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Tensor-based empirical interpolation method,\newline and its application in model reduction

Brij Nandan Tripathi, Hanumant Singh Shekhawat, Seip Weiland

TL;DR

A tensor-based interpolation method to approximate a matrix-valued function without transforming it into the vector form, an extension of the empirical interpolation method (EIM) for tensor bases is proposed.

Abstract

In general, matrix or tensor-valued functions are approximated using the method developed for vector-valued functions by transforming the matrix-valued function into vector form. This paper proposes a tensor-based interpolation method to approximate a matrix-valued function without transforming it into the vector form. The tensor-based technique has the advantage of reducing offline and online computation without sacrificing much accuracy. The proposed method is an extension of the empirical interpolation method (EIM) for tensor bases. This paper presents a necessary theoretical framework to understand the method's functioning and limitations. Our mathematical analysis establishes a key characteristic of the proposed method: it consistently generates interpolation points in the form of a rectangular grid. This observation underscores a fundamental limitation that applies to any matrix-based approach relying on widely used techniques like EIM or DEIM method. It has also been theoretically shown that the proposed method is equivalent to the DEIM method applied in each direction due to the rectangular grid structure of the interpolation points. The application of the proposed method is shown in the model reduction of the semi-linear matrix differential equation. We have compared the approximation result of our proposed method with the DEIM method used to approximate a vector-valued function. The comparison result shows that the proposed method takes less time, albeit with a minor compromise with accuracy.

Tensor-based empirical interpolation method,\newline and its application in model reduction

TL;DR

A tensor-based interpolation method to approximate a matrix-valued function without transforming it into the vector form, an extension of the empirical interpolation method (EIM) for tensor bases is proposed.

Abstract

In general, matrix or tensor-valued functions are approximated using the method developed for vector-valued functions by transforming the matrix-valued function into vector form. This paper proposes a tensor-based interpolation method to approximate a matrix-valued function without transforming it into the vector form. The tensor-based technique has the advantage of reducing offline and online computation without sacrificing much accuracy. The proposed method is an extension of the empirical interpolation method (EIM) for tensor bases. This paper presents a necessary theoretical framework to understand the method's functioning and limitations. Our mathematical analysis establishes a key characteristic of the proposed method: it consistently generates interpolation points in the form of a rectangular grid. This observation underscores a fundamental limitation that applies to any matrix-based approach relying on widely used techniques like EIM or DEIM method. It has also been theoretically shown that the proposed method is equivalent to the DEIM method applied in each direction due to the rectangular grid structure of the interpolation points. The application of the proposed method is shown in the model reduction of the semi-linear matrix differential equation. We have compared the approximation result of our proposed method with the DEIM method used to approximate a vector-valued function. The comparison result shows that the proposed method takes less time, albeit with a minor compromise with accuracy.

Paper Structure

This paper contains 22 sections, 8 theorems, 86 equations, 5 figures, 9 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Notations
  3. Non-linear matrix approximation using tensors
  4. POD Using Tensor
  5. Computationally Efficient form of $\mathcal{U}$
  6. Inverse of the operator $\mathbf{M\mathcal{U}}$
  7. Tensor Empirical Interpolation Method(TEIM)
  8. Computationally Efficient Form of $M\mathcal{U}$
  9. Final Form of Approximated Matrix-valued Function
  10. Computational Complexity
  11. Comparative Analysis of Accuracy: DEIM vs. TEIM
  12. Model Reduction and Numerical Experimentation
  13. Approximation of matrix-valued function
  14. Example 1
  15. Example 2
  16. ...and 7 more sections

Key Result

Lemma 1

Let T : $\mathbb{R}^{m_1 \times m_2} \rightarrow \mathbb{R}^{n_1 \times n_2}$ be an operator given by then where $U_k$ =$$ and $V_k$ =$$ for $k \in \{1,2\}$.

Figures (5)

  • Figure 1: HOSVD Basis and POD Basis
  • Figure 2: Average Computation Time for a different number of interpolation points.
  • Figure 3: Original and Approximated function for $\mu=(-0.7879,-0.7171)$.
  • Figure 4: Usual POD basis and tensor basis.
  • Figure 5: Original and Approximated function for the $t=0.5$.

Theorems & Definitions (16)

  • Definition 1
  • Lemma 1
  • proof
  • Corollary 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • Theorem 1
  • ...and 6 more