Dynamical low-rank approximation: applications and numerical experiments
Mathematics and Computers in Simulation
Locating coalescing singular values of large two-parameter matrices
Mathematics and Computers in Simulation
Dynamical Tensor Approximation
SIAM Journal on Matrix Analysis and Applications
A Riemannian Optimization Approach for Computing Low-Rank Solutions of Lyapunov Equations
SIAM Journal on Matrix Analysis and Applications
The Alternating Linear Scheme for Tensor Optimization in the Tensor Train Format
SIAM Journal on Scientific Computing
Geometric multiscale decompositions of dynamic low-rank matrices
Computer Aided Geometric Design
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For the low-rank approximation of time-dependent data matrices and of solutions to matrix differential equations, an increment-based computational approach is proposed and analyzed. In this method, the derivative is projected onto the tangent space of the manifold of rank-$r$ matrices at the current approximation. With an appropriate decomposition of rank-$r$ matrices and their tangent matrices, this yields nonlinear differential equations that are well suited for numerical integration. The error analysis compares the result with the pointwise best approximation in the Frobenius norm. It is shown that the approach gives locally quasi-optimal low-rank approximations. Numerical experiments illustrate the theoretical results.