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

Dibyendu Adak (), Duc P. Truong, Radoslav Vuchkov, Saibal De, Derek DeSantis, Nathan V. Roberts, Kim Ø. Rasmussen and Boian S. Alexandrov
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Dibyendu Adak: Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
Duc P. Truong: Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
Radoslav Vuchkov: Sandia National Laboratories, Albuquerque, NM 87185, USA
Saibal De: Sandia National Laboratories, Livermore, CA 94551, USA
Derek DeSantis: Computer, Computational and Statistical Sciences Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
Nathan V. Roberts: Sandia National Laboratories, Albuquerque, NM 87185, USA
Kim Ø. Rasmussen: Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
Boian S. Alexandrov: Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

Mathematics, 2025, vol. 13, issue 14, 1-41

Abstract: In this paper, we present a new space-time Galerkin-like method, where we treat the discretization of spatial and temporal domains simultaneously. This method utilizes a mixed formulation of the tensor-train (TT) and quantized tensor-train (QTT) (please see Section Tensor-Train Decomposition), designed for the finite element discretization (Q1-FEM) of the time-dependent convection–diffusion–reaction (CDR) equation. We reformulate the assembly process of the finite element discretized CDR to enhance its compatibility with tensor operations and introduce a low-rank tensor structure for the finite element operators. Recognizing the banded structure inherent in the finite 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.

Keywords: space-time finite element; tensor network approach; time-dependent problem; convection–diffusion–reaction equation; variable coefficients (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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