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A rational Krylov methods for large scale linear multidimensional dynamical systems

Houda Barkouki, Khalide Jbilou

Abstract

In this paper, we investigate the use of multilinear algebra for reducing the order of multidimensional linear time-invariant (MLTI) systems. Our main tools are tensor rational Krylov subspace methods, which enable us to approximate the systems solution within a low-dimensional subspace. We introduce the tensor rational block Arnoldi and tensor rational block Lanczos algorithms. By utilizing these methods, we develop a model reduction approach based on projection techniques. Additionally, we demonstrate how these approaches can be applied to large-scale Lyapunov tensor equations, which are critical for the balanced truncation method, a well-known technique for order reduction. An adaptive method for choosing the interpolation points is also introduced. Finally, some numerical experiments are reported to show the effectiveness of the proposed adaptive approaches.

A rational Krylov methods for large scale linear multidimensional dynamical systems

Abstract

In this paper, we investigate the use of multilinear algebra for reducing the order of multidimensional linear time-invariant (MLTI) systems. Our main tools are tensor rational Krylov subspace methods, which enable us to approximate the systems solution within a low-dimensional subspace. We introduce the tensor rational block Arnoldi and tensor rational block Lanczos algorithms. By utilizing these methods, we develop a model reduction approach based on projection techniques. Additionally, we demonstrate how these approaches can be applied to large-scale Lyapunov tensor equations, which are critical for the balanced truncation method, a well-known technique for order reduction. An adaptive method for choosing the interpolation points is also introduced. Finally, some numerical experiments are reported to show the effectiveness of the proposed adaptive approaches.

Paper Structure

This paper contains 23 sections, 9 theorems, 94 equations, 5 figures, 2 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. Background and definitions
  3. Tensor unfolding
  4. MLTI system theory
  5. Block tensors
  6. QR decomposition
  7. SVD decomposition
  8. Tensor classic Krylov subspace
  9. Tensor block Arnoldi algorithm
  10. Tensor block Lanczos algorithm
  11. Projection based tensor rational block methods
  12. Tensor rational block Arnoldi method
  13. Rational approximation
  14. Error estimation for the transfer function
  15. Tensor rational block Lanczos method
  16. ...and 8 more sections

Key Result

Proposition 2.3

Consider $\mathcal{A} \in \mathbb{R}^{J_1 \times \ldots \times J_N \times K_1 \times \ldots \times K_M}$ and $\mathcal{B} \in \mathbb{R}^{K_1 \times \ldots \times K_M \times I_1 \times \ldots \times I_L}$. Then we have the following relations:

Figures (5)

  • Figure 7.1: Example 1.1. Left: $\Vert {\mathcal{F}(j \omega)} \Vert _{2}$ and it's approximation $\Vert \mathcal{F}_m(j \omega) \Vert _{2}$. Right: the exact error $\Vert \mathcal{F}(j \omega)- \mathcal{F}_m(j \omega) \Vert _{2}$ using the TRBA algorithm with $m=5$.
  • Figure 7.2: Example 1.1. Left: $\Vert {H(j \omega)} \Vert _{2}$ and it's approximation $\Vert H_m(j \omega) \Vert _{2}$. Right: the exact error $\Vert H(j \omega)- H_m(j \omega) \Vert _{2}$ using the TRBL algorithm with $m=10$.
  • Figure 7.3: Example 1.2. Left: $\Vert {H(j \omega)} \Vert _{2}$ and it's approximation $\Vert H_m(j \omega) \Vert _{2}$. Right: the exact error $\Vert H(j \omega)- H_m(j \omega) \Vert _{2}$ using the TRBA algorithm with $m=10$.
  • Figure 7.4: Example 1.2. Left: $\Vert {\mathcal{F}(j \omega)} \Vert _{2}$ and it's approximation $\Vert \mathcal{F}_m(j \omega) \Vert _{2}$. Right: the exact error $\Vert \mathcal{F}(j \omega)- \mathcal{F}_m(j \omega) \Vert _{2}$ using the TRBL algorithm with $m=15$.
  • Figure 7.5: Left: $\Vert {\mathcal{F}(j \omega)} \Vert _{2}$ and it's approximation $\Vert \mathcal{F}_m(j \omega) \Vert _{2}$. Right: the exact error $\Vert \mathcal{F}(j \omega)- \mathcal{F}_m(j \omega) \Vert _{2}$ for m=5.

Theorems & Definitions (21)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.7
  • Definition 2.8
  • Theorem 2.9
  • Definition 2.10
  • ...and 11 more