Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Active Sampling of Interpolation Points to Identify Dominant Subspaces for Model Reduction

Celine Reddig, Pawan Goyal, Igor Pontes Duff, Peter Benner

TL;DR

The proposed active sampling scheme to obtain reduced‐order models via dominant reachable and observable subspaces via dominant reachable and observable subspaces is illustrated and it is shown that the active sample strategy can provide a speed‐up by one order of magnitude without sacrificing any noticeable accuracy.

Abstract

Model reduction is an active research field to construct low-dimensional surrogate models of high fidelity to accelerate engineering design cycles. In this work, we investigate model reduction for linear structured systems using dominant reachable and observable subspaces. When the training set $-$ containing all possible interpolation points $-$ is large, then these subspaces can be determined by solving many large-scale linear systems. However, for high-fidelity models, this easily becomes computationally intractable. To circumvent this issue, in this work, we propose an active sampling strategy to sample only a few points from the given training set, which can allow us to estimate those subspaces accurately. To this end, we formulate the identification of the subspaces as the solution of the generalized Sylvester equations, guiding us to select the most relevant samples from the training set to achieve our goals. Consequently, we construct solutions of the matrix equations in low-rank forms, which encode subspace information. We extensively discuss computational aspects and efficient usage of the low-rank factors in the process of obtaining reduced-order models. We illustrate the proposed active sampling scheme to obtain reduced-order models via dominant reachable and observable subspaces and present its comparison with the method where all the points from the training set are taken into account. It is shown that the active sample strategy can provide us $17$x speed-up without sacrificing any noticeable accuracy.

Active Sampling of Interpolation Points to Identify Dominant Subspaces for Model Reduction

TL;DR

The proposed active sampling scheme to obtain reduced‐order models via dominant reachable and observable subspaces via dominant reachable and observable subspaces is illustrated and it is shown that the active sample strategy can provide a speed‐up by one order of magnitude without sacrificing any noticeable accuracy.

Abstract

Model reduction is an active research field to construct low-dimensional surrogate models of high fidelity to accelerate engineering design cycles. In this work, we investigate model reduction for linear structured systems using dominant reachable and observable subspaces. When the training set containing all possible interpolation points is large, then these subspaces can be determined by solving many large-scale linear systems. However, for high-fidelity models, this easily becomes computationally intractable. To circumvent this issue, in this work, we propose an active sampling strategy to sample only a few points from the given training set, which can allow us to estimate those subspaces accurately. To this end, we formulate the identification of the subspaces as the solution of the generalized Sylvester equations, guiding us to select the most relevant samples from the training set to achieve our goals. Consequently, we construct solutions of the matrix equations in low-rank forms, which encode subspace information. We extensively discuss computational aspects and efficient usage of the low-rank factors in the process of obtaining reduced-order models. We illustrate the proposed active sampling scheme to obtain reduced-order models via dominant reachable and observable subspaces and present its comparison with the method where all the points from the training set are taken into account. It is shown that the active sample strategy can provide us x speed-up without sacrificing any noticeable accuracy.
Paper Structure (21 sections, 2 theorems, 53 equations, 5 figures, 2 algorithms)

This paper contains 21 sections, 2 theorems, 53 equations, 5 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Linear structured systems
  4. Interpolation-based MOR for LSS
  5. Dominant Reachable and Observable Subspaces and Interpolation
  6. Generalized Sylvester equations and computation of subspaces
  7. Active Sampling of Interpolation Points
  8. Computational Aspects
  9. Solving the projected minimization problem
  10. Residual computation
  11. Selecting multiple interpolation points
  12. Accelerating the SVD in Algorithm 1 by using low-rank factorization
  13. Numerical Experiments
  14. An LTI model
  15. Results:
  16. ...and 6 more sections

Key Result

Theorem 1

Let $\mathbf{H}\left(s\right)$ be a transfer function as in eq:structured_sys. Consider a set of interpolation points $\left\{\sigma_1,\dots,\sigma_r\right\}$ and $\left\{\mu_1,\dots,\mu_r\right\}$ such that $\mathcal{K}\left(s\right)$ is invertible at these interpolation points. Furthermore, let $\ If the reduced matrices are computed as in eq:redmat using these $\mathbf{V}$ and $\mathbf{W}$, the

Figures (5)

  • Figure 1: LTI system: A comparison of the performance of gDROP and DROP.
  • Figure 2: LTI system: relative decay of the singular values for DROP and gDROP.
  • Figure 3: Heating rod model: A comparison of the performance of gDROP and DROP.
  • Figure 4: Heat Equation: A comparison of the performance of gDROP and DROP.
  • Figure 5: Butterfly Gyroscope: A comparison of the performance of gDROP and DROP.

Theorems & Definitions (4)

  • Theorem 1: morBeaG09
  • Remark 1
  • Theorem 2
  • proof