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Cross Interpolation for Solving High-Dimensional Dynamical Systems on Low-Rank Tucker and Tensor Train Manifolds

Behzad Ghahremani, Hessam Babaee

Abstract

We present a novel tensor interpolation algorithm for the time integration of nonlinear tensor differential equations (TDEs) on the tensor train and Tucker tensor low-rank manifolds, which are the building blocks of many tensor network decompositions. This paper builds upon our previous work (Donello et al., Proceedings of the Royal Society A, Vol. 479, 2023) on solving nonlinear matrix differential equations on low-rank matrix manifolds using CUR decompositions. The methodology we present offers multiple advantages: (i) It delivers near-optimal computational savings both in terms of memory and floating-point operations by leveraging cross algorithms based on the discrete empirical interpolation method to strategically sample sparse entries of the time-discrete TDEs to advance the solution in low-rank form. (ii) Numerical demonstrations show that the time integration is robust in the presence of small singular values. (iii) High-order explicit Runge-Kutta time integration schemes are developed. (iv) The algorithm is easy to implement, as it requires the evaluation of the full-order model at strategically selected entries and does not use tangent space projections, whose efficient implementation is intrusive. We demonstrate the efficiency of the presented algorithm for several test cases, including a nonlinear 100-dimensional TDE for the evolution of a tensor of size $70^{100} \approx 3.2 \times 10^{184}$ and a stochastic advection-diffusion-reaction equation with a tensor of size $4.7 \times 10^9$.

Cross Interpolation for Solving High-Dimensional Dynamical Systems on Low-Rank Tucker and Tensor Train Manifolds

Abstract

We present a novel tensor interpolation algorithm for the time integration of nonlinear tensor differential equations (TDEs) on the tensor train and Tucker tensor low-rank manifolds, which are the building blocks of many tensor network decompositions. This paper builds upon our previous work (Donello et al., Proceedings of the Royal Society A, Vol. 479, 2023) on solving nonlinear matrix differential equations on low-rank matrix manifolds using CUR decompositions. The methodology we present offers multiple advantages: (i) It delivers near-optimal computational savings both in terms of memory and floating-point operations by leveraging cross algorithms based on the discrete empirical interpolation method to strategically sample sparse entries of the time-discrete TDEs to advance the solution in low-rank form. (ii) Numerical demonstrations show that the time integration is robust in the presence of small singular values. (iii) High-order explicit Runge-Kutta time integration schemes are developed. (iv) The algorithm is easy to implement, as it requires the evaluation of the full-order model at strategically selected entries and does not use tangent space projections, whose efficient implementation is intrusive. We demonstrate the efficiency of the presented algorithm for several test cases, including a nonlinear 100-dimensional TDE for the evolution of a tensor of size and a stochastic advection-diffusion-reaction equation with a tensor of size .
Paper Structure (20 sections, 44 equations, 11 figures, 6 algorithms)

This paper contains 20 sections, 44 equations, 11 figures, 6 algorithms.

Table of Contents

  1. Introduction
  2. Methodology
  3. Preliminaries
  4. Time integration using tangent space projection
  5. Time discrete variational principle and cross low-rank approximation
  6. Cross low-rank approximations based on DEIM
  7. CUR-DEIM algorithm for matrix low-rank approximation
  8. CUR-DEIM algorithm for tensor train low-rank approximation
  9. Runge-Kutta temporal integration
  10. Rank adaptivity
  11. Extension to Tucker tensor low-rank approximation
  12. Summary of computational advantages
  13. Demonstrations
  14. Toy Examples
  15. High-dimensional nonlinear TDE
  16. ...and 5 more sections

Figures (11)

  • Figure 1: (a) Schematic of the cross algorithm for time integration of tensor differential equations on the manifold of low-rank tensors without utilizing tangent space projection. (b) Graphical representation of tensor train and Tucker tensor trees for a five-dimensional tensor $V(i_1,i_2,i_3,i_4,i_5)$.
  • Figure 2: Graphical representation of CUR-DEIM algorithm applied to the first unfolding of a fifth-order tensor $V(i_1,i_2,i_3,i_4,i_5)$, where $V(i_1,i_2,i_3,i_4,i_5) \approx \sum_{\alpha=1}^r \mathbf G(i_1,\alpha) R(\alpha,i_2,i_3,i_4,i_5)$. In this representation, the single index belongs to the integer set of $[1,2,\dots, n_k]$ and the combined indices $i_2i_3 i_4 i_5$ belongs to the integer set of $[1,2,\dots, n_2 n_3 n_4 n_5]$.
  • Figure 3: Graphical representation of TT-CUR-DEIM algorithm for a fifth-order tensor $V(i_1,i_2,i_3,i_4,i_5)$.
  • Figure 4: Schematic of the TT-CUR-DEIM algorithm for a three-dimensional tensor $V$ of size $n_1 \times n_2 \times n_3$. Note that $\bm R_1$ is not computed or stored and $\bm C_2$ and $\bm R_2$ are obtained directly from $V_{(1)}$.
  • Figure 5: Toy Examples: Comparison of the TT-CUR-DEIM (iterative), TT-Cross-maxvol, and TT-SVD: (a) Approximation error of $F_1$ versus rank; (b) Approximation error of $F_2, (b=3,5)$ versus rank; (c) Effect of random fiber initialization on the approximation error of $F_2, (b=3)$ for three different ranks ($r=20$, $r=25$, $r=30$).
  • ...and 6 more figures

Theorems & Definitions (3)

  • Definition 1: Tensor train format
  • Definition 2: Tucker tensor format
  • Definition 3: Low-rank tensor manifolds