In this paper, we present a new space-time Petrov-Galerkin-like method. This method utilizes a mixed formulation of Tensor Train (TT) and Quantized Tensor Train (QTT), designed for the spectral element discretization (Q1-SEM) of the time-dependent convection-diffusion-reaction (CDR) equation. We reformulate the assembly process of the spectral element discretized CDR to enhance its compatibility with tensor operations and introduce a low-rank tensor structure for the spectral element operators. Recognizing the banded structure inherent in the spectral element framework's discrete operators, we further exploit the QTT format of the CDR to achieve greater speed and compression. Additionally, we present a comprehensive approach for integrating variable coefficients of CDR into the global discrete operators within the TT/QTT framework. The effectiveness of the proposed method, in terms of memory efficiency and computational complexity, is demonstrated through a series of numerical experiments, including a semi-linear example.

翻译：本文提出了一种新的时空类Petrov-Galerkin方法。该方法采用张量列(TT)与量化张量列(QTT)的混合格式，专为时间依赖对流-扩散-反应(CDR)方程的谱元离散化(Q1-SEM)而设计。我们重构了谱元离散化CDR的组装过程以增强其与张量运算的兼容性，并为谱元算子引入了低秩张量结构。基于谱元框架离散算子固有的带状结构特性，我们进一步利用CDR方程的QTT格式以实现更高的计算速度与压缩效率。此外，我们提出了一种在TT/QTT框架内将CDR方程变系数整合至全局离散算子的系统方法。通过一系列数值实验（包括半线性算例），验证了所提方法在内存效率与计算复杂度方面的优越性。