A parallel Basis Update and Galerkin Integrator for Tree Tensor Networks
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866915042521251840 |
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| author | Ceruti, Gianluca Kusch, Jonas Lubich, Christian Sulz, Dominik |
| author_facet | Ceruti, Gianluca Kusch, Jonas Lubich, Christian Sulz, Dominik |
| contents | Computing the numerical solution to high-dimensional tensor differential equations can lead to prohibitive computational costs and memory requirements. To reduce the memory and computational footprint, dynamical low-rank approximation (DLRA) has proven to be a promising approach. DLRA represents the solution as a low-rank tensor factorization and evolves the resulting low-rank factors in time. A central challenge in DLRA is to find time integration schemes that are robust to the arising small singular values. A robust parallel basis update & Galerkin integrator, which simultaneously evolves all low-rank factors, has recently been derived for matrix differential equations. This work extends the parallel low-rank matrix integrator to Tucker tensors and general tree tensor networks, yielding an algorithm in which all bases and connecting tensors are evolved in parallel over a time step. We formulate the algorithm, provide a robust error bound, and demonstrate the efficiency of the new integrators for problems in quantum many-body physics, uncertainty quantification, and radiative transfer. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_00858 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A parallel Basis Update and Galerkin Integrator for Tree Tensor Networks Ceruti, Gianluca Kusch, Jonas Lubich, Christian Sulz, Dominik Numerical Analysis 65L05, 65L20, 65L70, 15A69 Computing the numerical solution to high-dimensional tensor differential equations can lead to prohibitive computational costs and memory requirements. To reduce the memory and computational footprint, dynamical low-rank approximation (DLRA) has proven to be a promising approach. DLRA represents the solution as a low-rank tensor factorization and evolves the resulting low-rank factors in time. A central challenge in DLRA is to find time integration schemes that are robust to the arising small singular values. A robust parallel basis update & Galerkin integrator, which simultaneously evolves all low-rank factors, has recently been derived for matrix differential equations. This work extends the parallel low-rank matrix integrator to Tucker tensors and general tree tensor networks, yielding an algorithm in which all bases and connecting tensors are evolved in parallel over a time step. We formulate the algorithm, provide a robust error bound, and demonstrate the efficiency of the new integrators for problems in quantum many-body physics, uncertainty quantification, and radiative transfer. |
| title | A parallel Basis Update and Galerkin Integrator for Tree Tensor Networks |
| topic | Numerical Analysis 65L05, 65L20, 65L70, 15A69 |
| url | https://arxiv.org/abs/2412.00858 |