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arxiv: 2606.30448 · v1 · pith:N6SA5MM7new · submitted 2026-06-29 · 🧮 math.NA · cs.NA

Iterated Tikhonov regularization of large linear problems

Davide Furch\`i , Lothar Reichel This is my paper

Pith reviewed 2026-06-30 04:44 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords iterated Tikhonov regularizationGolub-Kahan bidiagonalizationGauss quadraturediscrepancy principleill-posed problemsregularization parameter selectionlarge-scale linear systems
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The pith

Partial Golub-Kahan bidiagonalization determines the regularization parameter for iterated Tikhonov regularization via Gauss quadrature without computing multiple solutions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper addresses the choice of regularization parameter in iterated Tikhonov regularization for large linear discrete ill-posed problems when an estimate of the data error norm is available. It shows that partial Golub-Kahan bidiagonalization can be combined with the link to Gauss quadrature to select a parameter that satisfies the discrepancy principle. This avoids the usual requirement of computing and testing several trial solutions. The result is a reduction in computational work while still producing an approximate solution of the quality expected from iterated regularization.

Core claim

Iterated Tikhonov regularization based on partial Golub-Kahan bidiagonalization can determine the regularization parameter without computing several approximate solutions by using the connection between Golub-Kahan bidiagonalization and Gauss quadrature. This approach reduces the computational effort required to compute a desired solution that satisfies the discrepancy principle.

What carries the argument

The connection between partial Golub-Kahan bidiagonalization and Gauss quadrature, used to estimate the value of the regularization parameter that makes the residual norm match the known data error norm.

If this is right

  • A single run of the bidiagonalization process yields both an approximate solution and the needed parameter value.
  • The method inherits the higher accuracy typical of iterated Tikhonov regularization while lowering the cost of parameter selection.
  • The approach scales to large problems because only a modest number of bidiagonalization steps are required.
  • No explicit formation of the full normal equations or repeated factorizations is needed.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same quadrature link might be adapted to other iterative regularization families that rely on bidiagonalization.
  • In applications where the error norm estimate is updated during computation, the method could be re-run with modest extra cost.
  • The technique could be combined with early stopping criteria to further limit the number of matrix-vector products.

Load-bearing premise

The Gauss quadrature connection supplied by the bidiagonalization process accurately identifies a regularization parameter that satisfies the discrepancy principle.

What would settle it

A numerical test on a standard ill-posed problem where the quadrature estimate produces a residual norm that deviates from the known error norm by more than the discrepancy tolerance.

Figures

Figures reproduced from arXiv: 2606.30448 by Davide Furch\`i, Lothar Reichel.

Figure 1
Figure 1. Figure 1: Example 5.1: Solution ̂ of the error-free linear system (3) (continuous curve) and computed approximate solution , (dotted curve) by the method of the present paper for = 8 and = 10 [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Example 5.1 - The function () and its approximations for 102 ≤ ≤ 105 . From top to bottom at the right-hand side of the figures: ̄ +1, (dashed curve), (continuous curve), , (dotted curves) for = 8 and = 10 [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Example 5.2 - Solution ̂ of the error-free linear system (3) (continuous curve) and computed approximate solution , (dotted curve) by the method of the present paper for = 5 and = 10. D. Furchì and L. Reichel: Preprint submitted to Elsevier Page 14 of 13 [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Example 5.2 - The function () and its approximations for 102 ≤ ≤ 105 . From top to bottom at the right-hand side of the figure: ̄ +1, (dashed curve), (continuous curve), , (dotted curve) for = 5 and = 10 [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Example 5.3 - Solution ̂ (left) of the error-free linear system (3) and computed approximate solution , computed by the method of the present paper for = 1 (center) and = 20 (right) for = 50. 10−1 100 101 102 103 10−2 100 102 104 [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Example 5.3 - Approximations of the function () for 102 ≤ ≤ 105 . From top to bottom at the right-hand side of the figure: ̄ +1, (dashed curve), , (dotted curve) for = 50 and = 10. D. Furchì and L. Reichel: Preprint submitted to Elsevier Page 15 of 13 [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
read the original abstract

Many solution methods for linear discrete ill-posed problems with error-contaminated data (right-hand side) apply Tikhonov regularization to compute a meaningful approximate solution. This solution depends on a regularization parameter. It is well known that iterated Tikhonov regularization often determines an approximate solution of higher quality than (standard) Tikhonov regularization. We consider the situation when an estimate of the norm of the error in the data is known and would like to apply iterative Tikhonov regularization to determine an approximate solution that satisfies the discrepancy principle. This requires a suitable choice of a regularization parameter. The standard approach to determine this parameter is to compute solutions for several values of the regularization parameter and choose a computed approximate solution that satisfies the discrepancy principle. This paper discusses iterated Tikhonov regularization based on partial Golub-Kahan bidiagonalization and describes how the regularization parameter can be determined without computing several approximate solutions by using the connection between Golub-Kahan bidiagonalization and Gauss quadrature. This approach reduces the computational effort required to compute a desired solution.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper proposes iterated Tikhonov regularization for large linear discrete ill-posed problems with known noise level. It employs partial Golub-Kahan bidiagonalization and the link to Gauss quadrature to select the regularization parameter satisfying the discrepancy principle, avoiding explicit computation of multiple approximate solutions for different parameter values and thereby reducing computational effort compared to the standard approach.

Significance. If the central derivation holds, the work extends the established Golub-Kahan/Gauss-quadrature connection (previously used for standard Tikhonov and LSQR) to the iterated setting. This yields a parameter-free way to evaluate the discrepancy functional from bidiagonalization quantities alone, which is a practical efficiency gain for large-scale inverse problems where repeated solves are prohibitive.

minor comments (2)
  1. [Abstract / Method description] The abstract and method description would benefit from an explicit statement of the iterated filter factors in terms of the bidiagonalization quantities (e.g., a displayed equation analogous to the standard Tikhonov case) to make the quadrature reuse immediately verifiable.
  2. A short numerical example or complexity count comparing the new approach to the conventional multi-λ search would strengthen the claim of reduced effort; its absence leaves the efficiency statement qualitative.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and significance assessment of our work on extending the Golub-Kahan/Gauss-quadrature approach to iterated Tikhonov regularization. The recommendation of minor revision is noted.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper extends the established Golub-Kahan bidiagonalization–Gauss quadrature connection (previously used for standard Tikhonov) to iterated Tikhonov so that the discrepancy principle can be evaluated from bidiagonalization quantities alone. The abstract and method description indicate that iterated filter factors are expressed via the same tridiagonal structure, allowing quadrature reuse. No self-definitional equations, fitted inputs renamed as predictions, or load-bearing self-citation chains appear; the central claim rests on an independent, externally verifiable mathematical link applied to a new regularization variant. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract does not mention or rely on any specific free parameters, axioms, or invented entities.

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