Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Space-Time Spectral Element Tensor Network Approach for 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, Boian S. Alexandrov

Abstract

In this paper, we present a new space-time Petrov-Galerkin-like method. This method utilizes a mixed formulation of Tensor Train (TT) and Quantized Tensor Train (QTT), designed for the spectral element discretization (Q1-SEM) of the time-dependent convection-diffusion-reaction (CDR) equation. We reformulate the assembly process of the spectral element discretized CDR to enhance its compatibility with tensor operations and introduce a low-rank tensor structure for the spectral element operators. Recognizing the banded structure inherent in the spectral 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.

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

Abstract

In this paper, we present a new space-time Petrov-Galerkin-like method. This method utilizes a mixed formulation of Tensor Train (TT) and Quantized Tensor Train (QTT), designed for the spectral element discretization (Q1-SEM) of the time-dependent convection-diffusion-reaction (CDR) equation. We reformulate the assembly process of the spectral element discretized CDR to enhance its compatibility with tensor operations and introduce a low-rank tensor structure for the spectral element operators. Recognizing the banded structure inherent in the spectral 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.

Paper Structure

This paper contains 45 sections, 2 theorems, 93 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Model Time-dependent 3D Convection-Reaction-Diffusion Problem
  3. Classical Methods for Solving CDR
  4. Mitigating the Curse of Dimensionality
  5. Space-Time Finite Element Formulation of CDR
  6. Space-Time Weak Formulation
  7. Finite Element Approximation
  8. Local Estimates
  9. Local and Global Interpolation Operators
  10. Tensor Train Decomposition
  11. Tensor Train
  12. Linear Operators in TT-matrix format
  13. TT Cross Interpolation
  14. Quantized Tensor Train format
  15. Tensorization of the SEM
  16. ...and 30 more sections

Key Result

Theorem 2.1

Let us assume that $f \in L^2(0,T; H^{-1}(\Omega))$, and the bilinear form $\mathcal{D}(\cdot,\cdot)$Biinear:FM is bounded as discussed in bound:D and satisfies stability condition stability:D. Then, there exists a unique solution $u \in U$ of Weak:Form satisfying where $C$ is positive constant independent of mesh size $h$ but depends on $\kappa(t,\bold{x})$, $\bold{b}(t,\bold{x})$, and $c(t,\bol

Figures (6)

  • Figure 1: TT format of a 4D tensor with TT ranks $\mathbf{r} = [r_1,r_2,r_3]$ and approximation error $\varepsilon$, in accordance with Eq. \ref{['eqn:TT_def_element']}.
  • Figure 2: 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 $\mathcal{A}^{TT{}}$.
  • Figure 3: CUR matrix decomposition.
  • Figure 4: Performance of full-grid, TT and QTT solvers on 3D Poisson equation.
  • Figure 5: Performance of full-grid, TT and QTT solvers on 3D CDR equation with manufactured solution.
  • ...and 1 more figures

Theorems & Definitions (2)

  • Theorem 2.1
  • Theorem 2.2