A Fast, Accurate and Oscillation-free Spectral Collocation Solver for High-dimensional Transport Problems
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866908394446651392 |
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| author | Cavallini, Nicola Manzini, Gianmarco Funaro, Daniele Favalli, Andrea |
| author_facet | Cavallini, Nicola Manzini, Gianmarco Funaro, Daniele Favalli, Andrea |
| contents | Transport phenomena-describing the movement of particles, energy, or other physical quantities-are fundamental in various scientific disciplines, including nuclear physics, plasma physics, astrophysics, engineering, and the natural sciences.
However, solving the associated seven-dimensional transport equations poses a significant computational challenge due to the curse of dimensionality.
We introduce the Tensor Train Superconsistent Spectral (T${^2}$S${^2}$) solver to address this challenge, integrating Spectral Collocation for exponential convergence, Superconsistency for stabilization in transport-dominated regimes, and Tensor Train format for substantial data compression. T${^2}$S${^2}$ enforces a dimension-wise superconsistent condition compatible with tensor structures, achieving extremely low compression ratios, in the order of $(10^{-12})$, while preserving spectral accuracy. Numerical experiments on linear problems demonstrate that T${^2}$S${^2}$ can solve high-dimensional transport problems in minutes on standard hardware, making previously intractable problems computationally feasible. This advancement opens new avenues for efficiently and accurately modeling complex transport phenomena. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_04732 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Fast, Accurate and Oscillation-free Spectral Collocation Solver for High-dimensional Transport Problems Cavallini, Nicola Manzini, Gianmarco Funaro, Daniele Favalli, Andrea Numerical Analysis Transport phenomena-describing the movement of particles, energy, or other physical quantities-are fundamental in various scientific disciplines, including nuclear physics, plasma physics, astrophysics, engineering, and the natural sciences. However, solving the associated seven-dimensional transport equations poses a significant computational challenge due to the curse of dimensionality. We introduce the Tensor Train Superconsistent Spectral (T${^2}$S${^2}$) solver to address this challenge, integrating Spectral Collocation for exponential convergence, Superconsistency for stabilization in transport-dominated regimes, and Tensor Train format for substantial data compression. T${^2}$S${^2}$ enforces a dimension-wise superconsistent condition compatible with tensor structures, achieving extremely low compression ratios, in the order of $(10^{-12})$, while preserving spectral accuracy. Numerical experiments on linear problems demonstrate that T${^2}$S${^2}$ can solve high-dimensional transport problems in minutes on standard hardware, making previously intractable problems computationally feasible. This advancement opens new avenues for efficiently and accurately modeling complex transport phenomena. |
| title | A Fast, Accurate and Oscillation-free Spectral Collocation Solver for High-dimensional Transport Problems |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2506.04732 |