A Fast, Accurate and Oscillation-free Spectral Collocation Solver for High-dimensional Transport Problems

TL;DR

This paper presents the T²S² solver, a novel spectral collocation method using tensor train formats that efficiently solves high-dimensional transport equations with high accuracy and low computational cost.

Contribution

The T²S² solver combines spectral collocation, superconsistency, and tensor train compression to enable fast, accurate, and stable solutions for high-dimensional transport problems.

Findings

Achieves spectral accuracy with extremely low data compression ratios (~10^{-12})

Solves high-dimensional transport problems in minutes on standard hardware

Enables tractable computation of previously intractable high-dimensional transport equations

Abstract

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.