolds
• A rank-adaptive robust integrator for dynamical low-rank approximation
• Rank-adaptive tensor methods for high-dimensional nonlinear PDEs
• An unconventional robust integrator for dynamical low-rank approximation
• On the convergence of Lawson methods for semilinear stiff problems
• Iterative Methods Based on Soft Thresholding of Hierarchical Tensors
• A literature survey of low-rank tensor approximation techniques
• A projector-splitting integrator for dynamical low-rank approximation
• Fourth-Order Time-Stepping for Stiff PDEs
• Calculation of Gauss quadrature rules
• Generalized Runge-Kutta Processes for Stable Systems with Large Lipschitz Constants

Related

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• CUR for Implicit Time Integration of Random Partial Differential Equations on Low-Rank Matrix Manifolds