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Two-sided uniformly randomized GSVD for large-scale discrete ill-posed problems with Tikhonov regularizations

Weiwei Xu, Weijie Shen, Zheng-Jian Bai

TL;DR

This work introduces a two-sided uniformly randomized GSVD methodology to efficiently solve large-scale discrete ill-posed problems with general Tikhonov regularization. By employing uniform random sampling in a two-stage range decomposition, the method forms small projected GMPs and computes an approximate GSVD, enabling a compact and accurate regularized solution ${\mathbf{x}_{\lambda}}$. The authors provide perturbation-based error analyses for both overdetermined and underdetermined cases, and demonstrate through numerical experiments that the approach achieves comparable accuracy to classical GSVD-based methods with substantially reduced computational cost and memory usage. The technique is particularly beneficial for very large problems where the full GSVD is prohibitive, and it supports parameter selection via GCV or L-curve rules using the randomized GSVD basis. Overall, the paper advances scalable regularization for large-scale ill-posed problems by integrating randomized range finders with GSVD in a rigorous, two-sided framework.

Abstract

The generalized singular value decomposition (GSVD) is a powerful tool for solving discrete ill-posed problems. In this paper, we propose a two-sided uniformly randomized GSVD algorithm for solving the large-scale discrete ill-posed problem with the general Tikhonov regularization. Based on two-sided uniform random sampling, the proposed algorithm can improve the efficiency with less computing time and memory requirement and obtain expected accuracy. The error analysis for the proposed algorithm is also derived. Finally, we report some numerical examples to illustrate the efficiency of the proposed algorithm.

Two-sided uniformly randomized GSVD for large-scale discrete ill-posed problems with Tikhonov regularizations

TL;DR

This work introduces a two-sided uniformly randomized GSVD methodology to efficiently solve large-scale discrete ill-posed problems with general Tikhonov regularization. By employing uniform random sampling in a two-stage range decomposition, the method forms small projected GMPs and computes an approximate GSVD, enabling a compact and accurate regularized solution . The authors provide perturbation-based error analyses for both overdetermined and underdetermined cases, and demonstrate through numerical experiments that the approach achieves comparable accuracy to classical GSVD-based methods with substantially reduced computational cost and memory usage. The technique is particularly beneficial for very large problems where the full GSVD is prohibitive, and it supports parameter selection via GCV or L-curve rules using the randomized GSVD basis. Overall, the paper advances scalable regularization for large-scale ill-posed problems by integrating randomized range finders with GSVD in a rigorous, two-sided framework.

Abstract

The generalized singular value decomposition (GSVD) is a powerful tool for solving discrete ill-posed problems. In this paper, we propose a two-sided uniformly randomized GSVD algorithm for solving the large-scale discrete ill-posed problem with the general Tikhonov regularization. Based on two-sided uniform random sampling, the proposed algorithm can improve the efficiency with less computing time and memory requirement and obtain expected accuracy. The error analysis for the proposed algorithm is also derived. Finally, we report some numerical examples to illustrate the efficiency of the proposed algorithm.

Paper Structure

This paper contains 13 sections, 10 theorems, 57 equations, 4 figures, 4 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A randomized scheme for the range approximation via uniform sampling
  4. A GSVD algorithm of a full-rank GMP
  5. Tikhonov regularization via two-sided uniformly randomized GSVD
  6. Overdetermined case ($m\ge n$)
  7. Underdetermined case ($m< n$)
  8. Choice of the regularization parameter $\lambda$
  9. Error Analysis
  10. Overdetermined case
  11. Underdetermined case
  12. Numerical examples
  13. Concluding remarks

Key Result

Lemma 2.1

Let $B\in {\mathbb{R}}^{m\times \ell}$ and $C\in {\mathbb{R}}^{m\times n}$ be two constant matrices with $C$ being of full row rank. Let $\Omega\in{\mathbb{R}}^{n\times \ell}$ ($n\ge \ell$) be a random matrix whose entries are all independent random variables such that, for any $c\in{\mathbb{R}}$, Then for any $s\times t$ submatrix $\Phi$ of the random matrix $M=B+C \Omega$, one has ${\rm rank}(\

Figures (4)

  • Figure 6.1: Algorithm \ref{['alg:alg4']} (GCV) for Example \ref{['ex1']} with $m=n=2048$
  • Figure 6.2: Algorithm \ref{['alg:alg4']} (L-curve) for Example \ref{['ex1']} with $m=n=2048$
  • Figure 6.3: Algorithm \ref{['alg:alg5']} (GCV) for Example \ref{['ex2']} with $n=2048$
  • Figure 6.4: Images of Example \ref{['ex3']}: ppower$(3000 \times 2500)$

Theorems & Definitions (17)

  • Lemma 2.1
  • Remark 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Proposition 2.5
  • Theorem 3.1
  • Definition 4.1
  • Definition 4.2
  • Lemma 5.1: s1977, Theorem 3.3
  • Lemma 5.2: s1977, Theorem 3.4
  • ...and 7 more