Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Enhanced Error Bounds For The Masked Projection Techniques via Cosine-Sine Decomposition

Brij Nandan Tripathi, Hanumant Singh Shekhawat

TL;DR

This work tackles the challenge of quantifying errors in masked projection for nonlinear dynamical-system reduction, where exact errors are intractable. It introduces two generalized error bounds using cosine-sine decomposition for uniquely determined masked projections, extending beyond DEIM and yielding tighter estimates than the existing QDEIM bound. The authors demonstrate, via two numerical examples, that the second bound is consistently tighter than the first and that both bounds outperform the state-of-the-art in average-error scenarios. The results enhance the reliability of reduced-order models that rely on masked projections and are broadly applicable to masked-projection techniques beyond DEIM.

Abstract

The masked projection techniques are popular in the area of non-linear model reduction. Quantifying and minimizing the error in model reduction, particularly from masked projections, is important. The exact error expressions are often infeasible. This leads to the use of error-bound expressions in the literature. In this paper, we derive two generalized error bounds using cosine-sine decomposition for uniquely determined masked projection techniques. Generally, the masked projection technique is employed to efficiently approximate non-linear functions in the model reduction of dynamical systems. The discrete empirical interpolation method (DEIM) is also a masked projection technique; therefore, the proposed error bounds apply to DEIM projection errors. Furthermore, the proposed error bounds are shown tighter than those currently available in the literature.

Enhanced Error Bounds For The Masked Projection Techniques via Cosine-Sine Decomposition

TL;DR

This work tackles the challenge of quantifying errors in masked projection for nonlinear dynamical-system reduction, where exact errors are intractable. It introduces two generalized error bounds using cosine-sine decomposition for uniquely determined masked projections, extending beyond DEIM and yielding tighter estimates than the existing QDEIM bound. The authors demonstrate, via two numerical examples, that the second bound is consistently tighter than the first and that both bounds outperform the state-of-the-art in average-error scenarios. The results enhance the reliability of reduced-order models that rely on masked projections and are broadly applicable to masked-projection techniques beyond DEIM.

Abstract

The masked projection techniques are popular in the area of non-linear model reduction. Quantifying and minimizing the error in model reduction, particularly from masked projections, is important. The exact error expressions are often infeasible. This leads to the use of error-bound expressions in the literature. In this paper, we derive two generalized error bounds using cosine-sine decomposition for uniquely determined masked projection techniques. Generally, the masked projection technique is employed to efficiently approximate non-linear functions in the model reduction of dynamical systems. The discrete empirical interpolation method (DEIM) is also a masked projection technique; therefore, the proposed error bounds apply to DEIM projection errors. Furthermore, the proposed error bounds are shown tighter than those currently available in the literature.
Paper Structure (10 sections, 3 theorems, 39 equations, 4 tables)

This paper contains 10 sections, 3 theorems, 39 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Notation and convention
  3. Projection and Masked Projection
  4. Error Analysis
  5. Comparison with state-of-the-art bound
  6. Numerical Experiments
  7. Example 1
  8. Example 2
  9. Conclusion
  10. Appendix

Key Result

Theorem 1

Suppose columns of $U= \in \mathbb{R}^{n \times n}$ form a complete basis of the range of the function $f:\mathbb{S} \rightarrow \mathbb{R}^n$ for all possible values of $\mu \in \mathbb{S} \subset{\mathbb{R}}$ where $U_1 \in \mathbb{R}^{n \times m}$, $U_1^TU_1=\mathbb{I}_m$, $U_2 \in \mathbb{R}^{n where $\sigma_i$ is the $i^{th}$ singular value of $P^TU_1$. Moreover, the distance between the mas

Theorems & Definitions (6)

  • Definition 1: Mask/Selection Operator
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3