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Structured Backward Errors for Special Classes of Saddle Point Problems with Applications

Sk. Safique Ahmad, Pinki Khatun

TL;DR

This paper develops a comprehensive framework for structured backward errors in saddle point problems whose block matrices are circulant, Toeplitz, or symmetric-Toeplitz, while preserving sparsity in perturbations. It derives explicit BE formulas for each structured class, along with minimal perturbation matrices that retain the original structure (and sparsity), and also provides unstructured BE results when sparsity is not enforced. The authors apply the theory to weighted regularized least squares (WRLS) problems and demonstrate the practical utility through numerical experiments that assess backward stability and strong backward stability of solvers. Overall, the work offers new, structure-preserving tools to quantify and certify the robustness of algorithms solving structured SPPs and WRLS problems.

Abstract

In the realm of numerical analysis, the study of structured backward errors (BEs) in saddle point problems (SPPs) has shown promising potential for development. However, these investigations overlook the inherent sparsity pattern of the coefficient matrix of the SPP. Moreover, the existing techniques are not applicable when the block matrices have circulant, Toeplitz, or symmetric-Toeplitz structures and do not even provide structure preserving minimal perturbation matrices for which the BE is attained. To overcome these limitations, we investigate the structured BEs of SPPs when the perturbation matrices exploit the sparsity pattern as well as circulant, Toeplitz, and symmetric-Toeplitz structures. Furthermore, we construct minimal perturbation matrices that preserve the sparsity pattern and the aforementioned structures. Applications of the developed frameworks are utilized to compute BEs for the weighted regularized least squares problem. Finally, numerical experiments are performed to validate our findings, showcasing the utility of the obtained structured BEs in assessing the strong backward stability of numerical algorithms.

Structured Backward Errors for Special Classes of Saddle Point Problems with Applications

TL;DR

This paper develops a comprehensive framework for structured backward errors in saddle point problems whose block matrices are circulant, Toeplitz, or symmetric-Toeplitz, while preserving sparsity in perturbations. It derives explicit BE formulas for each structured class, along with minimal perturbation matrices that retain the original structure (and sparsity), and also provides unstructured BE results when sparsity is not enforced. The authors apply the theory to weighted regularized least squares (WRLS) problems and demonstrate the practical utility through numerical experiments that assess backward stability and strong backward stability of solvers. Overall, the work offers new, structure-preserving tools to quantify and certify the robustness of algorithms solving structured SPPs and WRLS problems.

Abstract

In the realm of numerical analysis, the study of structured backward errors (BEs) in saddle point problems (SPPs) has shown promising potential for development. However, these investigations overlook the inherent sparsity pattern of the coefficient matrix of the SPP. Moreover, the existing techniques are not applicable when the block matrices have circulant, Toeplitz, or symmetric-Toeplitz structures and do not even provide structure preserving minimal perturbation matrices for which the BE is attained. To overcome these limitations, we investigate the structured BEs of SPPs when the perturbation matrices exploit the sparsity pattern as well as circulant, Toeplitz, and symmetric-Toeplitz structures. Furthermore, we construct minimal perturbation matrices that preserve the sparsity pattern and the aforementioned structures. Applications of the developed frameworks are utilized to compute BEs for the weighted regularized least squares problem. Finally, numerical experiments are performed to validate our findings, showcasing the utility of the obtained structured BEs in assessing the strong backward stability of numerical algorithms.
Paper Structure (10 sections, 8 theorems, 80 equations, 1 figure, 1 table)

This paper contains 10 sections, 8 theorems, 80 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Structured BEs for circulant structured SPPs
  4. Structured BEs for Toeplitz structured SPPs
  5. Structured BEs for symmetric-Toeplitz structured SPPs
  6. Symmetric-Toeplitz structured SPPs
  7. Unstructured BEs with preserving sparsity pattern
  8. Application to derive the structured BEs for the WRLS problems
  9. Numerical experiments
  10. Conclusion

Key Result

Lemma 3.1

Let $A,B,M\in{\mathcal{C}}_n$ with generator vectors $\mathrm v\mathrm e \mathrm c_{{\mathcal{C}}}(A)=[a_1,\ldots,a_n]^T\in {\mathbb{C}}^n,\mathrm v\mathrm e \mathrm c_{{\mathcal{C}}}(B)=[b_1,\ldots,b_n]^T\in {\mathbb{C}}^n,$ and $\mathrm v\mathrm e \mathrm c_{{\mathcal{C}}}(M)=[m_1,\ldots,m_n]^T\in where $c(M): =\mathrm v\mathrm e \mathrm c_{{\mathcal{C}}}(\Theta_M)$ and $\mathcal{H}_y\in{\mathbb

Figures (1)

  • Figure 1: Different structured and unstructured BEs for $n=8:4:100$ .

Theorems & Definitions (14)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.1
  • Definition 2.3
  • Remark 2.2
  • Lemma 3.1
  • Corollary 3.1
  • Lemma 4.1
  • Corollary 4.1
  • Lemma 5.1
  • ...and 4 more