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

Dibyendu Adak, Duc P. Truong (), Gianmarco Manzini, 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
Gianmarco Manzini: Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, 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, 2024, vol. 12, issue 19, 1-20

Abstract: 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.

Keywords: space-time; collocation; tensor train; convection-diffusion-reaction PDE (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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