Tensor Network Space-Time Spectral Collocation Method for Time-Dependent Convection-Diffusion-Reaction Equations

Adak, Dibyendu; Truong, Duc P.; Manzini, Gianmarco; Rasmussen, Kim Ø.; Alexandrov, Boian S.

doi:10.3390/math12192988

Tensor Network Space-Time Spectral Collocation Method for Time-Dependent Convection-Diffusion-Reaction Equations

Journal Article · Wed Sep 25 00:00:00 EDT 2024 · Mathematics

DOI:https://doi.org/10.3390/math12192988· OSTI ID:2503531

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Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

Emerging tensor network techniques for solutions of partial differential equations (PDEs), known for their ability to break the curse of dimensionality, deliver new mathematical methods for ultra-fast numerical solutions of high-dimensional problems. Here, we introduce a Tensor Train (TT) Chebyshev spectral collocation method, in both space and time, for the solution of the time-dependent convection-diffusion-reaction (CDR) equation with inhomogeneous boundary conditions, in Cartesian geometry. Previous methods for numerical solution of time-dependent PDEs often used finite difference for time, and a spectral scheme for the spatial dimensions, which led to a slow linear convergence. Spectral collocation space-time methods show exponential convergence; however, for realistic problems they need to solve large four-dimensional systems. We overcome this difficulty by using a TT approach, as its complexity only grows linearly with the number of dimensions. We show that our TT space-time Chebyshev spectral collocation method converges exponentially, when the solution of the CDR is smooth, and demonstrate that it leads to a very high compression of linear operators from terabytes to kilobytes in TT-format, and a speedup of tens of thousands of times when compared to a full-grid space-time spectral method. These advantages allow us to obtain the solutions at much higher resolutions.

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Research Organization:: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

Sponsoring Organization:: USDOE; USDOE Laboratory Directed Research and Development (LDRD) Program; USDOE National Nuclear Security Administration (NNSA)

Grant/Contract Number:: 89233218CNA000001

OSTI ID:: 2503531

Alternate ID(s):: OSTI ID: 2476028

Report Number(s):: LA-UR--24-21752

Journal Information:: Mathematics, Journal Name: Mathematics Journal Issue: 19 Vol. 12; ISSN 2227-7390

Publisher:: MDPICopyright Statement

Country of Publication:: United States

Language:: English

References (34)

Nonnegative Tensor-Train Low-Rank Approximations of the Smoluchowski Coagulation Equation Manzini, Gianmarco; Skau, Erik; Truong, Duc P. Large-Scale Scientific Computing, p. 342-250 https://doi.org/10.1007/978-3-030-97549-4_39	book	March 2022
Spectral Elements for Transport-Dominated Equations Funaro, Daniele Lecture Notes in Computational Science and Engineering https://doi.org/10.1007/978-3-642-59185-3	book	April 1997
Legendre spectral collocation in space and time for PDEs Lui, S. H. Numerische Mathematik, Vol. 136, Issue 1 https://doi.org/10.1007/s00211-016-0834-x	journal	August 2016
Climate modeling with neural advection–diffusion equation Choi, Hwangyong; Choi, Jeongwhan; Hwang, Jeehyun Knowledge and Information Systems, Vol. 65, Issue 6 https://doi.org/10.1007/s10115-023-01829-2	journal	January 2023
Tensor Networks and Hierarchical Tensors for the Solution of High-Dimensional Partial Differential Equations Bachmayr, Markus; Schneider, Reinhold; Uschmajew, André Foundations of Computational Mathematics, Vol. 16, Issue 6 https://doi.org/10.1007/s10208-016-9317-9	journal	April 2016
Spectral collocation methods Hussaini, M. Y.; Kopriva, D. A.; Patera, A. T. Applied Numerical Mathematics, Vol. 5, Issue 3 https://doi.org/10.1016/0168-9274(89)90033-0	journal	May 1989
A theory of pseudoskeleton approximations Goreinov, S. A.; Tyrtyshnikov, E. E.; Zamarashkin, N. L. Linear Algebra and its Applications, Vol. 261, Issue 1-3 https://doi.org/10.1016/S0024-3795(96)00301-1	journal	August 1997
Tensor train versus Monte Carlo for the multicomponent Smoluchowski coagulation equation Matveev, Sergey A.; Zheltkov, Dmitry A.; Tyrtyshnikov, Eugene E. Journal of Computational Physics, Vol. 316 https://doi.org/10.1016/j.jcp.2016.04.025	journal	July 2016
Tensor methods for the Boltzmann-BGK equation Boelens, Arnout M. P.; Venturi, Daniele; Tartakovsky, Daniel M. Journal of Computational Physics, Vol. 421 https://doi.org/10.1016/j.jcp.2020.109744	journal	November 2020
TT-cross approximation for multidimensional arrays Oseledets, Ivan; Tyrtyshnikov, Eugene Linear Algebra and its Applications, Vol. 432, Issue 1 https://doi.org/10.1016/j.laa.2009.07.024	journal	January 2010
Quasioptimality of maximum-volume cross interpolation of tensors Savostyanov, Dmitry V. Linear Algebra and its Applications, Vol. 458 https://doi.org/10.1016/j.laa.2014.06.006	journal	October 2014
Rectangular maximum-volume submatrices and their applications Mikhalev, A.; Oseledets, I. V. Linear Algebra and its Applications, Vol. 538 https://doi.org/10.1016/j.laa.2017.10.014	journal	February 2018
The tensor-train mimetic finite difference method for three-dimensional Maxwell’s wave propagation equations Manzini, G.; Truong, P. M. D.; Vuchkov, R. Mathematics and Computers in Simulation, Vol. 210 https://doi.org/10.1016/j.matcom.2023.03.026	journal	August 2023
Quantized tensor networks for solving the Vlasov–Maxwell equations Ye, Erika; Loureiro, Nuno F. Journal of Plasma Physics, Vol. 90, Issue 3 https://doi.org/10.1017/S0022377824000503	journal	June 2024
Solving the time-independent Schrödinger equation for chains of coupled excitons and phonons using tensor trains Gelß, Patrick; Klein, Rupert; Matera, Sebastian The Journal of Chemical Physics, Vol. 156, Issue 2 https://doi.org/10.1063/5.0074948	journal	January 2022
CUR matrix decompositions for improved data analysis Mahoney, Michael W.; Drineas, Petros Proceedings of the National Academy of Sciences, Vol. 106, Issue 3 https://doi.org/10.1073/pnas.0803205106	journal	January 2009
Guaranteed a posteriori error bounds for low-rank tensor approximate solutions Dolgov, Sergey; Vejchodský, Tomáš IMA Journal of Numerical Analysis, Vol. 41, Issue 2 https://doi.org/10.1093/imanum/draa010	journal	June 2020
Quantum-inspired method for solving the Vlasov-Poisson equations Ye, Erika; Loureiro, Nuno F. G. Physical Review E, Vol. 106, Issue 3 https://doi.org/10.1103/PhysRevE.106.035208	journal	September 2022
Fast adaptive interpolation of multi-dimensional arrays in tensor train format Savostyanov, Dmitry; Oseledets, Ivan The 2011 International Workshop on Multidimensional (nD) Systems https://doi.org/10.1109/nDS.2011.6076873	conference	September 2011
Dynamic Programming Bellman, R. Science, Vol. 153, Issue 3731 https://doi.org/10.1126/science.153.3731.34	journal	July 1966
Tucker Dimensionality Reduction of Three-Dimensional Arrays in Linear Time Oseledets, I. V.; Savostianov, D. V.; Tyrtyshnikov, E. E. SIAM Journal on Matrix Analysis and Applications, Vol. 30, Issue 3 https://doi.org/10.1137/060655894	journal	January 2008
Tensor Decompositions and Applications Kolda, Tamara G.; Bader, Brett W. SIAM Review, Vol. 51, Issue 3 https://doi.org/10.1137/07070111X	journal	August 2009
Tensor-Train Decomposition Oseledets, I. V. SIAM Journal on Scientific Computing, Vol. 33, Issue 5 https://doi.org/10.1137/090752286	journal	January 2011
The Alternating Linear Scheme for Tensor Optimization in the Tensor Train Format Holtz, Sebastian; Rohwedder, Thorsten; Schneider, Reinhold SIAM Journal on Scientific Computing, Vol. 34, Issue 2 https://doi.org/10.1137/100818893	journal	January 2012
Low-Rank Explicit QTT Representation of the Laplace Operator and Its Inverse Kazeev, Vladimir A.; Khoromskij, Boris N. SIAM Journal on Matrix Analysis and Applications, Vol. 33, Issue 3 https://doi.org/10.1137/100820479	journal	January 2012
Solution of Linear Systems and Matrix Inversion in the TT-Format Oseledets, I. V.; Dolgov, S. V. SIAM Journal on Scientific Computing, Vol. 34, Issue 5 https://doi.org/10.1137/110833142	journal	January 2012
Fast Solution of Parabolic Problems in the Tensor Train/Quantized Tensor Train Format with Initial Application to the Fokker--Planck Equation Dolgov, S. V.; Khoromskij, B. N.; Oseledets, I. V. SIAM Journal on Scientific Computing, Vol. 34, Issue 6 https://doi.org/10.1137/120864210	journal	January 2012
Alternating Minimal Energy Methods for Linear Systems in Higher Dimensions Dolgov, Sergey V.; Savostyanov, Dmitry V. SIAM Journal on Scientific Computing, Vol. 36, Issue 5 https://doi.org/10.1137/140953289	journal	January 2014
A Semi-Lagrangian Vlasov Solver in Tensor Train Format Kormann, Katharina SIAM Journal on Scientific Computing, Vol. 37, Issue 4 https://doi.org/10.1137/140971270	journal	January 2015
Regularization of Poisson--Boltzmann Type Equations with Singular Source Terms Using the Range-Separated Tensor Format Benner, Peter; Khoromskaia, Venera; Khoromskij, Boris SIAM Journal on Scientific Computing, Vol. 43, Issue 1 https://doi.org/10.1137/19M1281435	journal	January 2021
How to Find a Good Submatrix Goreinov, S. A.; Oseledets, I. V.; Savostyanov, D. V. Matrix Methods: Theory, Algorithms and Applications https://doi.org/10.1142/9789812836021_0015	book	April 2010
Space-Time Finite Element Methods for Parabolic Problems Steinbach, Olaf Computational Methods in Applied Mathematics, Vol. 15, Issue 4 https://doi.org/10.1515/cmam-2015-0026	journal	October 2015
Feature Transportation Improves Graph Neural Networks Eliasof, Moshe; Haber, Eldad; Treister, Eran Proceedings of the AAAI Conference on Artificial Intelligence, Vol. 38, Issue 11 https://doi.org/10.1609/aaai.v38i11.29073	journal	March 2024
Graph Neural Convection-Diffusion with Heterophily Zhao, Kai; Kang, Qiyu; Song, Yang Proceedings of the Thirty-Second International Joint Conference on Artificial Intelligence https://doi.org/10.24963/ijcai.2023/518	conference	August 2023

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