Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2402.08607 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929242404552704 |
|---|---|
| author | Ceruti, Gianluca Einkemmer, Lukas Kusch, Jonas Lubich, Christian |
| author_facet | Ceruti, Gianluca Einkemmer, Lukas Kusch, Jonas Lubich, Christian |
| contents | Dynamical low-rank approximation has become a valuable tool to perform an on-the-fly model order reduction for prohibitively large matrix differential equations. A core ingredient is the construction of integrators that are robust to the presence of small singular values and the resulting large time derivatives of the orthogonal factors in the low-rank matrix representation. Recently, the robust basis-update & Galerkin (BUG) class of integrators has been introduced. These methods require no steps that evolve the solution backward in time, often have favourable structure-preserving properties, and allow for parallel time-updates of the low-rank factors. The BUG framework is flexible enough to allow for adaptations to these and further requirements. However, the BUG methods presented so far have only first-order robust error bounds. This work proposes a second-order BUG integrator for dynamical low-rank approximation based on the midpoint rule. The integrator first performs a half-step with a first-order BUG integrator, followed by a Galerkin update with a suitably augmented basis. We prove a robust second-order error bound which in addition shows an improved dependence on the normal component of the vector field. These rigorous results are illustrated and complemented by a number of numerical experiments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_08607 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A robust second-order low-rank BUG integrator based on the midpoint rule Ceruti, Gianluca Einkemmer, Lukas Kusch, Jonas Lubich, Christian Numerical Analysis 65L05, 65L20, 65L70 Dynamical low-rank approximation has become a valuable tool to perform an on-the-fly model order reduction for prohibitively large matrix differential equations. A core ingredient is the construction of integrators that are robust to the presence of small singular values and the resulting large time derivatives of the orthogonal factors in the low-rank matrix representation. Recently, the robust basis-update & Galerkin (BUG) class of integrators has been introduced. These methods require no steps that evolve the solution backward in time, often have favourable structure-preserving properties, and allow for parallel time-updates of the low-rank factors. The BUG framework is flexible enough to allow for adaptations to these and further requirements. However, the BUG methods presented so far have only first-order robust error bounds. This work proposes a second-order BUG integrator for dynamical low-rank approximation based on the midpoint rule. The integrator first performs a half-step with a first-order BUG integrator, followed by a Galerkin update with a suitably augmented basis. We prove a robust second-order error bound which in addition shows an improved dependence on the normal component of the vector field. These rigorous results are illustrated and complemented by a number of numerical experiments. |
| title | A robust second-order low-rank BUG integrator based on the midpoint rule |
| topic | Numerical Analysis 65L05, 65L20, 65L70 |
| url | https://arxiv.org/abs/2402.08607 |