Support identification for parameter variations in a PDE system via regularized methods

Chayma Nssibi; Houcine Meftahi

REVIEW 4 major objections 5 minor 28 references

Support identification for parameter variations in a PDE system via regularized methods

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read Boundary measurements can localize simultaneous changes in an elastic medium's Lamé parameters and density, even when the changes occupy disjoint regions.

desk verdict The paper's headline disjoint-support reconstruction is not supported by its own algorithm, which forces all three parameter supports to coincide. read the letter →

arxiv 2505.21663 v1 pith:EDTRFBFC submitted 2025-05-27 math.OC

Houcine Meftahi , Chayma Nssibi This is my paper

classification math.OC MSC 35R3065N2174B05

keywords inverseproblemslinearelasticitymonotonicitymethodtruncatedsingularvaluedecompositionsupportrecoveryNeumann-to-Dirichletoperatorparameteridentificationregularization

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

The paper studies the inverse problem of locating where the elastic parameters λ, μ, and density ρ of a body deviate from known background values, using only measurements of boundary forces and the resulting boundary displacements. Its central proposal is a two-stage reconstruction: a monotonicity-based inclusion test decides which candidate regions lie inside the true support, and a truncated singular value decomposition (TSVD) step damps noise before a constrained least-squares fit. The paper claims this hybrid recovers supports accurately under significant noise, including the previously difficult case where the supports of the three parameter variations are disjoint. If correct, the method gives a non-iterative, boundary-only imaging procedure that could serve in medical diagnostics, where such parameter variations are associated with tumors or lesions.

What carries the argument

The load-bearing object is the Neumann-to-Dirichlet operator $\Lambda(\lambda, \mu, \rho)$, which maps applied boundary forces to measured boundary displacements, together with its Loewner monotonicity: increasing all three parameters decreases the operator in the semidefinite order. The inclusion test evaluates the Fréchet derivative $\Lambda'$ on characteristic functions of test balls; Theorem 1 turns the geometric question $B \subseteq D$ into a positive-eigenvalue check on a self-adjoint matrix. Regularization proceeds by truncating the singular value decomposition of each sensitivity matrix $T_k$ before enforcing box constraints derived from the monotonicity inequalities, so the final reconstruction is the minimizer of a constrained low-rank residual.

What would settle it

Run the disjoint-support experiment of Figure 5 with two inclusions, one changing only ρ and one only λ, at 10% noise, and check which test balls pass the positive-eigenvalue test; if any ball outside the true supports is marked inside, or any ball inside is missed, the three-parameter monotonicity equivalence or its discretization is unsound.

Watch

Extended reading notes

Core claim

The central claim is that the support of simultaneous variations in λ, μ, and ρ can be identified from the Neumann-to-Dirichlet operator by testing candidate balls B through the semidefinite inequality $\Lambda(\lambda_0,\mu_0,\rho_0) - \Lambda(\lambda,\mu,\rho) + \Lambda'(\lambda_0,\mu_0,\rho_0)(C_\lambda\chi_B, C_\mu\chi_B, C_\rho\chi_B) \ge 0$: B lies inside the true support exactly when the matrix is positive semidefinite. The paper states this three-parameter equivalence as Theorem 1, with proof deferred to an analogous two-parameter argument in the literature. For noisy data the paper replaces exact measurements by noisy ones plus a $\delta I$ shift, and then solves a constrained least-squares problem in which the sensitivity matrices are truncated by TSVD. Numerical experiments on a square domain with 19 boundary loads and 100 test balls show the hybrid reconstructs common and disjoint supports more accurately than plain monotonicity constraints at 10% noise.

Load-bearing premise

The load-bearing premise is that the three-parameter support-inclusion equivalence of Theorem 1 is valid for this mixed boundary value problem; the paper does not prove it and instead refers to an analogous two-parameter result.

Editorial extensions

If this is right

Support identification in this three-parameter elastic model needs only boundary force and displacement data, not interior measurements.
The combined method handles disjoint supports for λ, μ, and ρ variations, a case where monotonicity constraints alone give less accurate reconstructions.
The regularization is convergent in the noise level: the noisy-data minimizer approaches the exact-data minimizer as δ tends to zero.
Each test ball only requires a handful of forward solves and an eigenvalue check, so the reconstruction avoids a full nonlinear iteration.
The TSVD threshold τ = 0.99 gives a data-dependent cut-off of small singular values in the sensitivity matrices, stabilizing the linearized update under noise.

Reading between the lines

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

An extension not explored in the paper would relax the assumption that the background and inclusion parameter values are known a priori, since the inclusion test uses the contrasts explicitly.
The single scalar ζ that ties the three parameter variations through fixed ratios τ1 and τ2 may limit reconstructions when the three perturbations have opposite signs; a vector-valued parametrization would be a natural next step.
Because the monotonicity structure is generic, the same hybrid recipe could be adapted to other positive-coefficient PDE systems, such as viscoelastic or poroelastic models, by supplying the corresponding Fréchet derivative.
A complete proof of Theorem 1 for three parameters under the mixed Dirichlet–Neumann boundary conditions would turn the numerical evidence into a rigorous foundation; the paper currently relies on an analogous two-parameter theorem.

Share X Bluesky LinkedIn Reddit HN

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

Colleague,

Two things to know about arXiv:2505.21663. First, the headline result—that the hybrid monotonicity–TSVD method recovers disjoint supports of δλ, δμ, δρ—is not supported by the algorithm the authors actually define. Second, the proof of the Fréchet derivative formula (Proposition 1) contains a wrong saddle-point computation. These aren't cosmetic issues.

The paper deserves credit for asking a useful question: can monotonicity tests identify supports of three elastic parameters from boundary data, especially when those supports don't overlap? The idea of adding TSVD to the monotonicity-constrained least-squares step is reasonable, and the common-support part builds on established monotonicity framework from Eberle and Harrach [10] and the authors' own stability preprint [24]. If the paper were only about the common-support case, the numerical illustrations might be worth a look.

But the disjoint-support demonstration doesn't match the math. In Section 4.2 the optimization is written as min over a single scalar field ζ with (α, β, γ) = (ζ, τ1ζ, τ2ζ), with τ1, τ2 fixed positive constants. That means in every pixel where ζ is nonzero, λ, μ, and ρ all vary simultaneously at prescribed relative strengths. Such an ansatz cannot represent a state where δμ and δρ have disjoint supports. Section 5's Eq. (19) keeps the same parametrization, yet Figure 5 claims to show exactly such disjoint reconstructions. Theorems 1 and 2, the only support-characterization results, are stated only for a common support D. So the central numerical claim is not a consequence of the described method; either the figures were produced by a different, unstated algorithm (independent α, β, γ per pixel), or they're not trustworthy.

Proposition 1's proof is also problematic. The Lagrangian L as defined does not satisfy L(λ,μ,ρ,u_g,v) = ⟨Λg,g⟩; plugging in the claimed saddle point (u_g, −2u_g) gives negative signs that don't match. The Fréchet derivative formula (5) is standard and probably true, but the derivation given is invalid. Minor but telling: Algorithm 1 is referenced but never provided, and there are no quantitative error metrics, code, or reproducibility details for the figures.

Bottom line: the paper is not ready for publication. The common-support material could survive a rewrite, but the disjoint-support claim, which is the main selling point, currently appears inconsistent with the method. I would want the authors to either reformulate the optimization with independent parameter fields and rerun the experiments, or remove the disjoint-support claim entirely. If they did that, the paper might have a chance; as it stands, I'd recommend against sending it on.

Referee Report

4 major / 5 minor

Summary. The manuscript studies the inverse problem of determining the support of perturbations of the Lamé parameters λ, µ and the density ρ in a linear elasticity model with mixed Dirichlet–Neumann boundary conditions, using measurements of the Neumann-to-Dirichlet map. The authors propose a linearized monotonicity inclusion test for the common-support case, a monotonicity-constrained least-squares reconstruction, and a combined monotonicity–TSVD procedure. They claim that the combined method stably recovers common and disjoint supports and is more accurate than monotonicity alone, with numerical experiments on a two-dimensional square.

Significance. If correct, the paper would make a useful numerical contribution: support localization from boundary data without a full nonlinear iteration, with a hybrid regularization that is plausible in practice. The paper contains several potentially valuable ingredients: a three-parameter monotonicity formulation, a box-constrained variational reformulation, and a TSVD truncation that is a standard and reproducible regularizer. However, the theoretical support is not established in the manuscript, and the disjoint-support numerical claim is not matched by the stated optimization problem. The significance therefore remains conditional on a substantial revision.

major comments (4)

[Section 2, Proposition 1] The proof of Proposition 1 is invalid. The claimed identity L(λ,µ,ρ,ug,v)=⟨Λ(λ,µ,ρ)g,g⟩ is false: with a(·,·) the bilinear form of (3), substituting φ=ug gives L(λ,µ,ρ,ug,v)=a(ug,v), because the term ∫ΓN g·ug cancels the quadratic term a(ug,ug). Consequently (ug,−2ug) is not a saddle point: ∂ψL at that point in a direction ψhat equals a(ug,ψhat)=⟨g,ψhat⟩, which does not vanish for general ψhat. The remainder estimate also relies on Lemma 3, whose proof contains an unjustified replacement of ∥u2∥ by ∥u1∥ in the final inequality; the stated lower bound with (λ2−λ2²/λ1)∥∇·u1∥² does not follow from the preceding expression. The Fréchet derivative formula (5) may be true, but a correct proof is needed.
[Section 3.2, Theorems 1 and 2] The central inclusion characterizations are not proved in the manuscript. The sentence that the proofs follow by arguments analogous to Corollaries 2 and 3 of [10] is not sufficient, because [10] treats a two-parameter problem and the present setting has three parameters with mixed Dirichlet–Neumann boundary conditions. Since every numerical inclusion test in Section 3.3 relies on these theorems, a complete proof or a detailed reduction to [10] must be supplied.
[Section 5, Eq. (19) and Figure 5] The disjoint-support experiment is not a consequence of the algorithm stated in Eq. (19). In (19)–(20) the optimization variable is a single scalar field ζ and the perturbations are constrained to be (ζ, τ1ζ, τ2ζ), so any pixel with nonzero ζ contributes to δλ, δµ and δρ simultaneously with fixed ratios. Such a parametrization cannot represent pairwise disjoint supports (the square, circle, and ellipse in Figure 5). If the actual solver uses independent coefficient fields α, β, γ, then Eq. (19), Algorithm 1, and the Figure 5 setup must be rewritten accordingly; otherwise the figure cannot have been produced by the described method. The common-support theory in Theorems 1–2 also does not cover the disjoint-support case.
[Section 4.2, Theorem 4] The convergence proof of Theorem 4 is incomplete. It asserts without proof that Cδ converges to C in the Hausdorff sense, and it passes from Jδ(ζ̂δ)≤Jδ(ζ̃δ) to J(ζ̂)≤J(ζ̃) although uniform convergence of Jδ to J on the admissible sets is not established. A rigorous proof should supply these steps.

minor comments (5)

[Section 3.3 and Figure 1] Figure 1 refers to Algorithm 1, but no Algorithm 1 appears anywhere in the manuscript; the bullet list in Section 3.3 is not an algorithm and omits the test-ball size and placement. The numerical experiments are therefore not reproducible as written.
[Throughout] There are numerous typos and incomplete sentences: 'In this this section', 'ρo' for ρ0 in Eqs. (10)–(11), the incomplete sentence 'If ∥(˜λ, ˜µ, ˜ρ)∥, then ...' in the proof of Proposition 1, 'subection', and 'obove'. These should be corrected.
[Section 5, Eq. (19)] In Eq. (19), the residual ~R is first defined with independent coefficients αk, βk, γk but the minimization is written over ζ with arguments (ζ, τ1ζ, τ2ζ); the notation should be made consistent.
[Section 3.3 and Theorem 2] The description of noisy data is inconsistent: Theorem 2 refers to a noisy operator Λδ(λ,µ,ρ) with noise level δ, while Section 3.3 defines Λδ as the noisy difference of the two Neumann-to-Dirichlet operators with a relative noise matrix E. The relationship between these two noise models should be clarified.
[Section 2.1] The statement of Lemma 1 is not used later, and Definition 2(c) contains the awkward phrase 'out ∂Ω supp(φ)'; the notation should be simplified or the lemma omitted if it is not needed.

Circularity Check

0 steps flagged · score 2.0 of 10

No derivation reduces to its own inputs; the numerical reconstructions are benchmarked against synthetic ground truth and the TSVD regularization is generic, so the core claim has independent content.

full rationale

The central reconstruction chain is not circular: data U is assembled from boundary measurements and reference solutions, the sensitivity matrices T_k are computed from the forward problem, and the output (ζ or, in the linearized system, (α,β,γ)) is compared with synthetic ground truth rather than being inserted as an input. The TSVD truncation is a generic regularizer, and the monotonicity constraints enter through β_k bounds derived from U, but the recovered field still minimizes a data-misfit functional and is not equal to the data by construction. The paper's reliance on external results is real but not self-referential: Lemma 2 is cited from [24], a companion paper by the same authors, and Theorems 1-2 are delegated to "arguments analogous" to [10] (Eberle-Harrach, independent work); these are omitted proofs and a completeness burden, not re-importations of the target conclusion. The fixed-ratio ansatz in Eqs. (17)-(20), with μ = τ1ζ and ρ = τ2ζ, forces the three reconstructed supports to coincide, so the abstract's and Section 5.1's disjoint-support claim is not a consequence of the stated minimization; this is a correctness/completeness gap in the presented algorithm, not a circularity in which the output is equivalent to the input. Therefore no concrete circular step can be exhibited, and the modest score reflects only the borrowed-support burden.

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

The central reconstruction depends on several unproved or borrowed ingredients: the differentiability of the NtD map, the monotonicity inclusions (from the authors' companion preprint [24] and from [10]), the box-constraint formulas, and the piecewise constant ansatz. The new contribution (TSVD truncation) is a regularization choice with a hand-set threshold rather than a derived identity.

free parameters (5)

test constants Cλ, Cμ, Cρ = 0.5, 0.5, 0.5
Chosen a priori in Sections 3.3 and 4.2; must satisfy Cλ ≤ λ0/λ1 (λ1−λ0), etc., but no sensitivity analysis is given.
TSVD energy threshold τ = 0.99
Set to 0.99 in Section 5.1; the truncation cutoff is a design choice that affects the reconstruction.
noise level δ = 0.001 (Section 3.3), 0.1 (Sections 4.3, 5.1)
The method needs a noise level to define the regularization term δI and the feasible sets; the paper assumes it is known.
number of test balls and measurements = 100 test balls, m=19
Computational parameters chosen for the experiments; no study of their influence.
a priori parameter bounds λmin, λmax, μmin, μmax, ρmin, ρmax = not specified
Used to define amax, bmax, cmax and the admissible set C; assumed known but values are not stated.

assumptions (5)

standard math Existence and uniqueness of the direct problem by Lax-Milgram for λ, μ, ρ ∈ L∞+
Section 2, weak formulation (3).
domain assumption Fréchet differentiability of the Neumann-to-Dirichlet operator with derivative formula (5)
Proposition 1 attempts a proof, but the minimax saddle-point argument contains mathematical errors; the result is standard but not rigorously established here.
domain assumption Monotonicity lemma (Lemma 2) and the three-parameter shape characterization (Theorems 1-2)
Lemma 2 is cited from the authors' companion paper [24]; Theorems 1-2 are stated with proofs deferred 'by analogy' to [10]. The three-parameter extension is not verified.
domain assumption Formula for βk and βδk via the most negative eigenvalue of the Cholesky-scaled matrix (Section 4.2)
Taken from [10] without proof; used to construct the admissible set C.
domain assumption Piecewise constant ansatz on L disjoint pixels Bk and known background/interior parameter values
Section 4.1; the inversion is restricted to this parameterization.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Support identification for parameter variations in a PDE system via regularized methods." pith.science (2026). https://pith.science/paper/EDTRFBFC

@misc{pith2026250521663,
  author       = {Pith},
  title        = {Pith review of: Support identification for parameter variations in a PDE system via regularized methods},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EDTRFBFC}},
  note         = {Machine review of arXiv:2505.21663}
}

read the original abstract

We study the inverse problem of recovering the spatial support of parameter variations in a system of partial differential equations (PDEs) from boundary measurements. A reconstruction method is developed based on the monotonicity properties of the Neumann-to-Dirichlet operator, which provides a theoretical foundation for stable support identification. To improve reconstruction accuracy, particularly when parameters have disjoint supports, we propose a combined regularization approach integrating monotonicity principles with Truncated Singular Value Decomposition (TSVD) regularization. This hybrid strategy enhances robustness against noise and ensures sharper support localization. Numerical experiments demonstrate the effectiveness of the proposed method, confirming its applicability in practical scenarios with varying parameter configurations.

Figures

Figures reproduced from arXiv: 2505.21663 by the authors.

**Figure 1.** Reconstruction of the shape D using Algorithm 1. The first row presents the reconstruction from noise-free data, while the second row shows the reconstruction from noisy data with a noise level of δ = 0.001 . 4 Reconstruction subject to monotonicity constraints The numerical experiments based on the linearized monotonicity test exhibit significant sensitivity to noise. To enhance the robustness of the reconstruction… view at source ↗

**Figure 2.** illustrates the reconstruction of the common support of the parameter variations using monotonicity constraints. The first row shows the reconstruction obtained from noise-free data, while the second row displays the results with a noise level of 10%. As observed, the reconstruction is more accurate when the support consists of a single connected subdomain, compared to the case where it is composed of two disjoint … view at source ↗

**Figure 3.** Reconstruction of the truncation point (highlighted in red): on the left, the [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Reconstruction of the shape D using combined monotonicity and TSVD regularization. The first row presents the reconstruction from noise-free data, while the second row shows the reconstruction from noisy data with a noise level of δ = 0.1 . 16 [PITH_FULL_IMAGE:figure…

**Figure 5.** Figure 5: The first column displays, from top to bottom, the reconstructions of supp [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages

[10]

Monotonicity-based regularization for shape reconstruction in linear elasticity

Sarah Eberle and Bastian Harrach. Monotonicity-based regularization for shape reconstruction in linear elasticity. Computational mechanics, 69(5):1069–1086, 2022

work page 2022
[24]

Stability analysis of an inverse coefficients problem in a system of partial differential equations

Houcine Meftahi and Chayma Nssibi. Stability analysis of an inverse coefficients prob- lem in a system of partial differential equations. arXiv preprint arXiv:2505.05116 , 2025

work page Pith review arXiv 2025
[1]

Topology opti- mization method with respect to the insertion of small coated inclusion

Zakaria Belhachmi, A Ben Abda, B Meftahi, and Houcine Meftahi. Topology opti- mization method with respect to the insertion of small coated inclusion. Asymptotic Analysis, 106(2):99–119, 2018

work page 2018
[2]

Level set-based shape op- timization approach for the inverse optical tomography problem

Zakaria Belhachmi, Rabeb Dhif, and Houcine Meftahi. Level set-based shape op- timization approach for the inverse optical tomography problem. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift f¨ ur Angewandte Mathematik und Mechanik, 103(3):e202200156, 2023

work page 2023
[3]

Shape sensitivity analysis for an inter- face problem via minimax differentiability

Zakaria Belhachmi and Houcine Meftahi. Shape sensitivity analysis for an inter- face problem via minimax differentiability. Applied Mathematics and Computation , 219(12):6828–6842, 2013

work page 2013
[4]

Fast methods for shape reconstruction in electrical resistance tomography

Flavio Calvano, Guglielmo Rubinacci, and Antonello Tamburrino. Fast methods for shape reconstruction in electrical resistance tomography. NDT and & E Interna- tional, 46:32–40, 2012

work page 2012
[5]

Topological and shape gradient strategy for solving geometrical inverse problems

S Chaabane, Mohamed Masmoudi, and Houcine Meftahi. Topological and shape gradient strategy for solving geometrical inverse problems. Journal of Mathematical Analysis and applications , 400(2):724–742, 2013

work page 2013
[6]

Directional derivative of a minimax function

Rafael Correa and Alberto Seeger. Directional derivative of a minimax function. Nonlinear Anal., 9(1):13–22, 1985

work page 1985

Show all 28 references

[7]

M. C. Delfour and J.-P. Zol´ esio. Shape sensitivity analysis via min max differentia- bility. SIAM J. Control Optim. , 26(4):834–862, 1988

1988
[8]

M. C. Delfour and J.-P. Zol´ esio. Shapes and geometries , volume 22 of Advances in Design and Control . Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, second edition, 2011. Metrics, analysis, differential calculus, and optimization

2011
[9]

Shape reconstruction in linear elasticity: standard and linearized monotonicity method

Sarah Eberle and Bastian Harrach. Shape reconstruction in linear elasticity: standard and linearized monotonicity method. Inverse Problems, 37(4):045006, 2021

2021
[11]

Resolution guarantees for the reconstruc- tion of inclusions in linear elasticity based on monotonicity methods

Sarah Eberle-Blick and Bastian Harrach. Resolution guarantees for the reconstruc- tion of inclusions in linear elasticity based on monotonicity methods. Inverse Prob- lems, 39(7):075006, 2023

2023
[12]

Analyse convexe et problemes variationnels

Ivar Ekeland and Roger Temam. Analyse convexe et problemes variationnels. (No Title), 1974

1974
[13]

Piecewise nonlinear materials and monotonicity principle

Antonio Corbo Esposito, Luisa Faella, Vincenzo Mottola, Gianpaolo Piscitelli, Ravi Prakash, and Antonello Tamburrino. Piecewise nonlinear materials and monotonicity principle. Inverse Problems, 40(8):085001, 2024

2024
[14]

Simplified reconstruction of layered materials in eit

Henrik Garde. Simplified reconstruction of layered materials in eit. Applied Mathe- matics Letters, 126:107815, 2022. 19

2022
[15]

Reconstruction of singular and degenerate inclu- sions in calder´ on’s problem.Inverse Problems and Imaging , 16(5):1219–1227, 2022

Henrik Garde and Nuutti Hyv¨ onen. Reconstruction of singular and degenerate inclu- sions in calder´ on’s problem.Inverse Problems and Imaging , 16(5):1219–1227, 2022

2022
[16]

Inverse medium scattering for a nonlinear helmholtz equation

Roland Griesmaier, Marvin Kn¨ oller, and Rainer Mandel. Inverse medium scattering for a nonlinear helmholtz equation. Journal of Mathematical Analysis and Applica- tions, 515(1):126356, 2022

2022
[17]

Monotonicity-based shape reconstruction in electrical impedance tomography

Bastian Harrach and Marcel Ullrich. Monotonicity-based shape reconstruction in electrical impedance tomography. SIAM Journal on Mathematical Analysis , 45(6):3382–3403, 2013

2013
[18]

Convexification with the viscocity term for electrical impedance tomography

Michael V Klibanov, Jingzhi Li, and Zhipeng Yang. Convexification with the viscocity term for electrical impedance tomography. arXiv preprint arXiv:2503.07916 , 2025

2025 arXiv
[19]

Shape and parameter reconstruction for the robin transmission inverse problem

Antoine Laurain and Houcine Meftahi. Shape and parameter reconstruction for the robin transmission inverse problem. Journal of Inverse and Ill-posed Problems , 24(6):643–662, 2016

2016
[20]

C.-L. Lin, G. Nakamura, Gunther Uhlmann, and J.-N. Wang. Quantitative strong unique continuation for the lam´ e system with lipschitz coefficients. Methods and Applications of Analysis , 13(2):183–198, 2006

2006
[21]

Monotonicity-based inversion of fractional semilinear elliptic equa- tions with power type nonlinearities

Yi-Hsuan Lin. Monotonicity-based inversion of fractional semilinear elliptic equa- tions with power type nonlinearities. Calculus of Variations and Partial Differential Equations, 61(5):188, 2022

2022
[22]

Sensitivity analysis for some inverse problems in lin- ear elasticity via minimax differentiability

H Meftahi and J-P Zol´ esio. Sensitivity analysis for some inverse problems in lin- ear elasticity via minimax differentiability. Applied Mathematical Modelling , 39(5- 6):1554–1576, 2015

2015
[23]

´Etudes th´ eoriques et num´ eriques de quelques probl` emes inverses

Houcine Meftahi. ´Etudes th´ eoriques et num´ eriques de quelques probl` emes inverses. PhD thesis, Lille 1, 2009

2009
[25]

Imaging of nonlinear materials via the monotonicity principle

Vincenzo Mottola, Antonio Corbo Esposito, Gianpaolo Piscitelli, and Antonello Tam- burrino. Imaging of nonlinear materials via the monotonicity principle. Inverse Problems, 40(3):035007, 2024

2024
[26]

Tamburrino

A. Tamburrino. Monotonicity based imaging methods for elliptic and parabolic in- verse problems. Journal of Numerical Mathematics , 14(6):633–642, 2006

2006
[27]

A new non-iterative inversion method for electrical resistance tomography

Antonello Tamburrino and Guglielmo Rubinacci. A new non-iterative inversion method for electrical resistance tomography. Inverse Problems, 18(6):1809, 2002

2002
[28]

Strong unique continuation for the lam´ e system with lipschitz coefficients in three dimensions

Hang Yu. Strong unique continuation for the lam´ e system with lipschitz coefficients in three dimensions. ESAIM: Control, Optimisation and Calculus of Variations , 17(3):761–770, 2011. 20

2011

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [10]

Monotonicity-based regularization for shape reconstruction in linear elasticity

Sarah Eberle and Bastian Harrach. Monotonicity-based regularization for shape reconstruction in linear elasticity. Computational mechanics, 69(5):1069–1086, 2022

work page 2022

[2] [24]

Stability analysis of an inverse coefficients problem in a system of partial differential equations

Houcine Meftahi and Chayma Nssibi. Stability analysis of an inverse coefficients prob- lem in a system of partial differential equations. arXiv preprint arXiv:2505.05116 , 2025

work page Pith review arXiv 2025

[3] [1]

Topology opti- mization method with respect to the insertion of small coated inclusion

Zakaria Belhachmi, A Ben Abda, B Meftahi, and Houcine Meftahi. Topology opti- mization method with respect to the insertion of small coated inclusion. Asymptotic Analysis, 106(2):99–119, 2018

work page 2018

[4] [2]

Level set-based shape op- timization approach for the inverse optical tomography problem

Zakaria Belhachmi, Rabeb Dhif, and Houcine Meftahi. Level set-based shape op- timization approach for the inverse optical tomography problem. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift f¨ ur Angewandte Mathematik und Mechanik, 103(3):e202200156, 2023

work page 2023

[5] [3]

Shape sensitivity analysis for an inter- face problem via minimax differentiability

Zakaria Belhachmi and Houcine Meftahi. Shape sensitivity analysis for an inter- face problem via minimax differentiability. Applied Mathematics and Computation , 219(12):6828–6842, 2013

work page 2013

[6] [4]

Fast methods for shape reconstruction in electrical resistance tomography

Flavio Calvano, Guglielmo Rubinacci, and Antonello Tamburrino. Fast methods for shape reconstruction in electrical resistance tomography. NDT and & E Interna- tional, 46:32–40, 2012

work page 2012

[7] [5]

Topological and shape gradient strategy for solving geometrical inverse problems

S Chaabane, Mohamed Masmoudi, and Houcine Meftahi. Topological and shape gradient strategy for solving geometrical inverse problems. Journal of Mathematical Analysis and applications , 400(2):724–742, 2013

work page 2013

[8] [6]

Directional derivative of a minimax function

Rafael Correa and Alberto Seeger. Directional derivative of a minimax function. Nonlinear Anal., 9(1):13–22, 1985

work page 1985