Second-order robust parallel integrators for dynamical low-rank approximation
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914702500560896 |
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| author | Kusch, Jonas |
| author_facet | Kusch, Jonas |
| contents | Due to its reduced memory and computational demands, dynamical low-rank approximation (DLRA) has sparked significant interest in multiple research communities. A central challenge in DLRA is the development of time integrators that are robust to the curvature of the manifold of low-rank matrices. Recently, a parallel robust time integrator that permits dynamic rank adaptation and enables a fully parallel update of all low-rank factors was introduced. Despite its favorable computational efficiency, the construction as a first-order approximation to the augmented basis-update & Galerkin integrator restricts the parallel integrator's accuracy to order one. In this work, an extension to higher order is proposed by a careful basis augmentation before solving the matrix differential equations of the factorized solution. A robust error bound with an improved dependence on normal components of the vector field together with a norm preservation property up to small terms is derived. These analytic results are complemented and demonstrated through a series of numerical experiments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_02834 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Second-order robust parallel integrators for dynamical low-rank approximation Kusch, Jonas Numerical Analysis 65L05, 65L20, 65L70 Due to its reduced memory and computational demands, dynamical low-rank approximation (DLRA) has sparked significant interest in multiple research communities. A central challenge in DLRA is the development of time integrators that are robust to the curvature of the manifold of low-rank matrices. Recently, a parallel robust time integrator that permits dynamic rank adaptation and enables a fully parallel update of all low-rank factors was introduced. Despite its favorable computational efficiency, the construction as a first-order approximation to the augmented basis-update & Galerkin integrator restricts the parallel integrator's accuracy to order one. In this work, an extension to higher order is proposed by a careful basis augmentation before solving the matrix differential equations of the factorized solution. A robust error bound with an improved dependence on normal components of the vector field together with a norm preservation property up to small terms is derived. These analytic results are complemented and demonstrated through a series of numerical experiments. |
| title | Second-order robust parallel integrators for dynamical low-rank approximation |
| topic | Numerical Analysis 65L05, 65L20, 65L70 |
| url | https://arxiv.org/abs/2403.02834 |