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

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 ultrafast numerical solutions of high-dimensional problems. Here, we introduce a Tensor Train (TT) Chebyshev spectral collocation method, in both space and time, for 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 use finite difference for time, and a spectral scheme for the spatial dimensions, which leads to 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 very high compression of linear operators from terabytes to kilobytes in TT-format, and tens of thousands times speedup when compared to full grid space-time spectral method. These advantages allow us to obtain the solutions at much higher resolutions.

Figures (10)

• Figure 1: 1D Space time grid with $N=4$ collocation nodes.
• Figure 2: TT decomposition of a 4D tensor $\mathcal{X}$, with TT rank, $\mathbf{r} = [r_1,r_2,r_3]$, and approximation error $\varepsilon$, in accordance with Eq. \ref{['eqn:TT_def_element']}.
• Figure 3: Representation of a linear matrix $\mathbf{A}$ in the TT-matrix format. First, we reshape the operation matrix $\bold{A}$ and permute its indices to create the tensor $\mathcal{A}$. Then, we factorize the tensor in the tensor-train matrix format according to Eq. \ref{['eqn:TT-matrix-componentwise']} to obtain $\mathcal{A}^{\text{TT}{}}$.
• Figure 4: CUR decomposition
• Figure 5: Test 1: Left Panel: Relative error curve in $L^2$ norm showing the exponential convergence of SP-SP schemes. Middle Panel: Elapsed time in seconds. Right Panel: Compression ratio of the solution. All plots are versus the number of points per dimension.