Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Discrete empirical interpolation in the tensor t-product framework

Sridhar Chellappa, Lihong Feng, Peter Benner

TL;DR

The tensor t-product Q-DEIM (t-Q-DEIM), an extension of the DEIM framework for dealing with tensor-valued data, is introduced and a rigorous, computable upper bound for the error resulting from the t-Q-DEIM approximation is developed.

Abstract

The discrete empirical interpolation method (DEIM) is a well-established approach, widely used for state reconstruction using sparse sensor/measurement data, nonlinear model reduction, and interpretable feature selection. We introduce the tensor t-product Q-DEIM (t-Q-DEIM), an extension of the DEIM framework for dealing with tensor-valued data. The proposed approach seeks to overcome one of the key drawbacks of DEIM, viz., the need for matricizing the data, which can distort any structural and/or geometric information. Our method leverages the recently developed tensor t-product algebra to avoid reshaping the data. In analogy with the standard DEIM, we formulate and solve a tensor-valued least-squares problem, whose solution is achieved through an interpolatory projection. We develop a rigorous, computable upper bound for the error resulting from the t-Q-DEIM approximation. Using five different tensor-valued datasets, we numerically illustrate the better approximation properties of t-Q-DEIM and the significant computational cost reduction it offers.

Discrete empirical interpolation in the tensor t-product framework

TL;DR

The tensor t-product Q-DEIM (t-Q-DEIM), an extension of the DEIM framework for dealing with tensor-valued data, is introduced and a rigorous, computable upper bound for the error resulting from the t-Q-DEIM approximation is developed.

Abstract

The discrete empirical interpolation method (DEIM) is a well-established approach, widely used for state reconstruction using sparse sensor/measurement data, nonlinear model reduction, and interpretable feature selection. We introduce the tensor t-product Q-DEIM (t-Q-DEIM), an extension of the DEIM framework for dealing with tensor-valued data. The proposed approach seeks to overcome one of the key drawbacks of DEIM, viz., the need for matricizing the data, which can distort any structural and/or geometric information. Our method leverages the recently developed tensor t-product algebra to avoid reshaping the data. In analogy with the standard DEIM, we formulate and solve a tensor-valued least-squares problem, whose solution is achieved through an interpolatory projection. We develop a rigorous, computable upper bound for the error resulting from the t-Q-DEIM approximation. Using five different tensor-valued datasets, we numerically illustrate the better approximation properties of t-Q-DEIM and the significant computational cost reduction it offers.

Paper Structure

This paper contains 46 sections, 7 theorems, 91 equations, 22 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related works
  3. Shortcomings of DEIM
  4. Our contributions
  5. Organization
  6. Mathematical preliminaries
  7. Notations
  8. The t-product algebra
  9. The tensor t-product Q-DEIM (t-Q-DEIM)
  10. t-Q-DEIM problem setup
  11. Computing the t-Q-DEIM basis and sampling tensor
  12. t-SVD to compute the basis
  13. Pivoted t-QR to compute the sampling tensor
  14. Error analysis
  15. A computable estimate for the orthogonal projection error
  16. ...and 31 more sections

Key Result

Theorem 2.1

Consider the third-order tensor $\mathcal{A} \in \mathbb{R}^{m \times \ell \times q}$. $\mathcal{A}$ can be factorized as with $\mathcal{U} \in \mathbb{R}^{m \times m \times q}$ and $\mathcal{W} \in \mathbb{R}^{\ell \times \ell \times q}$ being orthogonal tensors and $\mathcal{S} \in \mathbb{R}^{m \times \ell \times q}$ is an f-diagonal tensor, meaning each of its frontal slices is a diagonal mat

Figures (22)

  • Figure 1: Solution to the viscous Burgers' equation at viscosity $\mu = 0.004$.
  • Figure 2: Burgers' equation: influence of the sampling strategy on the approximation errors; sampling approach from \ref{['alg:tpqr']} and the method proposed in Asletal24 are compared.
  • Figure 3: Burgers' equation: comparison of the performance of t-Q-DEIM and Q-DEIM on the test parameter set for $n = 3$, $n = 10$, respectively.
  • Figure 4: t-Q-DEIM approximation of the Burgers' equation; each figure plots, from top to bottom, the true solution, the t-Q-DEIM approximation, and the pointwise errors. The black cross marks denote the location where the data is sampled.
  • Figure 5: Snapshots corresponding to the variables $w_{1}, w_{2}$ in \ref{['eq:fhn-pde']} and the limit cycle exhibited by the FitzHugh-Nagumo equations at parameter $\bm{\mu} := (\epsilon, c) = (0.022, 0.075)$
  • ...and 17 more figures

Theorems & Definitions (34)

  • Definition 2.1: Horizontal slices
  • Definition 2.2: Lateral slices
  • Definition 2.3: Frontal slices
  • Definition 2.4: Tube fiber Kiletal13
  • Definition 2.5: Block circulant matrix
  • Definition 2.6: Unfolding operation
  • Definition 2.7: Folding operation
  • Definition 2.8: t-product KilMP08
  • Remark 2.1
  • Definition 2.9: t-linearity Kiletal13
  • ...and 24 more