Convergence Analysis of Levenberg-Marquardt Method for Inverse Problem with H\"{o}lder Stability Estimate

Akari Ishida; Gen Nakamura; Sei Nagayasu

REVIEW 3 major objections 4 minor 17 references

Convergence Analysis of Levenberg-Marquardt Method for Inverse Problem with H\"{o}lder Stability Estimate

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The Levenberg-Marquardt method converges with explicit rates for nonlinear inverse problems satisfying a uniform Hölder stability estimate, and these local results yield global reconstruction algorithms for inverse problems with finitely…

desk verdict A mostly correct extension of LM convergence analysis to Hölder stability, with a genuine but fixable ball-radius gap in the main induction. read the letter →

arxiv 2501.08932 v1 pith:2DG52BLO submitted 2025-01-15 math.FA

Akari Ishida , Sei Nagayasu , Gen Nakamura This is my paper · ORCID

classification math.FA MSC 47J2565J2235R30

keywords InverseproblemsIll-posedoperatorequationsRegularizationLevenberg-MarquardtmethodHölderstabilityestimateConvergenceratesDiscrepancyprincipleFinitemeasurements

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves that the Levenberg-Marquardt (LM) iteration, a standard regularization method for nonlinear inverse problems, converges to a solution whenever the forward map satisfies a uniform Hölder stability estimate, with no additional source condition or tangential cone condition. For exact data, the iterates remain in a small error ball around the solution and converge, at a geometric rate if the Hölder exponent is $1$ and at an algebraic rate for smaller exponents. For noisy data, stopping by Morozov's discrepancy principle gives an approximate solution whose error is of order $\delta^2$, and under an extra condition the number of iterations grows only logarithmically in $1/\delta$. The same analysis, combined with a finite lattice search over a compact parameter set, yields global reconstruction algorithms for inverse problems with finitely many measurements. This matters because LM is known to be numerically faster than Landweber iteration, and the proof gives convergence rates for LM under a condition that holds for many coefficient identification problems.

What carries the argument

The engine of the proof is the LM step $x_{k+1}=x_k+(F'(x_k)^*F'(x_k)+\alpha_k I)^{-1}F'(x_k)^*(y-F(x_k))$ together with Morozov's discrepancy principle, which fixes $\alpha_k>0$ by the equation $\alpha_k\|(F'(x_k)F'(x_k)^*+\alpha_k I)^{-1}(y-F(x_k))\|_Y=q\|y-F(x_k)\|_Y$ for a fixed $q\in(0,1)$. The Hölder stability estimate is the load-bearing hypothesis: it implies a tangential-cone-type inequality (2.5) (with $\omega=2$ in the exact case), which in turn guarantees that $\alpha_k$ is well defined and yields the monotonicity estimate of Lemma 2.2. The rate argument rests on the differential inequality $\gamma_{k+1}-\gamma_k \le -c\,\gamma_k^{2/(1+\varepsilon)}$ for $\gamma_k=\frac12\|x_k-x^\dagger\|_X^2$, obtained by combining the stability estimate with the bound $\alpha_k\le \frac{q}{1-q}\|F'(x_k)\|^2$ from Lemma 2.3; solving this inequality gives the geometric rate when $\varepsilon=1$ and the algebraic rate (3.11) otherwise.

What would settle it

Construct or numerically simulate a forward map $F$ that satisfies Assumption 3.1 on a ball $B(x_0,\rho')$ with $\rho<\rho'<4\rho$, choose an initial guess $x_0$ with $\frac12\|x_0-x^\dagger\|^2=\rho$, and check whether the LM iterates leave $B$ or fail to converge to $x^\dagger$; if any iterate satisfies $\frac12\|x_k-x_0\|^2>\rho'$, the containment premise on which the proof relies is violated.

Watch

Extended reading notes

Core claim

The paper's central discovery is that the convergence theory of the LM method, previously built on the tangential cone condition plus a source condition, can be rerouted through the uniform Hölder stability estimate $\frac{1}{\sqrt{2}}\|x-\tilde{x}\|_X \le C_F \|F(x)-F(\tilde{x})\|_Y^{\frac{1+\varepsilon}{2}}$ holding on a ball. Under this assumption plus Lipschitz Fréchet differentiability of $F$, Theorem 3.3 shows that if the initial error satisfies $\frac12\|x_0-x^\dagger\|_X^2 \le \rho$ for a sufficiently small $\rho$, the LM iterates defined by (2.3) stay within that error level, converge to the solution $x^\dagger$, and satisfy the explicit rates (3.10) for $\varepsilon=1$ and (3.11) for $0<\varepsilon<1$. Theorem 4.1 extends this to noisy data with the discrepancy principle: the stopping index is finite, the error decreases before stopping, and the final error is $O(\delta^2)$ when $\rho\le C\delta^2$. Section 5 then shows how to make these local results global for inverse problems with finitely many measurements: a finite lattice over a compact set provides an initial guess within the required distance (Lemma 5.1), so Algorithms 1 and 2 reconstruct the unknown from exact or noisy finite data.

Load-bearing premise

The proof requires each iterate to stay in the ball $B=\{x:\frac12\|x-x_0\|^2\le \rho'\}$ on which Assumption 3.1 holds, yet the induction only establishes $\frac12\|x_k-x^\dagger\|^2\le \rho$, and together with the initial condition this gives $\frac12\|x_k-x_0\|^2\le 4\rho$, not $\rho'$; hence the stated assumption $\rho<\rho'$ is insufficient unless $4\rho<\rho'$ or the stability estimate is assumed on a ball centered at $x^\dagger$.

Editorial extensions

If this is right

For exact data with Hölder exponent $\varepsilon=1$, the LM error decays like $(1-c)^k$, a geometric rate reached without any source condition.
For $0<\varepsilon<1$, the rate is algebraic, $\gamma_k \lesssim k^{-(1+\varepsilon)/(1-\varepsilon)}$, which still gives useful a priori iteration counts.
For noisy data stopped by the discrepancy principle, the final error satisfies $\frac12\|x_{k_*}^\delta-x^\dagger\|^2 \le C'\delta^2$, so the reconstruction accuracy is of order $\delta^2$ in the noise level.
Because Lemma 5.1 supplies an initial guess within the required small distance, Algorithms 1 and 2 reconstruct solutions of inverse problems with finitely many measurements without requiring an a priori guess already close to the unknown.
The number of iterations needed to reach a prescribed accuracy can be computed in advance from the rate formulas, making the exact-data algorithm a constructive reconstruction method.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The $4\rho$ versus $\rho'$ gap is likely a fixable technical flaw: stating the smallness assumption as $4\rho<\rho'$, or assuming the Hölder estimate on a ball centered at the solution $x^\dagger$, would make the induction fully rigorous without changing the rates.
The proof discards the non-positive term $-\frac12\|F'(x_k)^*(F'(x_k)F'(x_k)^*+\alpha_k I)^{-1}(y-F(x_k))\|^2$ in (3.28); keeping it could yield sharper constants and may explain the numerically observed speed advantage of LM over Landweber, a comparison the authors flag as open.
The global algorithms inherit the finite-measurement stability framework from the lattice lemma; a natural stress test would be to run them on electrical impedance tomography or inverse scattering with piecewise-constant coefficients, where Lipschitz or Hölder stability is already known.
For noisy data, both the $O(\delta^2)$ rate and the logarithmic iteration bound depend on the discrepancy principle and on the choice of $\tau$ satisfying (4.2); an a priori stopping rule would require a different argument.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper does something real: it gives the first convergence and rate analysis for the Levenberg–Marquardt method under a Hölder stability estimate, extending the Landweber-based results of de Hoop–Qiu–Scherzer and Mittal–Giri. The proof is largely self-contained, with explicit geometric rates for ε=1 and algebraic rates for ε<1, plus a noisy-data variant and a clean adaptation of Alberti–Santacesaria’s global reconstruction framework. The authors honestly note that their rates are similar to those under Landweber and that the anticipated speed advantage remains a numerical question.

The main soft spot is real but not fatal. In Theorem 3.3 the induction proves γ_k ≤ ρ for γ_k = ½‖x_k−x†‖², and then the proof says x_m, x† ∈ B before applying Assumption 3.1. This does not follow from ρ<ρ′: the triangle inequality gives ½‖x_m−x0‖² ≤ 4ρ, not ρ′. The same gap appears in Theorem 4.1. It is straightforwardly fixable by strengthening the condition to 4ρ<ρ′ (or by centering the ball at x†), which is achievable by choosing q small, so the main theorems are likely correct after revision but are not fully proved as stated. A referee should insist this be repaired and the induction written so that membership in B is explicit.

Two smaller issues: the noisy-data theorem’s rate statement depends on condition (4.4) that relates ρ to δ², and the paper does not discuss how this interacts with the strengthened smallness condition. And while the algorithms in Section 5 are plausible, they are not tested numerically, so the practical promise of faster convergence is unverified. The citation pattern is fair; the paper builds on the right literature and does not oversell its novelty.

Who is this for? Researchers working on iterative regularization for nonlinear inverse problems, especially those interested in replacing Landweber by LM in the finite-measurement global reconstruction pipeline. It deserves a serious referee and likely publication after a careful revision. I would not cite the theorems in their current form until the ball-radius condition is fixed, but after that it should be a useful reference.

Referee Report

3 major / 4 minor

Summary. The paper studies the Levenberg–Marquardt iteration (2.3) for ill-posed operator equations F(x)=y in Hilbert spaces, assuming a uniform Hölder stability estimate (3.3) and Lipschitz differentiability of F on a ball B centered at the initial point x0. For exact data, Theorem 3.3 establishes local convergence and convergence rates: linear for ε=1 and algebraic for ε<1, under a small initial error condition (3.7). For noisy data, Theorem 4.1 provides an a priori stopping rule based on Morozov's discrepancy principle and an O(δ²) error estimate (4.5), together with a logarithmic bound on the stopping index in Proposition 4.2. Section 5 combines these local results with the finite-measurement stability estimates of Alberti–Santacesaria to propose global reconstruction algorithms (Algorithms 1 and 2) for exact and noisy data.

Significance. The significance of the paper is in extending the Hölder-stability-based convergence analysis, previously developed for Landweber iteration, to the Levenberg–Marquardt method, and in providing explicit rates and global reconstruction algorithms for inverse problems with finite measurements. The proof strategy is self-contained and follows standard energy estimates; the paper does not introduce free parameters or circular reasoning. However, the current statement of Theorems 3.3 and 4.1 contains a smallness-condition gap that affects the induction and the rate derivation, and Section 5's transfer of the lattice-based initial guess to the theorem hypotheses is not fully justified. These issues are local and repairable, and the central convergence claims appear likely to be correct after a strengthening of the hypotheses.

major comments (3)

[Theorem 3.3, proof around (3.24)–(3.29); Theorem 4.1, proof around (4.11)–(4.18)] The induction asserts that x_m and x† lie in B, where B = {x : (1/2)||x−x0||² ≤ ρ'}. The induction hypothesis only gives γ_m = (1/2)||x_m−x†||² ≤ ρ, and the initial condition gives γ_0 ≤ ρ. By the triangle inequality this yields (1/2)||x_m−x0||² ≤ 4ρ, not ρ'. Thus the condition ρ < ρ' in (3.6) (and in (4.1)) is insufficient to apply Assumption 3.1 at x_m; the same problem occurs at x_{m+1} and in Theorem 4.1 at x_m^δ and x_{m+1}^δ. The induction can be repaired by strengthening the smallness condition to 4ρ ≤ ρ', which can be achieved by choosing q sufficiently small in (3.5) and (4.1). As stated, however, the theorems are not fully proved.
[Theorem 3.3, equation (3.34)] The derivation of the algebraic rate for ε∈(0,1) invokes the inequality (1−s)^{-β} ≥ 1+βs for 0≤s<1 with s = c γ_k^{(1−ε)/(1+ε)}. The proof does not verify that s<1 under the stated assumptions (3.5)–(3.7). For arbitrary q satisfying (3.5), c γ_k^{(1−ε)/(1+ε)} can exceed 1, in which case the expression (1−s)^{-β} is not real and the inequality cannot be applied. This gap can be fixed either by adding the smallness condition already needed for ball containment, so that s<1, or by treating the case s≥1 separately, where γ_{k+1}=0 and the claimed bound is trivial. As written, the rate estimate (3.11) lacks a complete proof.
[Section 5, Lemma 5.1 and condition (3.7)] Lemma 5.1(ii) concludes that ||x0−x†||_X < ρ for the lattice point chosen as initial guess. However, condition (3.7) requires (1/2)||x0−x†||² ≤ ρ, i.e., ||x0−x†|| ≤ √(2ρ). The implication from ||x0−x†|| < ρ to (3.7) is valid only if ρ ≤ 2. Since ρ in (3.6) and (4.1) is not assumed to satisfy ρ ≤ 2, the global algorithms may start from an x0 that does not satisfy the local convergence theorem's hypothesis. The lattice radius in (5.2) should be adjusted to √(2ρ)/(2~L~C||Q||) (or one should explicitly require ρ ≤ 2 and 4ρ ≤ ρ'), so that Lemma 5.1 yields the actual smallness condition (3.7).

minor comments (4)

[Algorithm 2, lines 14–19] The control flow contains a redundant 'else if the stopping criterion is satisfied' immediately after the same condition has been tested; the second branch appears intended for a different stopping condition (such as reaching the maximal iteration count M) and should be clarified.
[Section 5, paragraph before (5.2)] The paper states that Q∘F satisfies Assumption 3.1 but does not give the explicit correspondence between the constants ~C, ~L, ||Q|| and C_F, L, Lhat. A short derivation of (3.3) with ε=1 from (5.1) would make the applicability of Theorem 3.3 and Theorem 4.1 in the finite-measurement setting more transparent.
[Theorem 4.1, equation (4.5)] The constant C' in (4.5) depends on k*; the proof shows via (4.21) and ρ ≤ Cδ² that C' ≥ 0, but this is not stated explicitly and would be worth a brief remark.
[Throughout] Several displayed equations, especially (2.3) and (2.4), are typeset with dense parentheses that make the structure of the inverse operators harder to read; adding line breaks or larger parentheses would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the LM convergence proof proceeds from the stated Hölder stability and smoothness assumptions, and the external citations are not author-generated.

full rationale

The paper's central claim is Theorem 3.3: under Assumption 3.1, which includes the Hölder stability estimate (3.3), Lipschitz continuity of F′ and boundedness of F′, LM iterates converge with explicit rates. The proof derives the key inequalities (3.23), (3.28) and (3.33) from these assumptions together with standard identities for the LM step and the discrepancy principle. No parameter is fitted to data and later renamed a prediction; q and ρ are chosen to satisfy explicit inequalities (3.5)–(3.6) and (4.1)–(4.2), and the convergence rates are consequences of those choices. Section 5 uses Alberti-Santacesaria [1] for finite-measurement stability and for the lattice-based initial guess; that is an external cited result, not a self-citation by the present authors, and it is not used to redefine the paper's conclusion as an input. The paper also does not invoke any author-specific uniqueness theorem or smuggle in an ansatz through self-citation. The reviewer-identified gap concerning the induction ball radius—where only γk ≤ ρ is proved but Assumption 3.1 is stated on a ball of radius ρ′—is a proof-strength issue about the sufficiency of the smallness condition ρ < ρ′ , not circularity: it does not identify the conclusion with the hypothesis. Therefore the derivation chain is self-contained and no circular step is exhibited.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central argument rests on the Hölder stability and smoothness assumptions, plus an imported finite-measurement stability condition in Section 5. There are no fitted parameters or invented entities. The main unstated technical premise is the ball-radius condition discussed in the weakest_assumption field.

assumptions (5)

domain assumption F has a continuous Fréchet derivative, F' is Lipschitz on B with constant L, and ||F'(x)||≤hat_L on B (Assumption 3.1(a,b)).
These smoothness and boundedness conditions are the basis for the Taylor estimates and for bounding the LM regularization parameter.
domain assumption Uniform Hölder stability estimate (3.3): (1/√2)||x-~x||≤C_F||F(x)-F(~x)||^{(1+ε)/2} on B.
This is the key structural condition of the paper; it replaces the tangential cone and source conditions used elsewhere.
domain assumption There exists a solution x† with F(x†)=y and the initial guess satisfies the smallness condition (3.7).
Local convergence is only claimed near a solution.
domain assumption In Section 5, the finite-measurement stability estimate (5.1) from Alberti-Santacesaria holds, and K is compact with a finite lattice covering as in (5.2).
This imports the global initial-guess mechanism from [1].
standard math Standard Taylor expansion for Fréchet differentiable operators with Lipschitz derivative, cited as [14, Lemma A.63], and the uniqueness of α_k from the discrepancy principle under (2.5) given in [12].
These are unproved background tools used throughout the proofs.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Convergence Analysis of Levenberg-Marquardt Method for Inverse Problem with H\"{o}lder Stability Estimate." pith.science (2026). https://pith.science/paper/2DG52BLO

@misc{pith2026250108932,
  author       = {Pith},
  title        = {Pith review of: Convergence Analysis of Levenberg-Marquardt Method for Inverse Problem with H\"older Stability Estimate},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2DG52BLO}},
  note         = {Machine review of arXiv:2501.08932}
}

read the original abstract

We analyze convergence of the Levenberg-Marquardt method for solving nonlinear inverse problems in Hilbert spaces. Specifically, we establish local convergence and convergence rates for a class of inverse problems that satisfy H\"{o}lder stability estimate. Furthermore, based on what we found in the mentioned analysis, we develop global reconstruction algorithms for solving inverse problems with finite measurements for exact and noisy data, respectively.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

[1]

Alberti and Matteo Santacesaria

Giovanni S. Alberti and Matteo Santacesaria. Inﬁnite-dimensio nal inverse problems with ﬁnite measurements. Arch. Ration. Mech. Anal. , 243(1):1– 31, 2022

work page 2022
[2]

Stable determination of conductivity by bo undary measurements

Giovanni Alessandrini. Stable determination of conductivity by bo undary measurements. Appl. Anal. , 27(1-3):153–172, 1988

work page 1988
[3]

de Hoop, and Romina Gaburro

Giovanni Alessandrini, Maarten V. de Hoop, and Romina Gaburro. Unique- ness for the electrostatic inverse boundary value problem with piec ewise constant anisotropic conductivities. Inverse Problems , 33(12):125013, 24, 2017

work page 2017
[4]

de Hoop, Romina Gaburro, and Eva Sincich

Giovanni Alessandrini, Maarten V. de Hoop, Romina Gaburro, and Eva Sincich. Lipschitz stability for the electrostatic inverse boundary v alue problem with piecewise linear conductivities. J. Math. Pures Appl. (9) , 107(5):638–664, 2017

work page 2017
[5]

Lipschitz stability for t he inverse conductivity problem

Giovanni Alessandrini and Sergio Vessella. Lipschitz stability for t he inverse conductivity problem. Adv. in Appl. Math. , 35(2):207–241, 2005. 21

work page 2005
[6]

de Hoop, and Lingyun Qiu

Elena Beretta, Maarten V. de Hoop, and Lingyun Qiu. Lipschitz st ability of an inverse boundary value problem for a Schr¨ odinger-type equation. SIAM J. Math. Anal. , 45(2):679–699, 2013

work page 2013
[7]

de Hoop, Lingyun Qiu, and Otmar Scherzer

Maarten V. de Hoop, Lingyun Qiu, and Otmar Scherzer. Local an alysis of inverse problems: H¨ older stability and iterative reconstruction . Inverse Problems, 28(4):045001, 2012

work page 2012
[8]

A regularizing Levenberg-Marquardt scheme, wit h ap- plications to inverse groundwater ﬁltration problems

Martin Hanke. A regularizing Levenberg-Marquardt scheme, wit h ap- plications to inverse groundwater ﬁltration problems. Inverse Problems , 13(1):79–95, 1997

work page 1997

Show all 17 references

[9]

A conv ergence analysis of the Landweber iteration for nonlinear ill-posed problems

Martin Hanke, Andreas Neubauer, and Otmar Scherzer. A conv ergence analysis of the Landweber iteration for nonlinear ill-posed problems. Nu- mer. Math. , 72(1):21–37, 1995

1995
[10]

Uniqueness, stability and global convergenc e for a discrete inverse elliptic Robin transmission problem

Bastian Harrach. Uniqueness, stability and global convergenc e for a discrete inverse elliptic Robin transmission problem. Numer. Math. , 147(1):29–70, 2021

2021
[11]

Iterative regularization methods in inverse scattering

Thorsten Hohage. Iterative regularization methods in inverse scattering. In Proceedings of Inverse Problems in Engineering: Theory and Practice 3rd Int. Conference on Inverse Problems in Engineering, Port Lu dlow, WA, June 13–18 , 1999

1999
[12]

Iterative regularization methods for nonlinear ill-posed problems , volume 6 of Radon Series on Computational and Applied Mathematics

Barbara Kaltenbacher, Andreas Neubauer, and Otmar Scher zer. Iterative regularization methods for nonlinear ill-posed problems , volume 6 of Radon Series on Computational and Applied Mathematics . Walter de Gruyter GmbH & Co. KG, Berlin, 2008

2008
[13]

Iterative methods for nonlinear ill-posed problems in Banach spaces: converg ence and applications to parameter identiﬁcation problems

Barbara Kaltenbacher, Frank Sch¨ opfer, and Thomas Schus ter. Iterative methods for nonlinear ill-posed problems in Banach spaces: converg ence and applications to parameter identiﬁcation problems. Inverse Problems , 25(6):065003, 19, 2009

2009
[14]

An introduction to the mathematical theory of inverse problems, volume 120 of Applied Mathematical Sciences

Andreas Kirsch. An introduction to the mathematical theory of inverse problems, volume 120 of Applied Mathematical Sciences . Springer, Cham, third edition, 2021

2021
[15]

Improved local convergenc e anal- ysis of the Landweber iteration in Banach spaces

Gaurav Mittal and Ankik Kumar Giri. Improved local convergenc e anal- ysis of the Landweber iteration in Banach spaces. Arch. Math. (Basel) , 120(2):195–202, 2023

2023
[16]

Sch¨ opfer, A

F. Sch¨ opfer, A. K. Louis, and T. Schuster. Nonlinear iterativ e methods for linear ill-posed problems in Banach spaces. Inverse Problems , 22(1):311– 329, 2006

2006
[17]

Kazimierski

Thomas Schuster, Barbara Kaltenbacher, Bernd Hofmann, a nd Kamil S. Kazimierski. Regularization methods in Banach spaces , volume 10 of Radon Series on Computational and Applied Mathematics . Walter de Gruyter GmbH & Co. KG, Berlin, 2012. 22 Akari Ishida Graduate School of Mat...

2012

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Alberti and Matteo Santacesaria

Giovanni S. Alberti and Matteo Santacesaria. Inﬁnite-dimensio nal inverse problems with ﬁnite measurements. Arch. Ration. Mech. Anal. , 243(1):1– 31, 2022

work page 2022

[2] [2]

Stable determination of conductivity by bo undary measurements

Giovanni Alessandrini. Stable determination of conductivity by bo undary measurements. Appl. Anal. , 27(1-3):153–172, 1988

work page 1988

[3] [3]

de Hoop, and Romina Gaburro

Giovanni Alessandrini, Maarten V. de Hoop, and Romina Gaburro. Unique- ness for the electrostatic inverse boundary value problem with piec ewise constant anisotropic conductivities. Inverse Problems , 33(12):125013, 24, 2017

work page 2017

[4] [4]

de Hoop, Romina Gaburro, and Eva Sincich

Giovanni Alessandrini, Maarten V. de Hoop, Romina Gaburro, and Eva Sincich. Lipschitz stability for the electrostatic inverse boundary v alue problem with piecewise linear conductivities. J. Math. Pures Appl. (9) , 107(5):638–664, 2017

work page 2017

[5] [5]

Lipschitz stability for t he inverse conductivity problem

Giovanni Alessandrini and Sergio Vessella. Lipschitz stability for t he inverse conductivity problem. Adv. in Appl. Math. , 35(2):207–241, 2005. 21

work page 2005

[6] [6]

de Hoop, and Lingyun Qiu

Elena Beretta, Maarten V. de Hoop, and Lingyun Qiu. Lipschitz st ability of an inverse boundary value problem for a Schr¨ odinger-type equation. SIAM J. Math. Anal. , 45(2):679–699, 2013

work page 2013

[7] [7]

de Hoop, Lingyun Qiu, and Otmar Scherzer

Maarten V. de Hoop, Lingyun Qiu, and Otmar Scherzer. Local an alysis of inverse problems: H¨ older stability and iterative reconstruction . Inverse Problems, 28(4):045001, 2012

work page 2012

[8] [8]

A regularizing Levenberg-Marquardt scheme, wit h ap- plications to inverse groundwater ﬁltration problems

Martin Hanke. A regularizing Levenberg-Marquardt scheme, wit h ap- plications to inverse groundwater ﬁltration problems. Inverse Problems , 13(1):79–95, 1997

work page 1997