Low-cost non-iterative Sylvester solvers for dynamical low-rank approximation of reaction-diffusion equations
Abstract
In this work, we develop low-cost non-iterative Sylvester solvers for dynamical low-rank approximation of reaction-diffusion equations and incorporate them into a basis-update \& Galerkin mass-lumped finite element method (\mbox{BUG-MLFEM}). By incorporating mass lumping via Gauss-Lobatto quadrature, the spatial discretization yields a matrix differential equation equipped with diagonal positive-definite mass matrices. We establish the weighted dynamical low-rank approximation (WDLRA) within the Galerkin framework of the MLFEM system, proving the theoretical equivalence between the BUG integrator and MLFEM on low-rank subspaces, which guarantees the discrete well-posedness of the implicit BUG steps. To overcome the severe computational bottleneck of standard O(m3+n3) Sylvester solvers, we develop tailored, low-cost non-iterative solvers that exploit the inherent sparse-by-small algebraic structure. Under Dirichlet and Neumann boundary conditions (BCs), the large coefficient matrices are banded, enabling efficient diagonalization of the small r × r factor matrix and reducing the per-step computational complexity to O((m+n)r2) and storage to O((m+n)r). For periodic BCs, we combine this diagonalization strategy with a rank-2 Sherman-Morrison-Woodbury update to handle wrap-around corner entries at negligible additional cost. Numerical experiments demonstrate first-order temporal and (k+1)th-order spatial convergence of the proposed BUG-MLFEM in the discrete L2 norm, where k denotes the polynomial degree of the finite element space, together with significant computational speedups over existing low-rank integrators.
Related
- Sharp Error Estimates for a Fully Discrete Approximation of Semilinear Stochastic Convection--Diffusion Equations with Additive Noise and Nonsmooth Initial Data
- Numerical Solution of Large-Scale Low-Rank Coupled Generalized Sylvester Equations
- Factorized Krylov subspace methods for solving large Sylvester equations
- Grad-div stabilized finite element Picard-Newton iterative methods for the Navier-Stokes equations
- Iterative Methods for Solving New Generalized Absolute Value Equations
BibTeX
@misc{saha2026lowcost,
title = {{Low-cost non-iterative Sylvester solvers for dynamical low-rank approximation of reaction-diffusion equations}},
author = {Joy Saha and Jun Yang and Peimeng Yin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18419},
eprint = {2610.18419},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18419},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Low-cost non-iterative Sylvester solvers for dynamical low-rank approximation of reaction-diffusion equations
AU - Joy Saha
AU - Jun Yang
AU - Peimeng Yin
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.18419
UR - https://arxiv.org/abs/2610.18419
AB - In this work, we develop low-cost non-iterative Sylvester solvers for dynamical low-rank approximation of reaction-diffusion equations and incorporate them into a basis-update \& Galerkin mass-lumped finite element method (\mbox{BUG-MLFEM}). By incorporating mass lumping via Gauss-Lobatto quadrature, the spatial discretization yields a matrix differential equation equipped with diagonal positive-definite mass matrices. We establish the weighted dynamical low-rank approximation (WDLRA) within the Galerkin framework of the MLFEM system, proving the theoretical equivalence between the BUG integrator and MLFEM on low-rank subspaces, which guarantees the discrete well-posedness of the implicit BUG steps. To overcome the severe computational bottleneck of standard O(m^3+n^3) Sylvester solvers, we develop tailored, low-cost non-iterative solvers that exploit the inherent sparse-by-small algebraic structure. Under Dirichlet and Neumann boundary conditions (BCs), the large coefficient matrices are banded, enabling efficient diagonalization of the small r × r factor matrix and reducing the per-step computational complexity to O((m+n)r^2) and storage to O((m+n)r). For periodic BCs, we combine this diagonalization strategy with a rank-2 Sherman-Morrison-Woodbury update to handle wrap-around corner entries at negligible additional cost. Numerical experiments demonstrate first-order temporal and (k+1)th-order spatial convergence of the proposed BUG-MLFEM in the discrete L^2 norm, where k denotes the polynomial degree of the finite element space, together with significant computational speedups over existing low-rank integrators.
ER -