Space-Time Spectral Element Tensor Network Approach for Time Dependent Convection Diffusion Reaction Equation with Variable Coefficients

TL;DR

This paper introduces a novel space-time spectral element tensor network method combining TT and QTT formats to efficiently solve variable coefficient convection-diffusion-reaction equations, improving computational speed and memory use.

Contribution

It develops a new tensor network approach integrating TT and QTT formats with spectral element discretization for time-dependent PDEs with variable coefficients.

Findings

Enhanced computational efficiency demonstrated through numerical experiments.

Significant reduction in memory usage for large-scale problems.

Effective handling of variable coefficients within the tensor framework.

Abstract

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.