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Volume 13
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10.3390/math13142277
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This is an early access version, the complete PDF, HTML, and XML versions will be available soon.
Article

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

by
Dibyendu Adak
1,*,
Duc P. Truong
1,
Radoslav Vuchkov
2,
Saibal De
3,
Derek DeSantis
4,
Nathan V. Roberts
2,
Kim Ø. Rasmussen
1 and
Boian S. Alexandrov
1
1
Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
2
Sandia National Laboratories, Albuquerque, NM 87185, USA
3
Sandia National Laboratories, Livermore, CA 94551, USA
4
Computer, Computational and Statistical Sciences Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
*
Author to whom correspondence should be addressed.
Mathematics 2025, 13(14), 2277; https://doi.org/10.3390/math13142277
Submission received: 13 June 2025 / Revised: 4 July 2025 / Accepted: 9 July 2025 / Published: 15 July 2025
Versions Notes

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 space-time finite element; tensor network approach; time-dependent problem; convection–diffusion–reaction equation; variable coefficients

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MDPI and ACS Style

Adak, D.; Truong, D.P.; Vuchkov, R.; De, S.; DeSantis, D.; Roberts, N.V.; Rasmussen, K.Ø.; Alexandrov, B.S. Space-Time Finite Element Tensor Network Approach for the Time-Dependent Convection–Diffusion–Reaction Equation with Variable Coefficients. Mathematics 2025, 13, 2277. https://doi.org/10.3390/math13142277

AMA Style

Adak D, Truong DP, Vuchkov R, De S, DeSantis D, Roberts NV, Rasmussen KØ, Alexandrov BS. Space-Time Finite Element Tensor Network Approach for the Time-Dependent Convection–Diffusion–Reaction Equation with Variable Coefficients. Mathematics. 2025; 13(14):2277. https://doi.org/10.3390/math13142277

Chicago/Turabian Style

Adak, Dibyendu, Duc P. Truong, Radoslav Vuchkov, Saibal De, Derek DeSantis, Nathan V. Roberts, Kim Ø. Rasmussen, and Boian S. Alexandrov. 2025. "Space-Time Finite Element Tensor Network Approach for the Time-Dependent Convection–Diffusion–Reaction Equation with Variable Coefficients" Mathematics 13, no. 14: 2277. https://doi.org/10.3390/math13142277

APA Style

Adak, D., Truong, D. P., Vuchkov, R., De, S., DeSantis, D., Roberts, N. V., Rasmussen, K. Ø., & Alexandrov, B. S. (2025). Space-Time Finite Element Tensor Network Approach for the Time-Dependent Convection–Diffusion–Reaction Equation with Variable Coefficients. Mathematics, 13(14), 2277. https://doi.org/10.3390/math13142277

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