Dynamical Low-Rank Approximation

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.