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Regularization of linear inverse problems by rational Krylov methods

Stefan Kindermann

TL;DR

This work addresses solving ill-posed linear inverse problems by regularizing through aggregation of Tikhonov solutions and a RatCG variant, both reframed as rational Krylov space methods. By situating these approaches in K^n, R^n, and KR^n, and linking their residuals to CGNE-like polynomials and iterated-Tikhonov residuals, the authors derive convergence rates under Hölder source conditions when using the discrepancy principle and sufficiently large regularization parameters. The main contribution is proving optimal-order regularization for both aggregation and RatCG methods, with detailed analysis of polynomial zeros and discrepancy-based stopping, providing practical guidance on parameter choices. The results extend the spectral-filtering intuition to rational Krylov spaces and open avenues for nonlinear extensions while emphasizing the role of over-regularization in achieving robust convergence.

Abstract

For approximately solving linear ill-posed problems in Hilbert spaces, we investigate the regularization properties of the aggregation method and the RatCG method. These recent algorithms use previously calculated solutions of Tikhonov regularization (respectively, Landweber iterations) to set up a new search space on which the least-squares functional is minimized. We outline how these methods can be understood as rational Krylov space methods, i.e., based on the space of rational functions of the forward operator. The main result is that these methods form an optimal-order regularization schemes when combined with the discrepancy principle as stopping rule and when the underlying regularization parameters are sufficiently large.

Regularization of linear inverse problems by rational Krylov methods

TL;DR

This work addresses solving ill-posed linear inverse problems by regularizing through aggregation of Tikhonov solutions and a RatCG variant, both reframed as rational Krylov space methods. By situating these approaches in K^n, R^n, and KR^n, and linking their residuals to CGNE-like polynomials and iterated-Tikhonov residuals, the authors derive convergence rates under Hölder source conditions when using the discrepancy principle and sufficiently large regularization parameters. The main contribution is proving optimal-order regularization for both aggregation and RatCG methods, with detailed analysis of polynomial zeros and discrepancy-based stopping, providing practical guidance on parameter choices. The results extend the spectral-filtering intuition to rational Krylov spaces and open avenues for nonlinear extensions while emphasizing the role of over-regularization in achieving robust convergence.

Abstract

For approximately solving linear ill-posed problems in Hilbert spaces, we investigate the regularization properties of the aggregation method and the RatCG method. These recent algorithms use previously calculated solutions of Tikhonov regularization (respectively, Landweber iterations) to set up a new search space on which the least-squares functional is minimized. We outline how these methods can be understood as rational Krylov space methods, i.e., based on the space of rational functions of the forward operator. The main result is that these methods form an optimal-order regularization schemes when combined with the discrepancy principle as stopping rule and when the underlying regularization parameters are sufficiently large.
Paper Structure (12 sections, 28 theorems, 165 equations, 1 figure)

This paper contains 12 sections, 28 theorems, 165 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Representation
  4. Collection of results for orthogonal polynomials
  5. Residual polynomials for the CGNE method
  6. Iterated Tikhonov method
  7. Convergence results
  8. Estimates for the discrepancy principle.
  9. Roots of polynomials
  10. The main convergence result
  11. Conclusion
  12. Results for the roots of the derived polynomials

Key Result

Proposition 1

With the previous definitions, for any $\mathcal{X} \in \{\mathcal{K}^n, \mathcal{R}^n,\mathcal{KR}^n\}$, it holds that $n = n_{bd,\mathcal{X}}$ if and only if $A^*y_\delta$ can be written as a linear combination of $n-1$ nonzero eigenvectors of $A^*A$. In particular

Figures (1)

  • Figure 1: Schematic description of the steps of the aggregation method.

Theorems & Definitions (49)

  • Definition 1
  • Proposition 1
  • Definition 2
  • Proposition 2
  • Lemma 1
  • Lemma 2
  • proof
  • Definition 3
  • Proposition 3
  • proof
  • ...and 39 more