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

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Pith's one-line read For parameters in a finite-dimensional class, the boundary force-to-displacement map determines the density—and, under an ordering assumption, the Lamé parameters—so that parameter error is bounded by a constant times the boundary-map…

desk verdict New Lipschitz stability results for elasticity via monotonicity and localized potentials, but Theorem 2's proof assumes a sign-definiteness property that its L∞ assumption does not deliver. read the letter →

arxiv 2505.05116 v1 pith:QT6KPJHY submitted 2025-05-08 math.OC

Houcine Meftahi , Chayma Nssibi This is my paper · ORCID
classification math.OC MSC 35R3074B0535J47
keywords linearelasticityinversecoefficientsproblemNeumann-to-DirichletmapLipschitzstabilitymonotonicitylocalizedpotentialsLaméparametersdensityrecovery
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 studies the inverse problem of recovering the density $\rho$, or simultaneously the Lamé parameters $\lambda,\mu$ and density $\rho$, of an elastic body from its Neumann-to-Dirichlet map—the boundary operator mapping an applied surface force to the resulting boundary displacement. The central claim is that, inside an a-priori known finite-dimensional class of parameters with fixed upper and lower bounds, the boundary map determines the parameters with Lipschitz stability: the parameter error is bounded by a constant times the operator-norm error of the measured boundary map. For the density-only problem with piecewise constant parameters, the authors give a constructive version whose stability constant $\alpha$ is explicit and whose boundary loads are produced by finitely many well-posed forward solves. If the estimates are correct, small measurement noise in elastography and non-destructive testing leads to errors in recovered tissue stiffness and density that grow only linearly, which is the property needed for stable numerical reconstruction.

What carries the argument

The load-bearing object is the Neumann-to-Dirichlet operator $\Lambda_{\lambda,\mu,\rho}$, which maps a boundary load $g$ to the boundary displacement $u|_{\Gamma_N}$. The argument turns on two tools. First, the monotonicity relations of Lemma 1 and Lemma 6 bracket the quadratic-form difference of two boundary maps between integrals of the coefficient differences against the energies of the two corresponding solutions. Second, the localized-potentials theorems (Theorem 1 and Theorem 4) supply boundary loads whose solutions concentrate energy in a chosen open set $D_1$ while making it vanish on a disjoint set $D_2$. Combining the monotonicity bracket with localized potentials, and using compactness of the finite-dimensional parameter set, produces a uniform positive lower bound for the ratio of boundary-map difference to parameter difference—which is precisely Lipschitz stability.

What would settle it

Let $\Omega$ be the unit square and let $E$ be the one-dimensional subspace spanned by a bounded function $\varphi$ that takes the value $+1$ on a dense set and $-1$ on its dense complement. For $\zeta=\varphi$, the proof of inequality (18) requires an open set $D_1$ with $|\zeta|\ge\beta>0$, which does not exist; computing the infimum in Lemma 4 for this $E$—analytically or by high-resolution approximation—decides whether the positivity of the lower bound survives without the sign-definite-region condition.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that the severe ill-posedness of this inverse elasticity problem can be tamed by imposing finite-dimensional structure. Theorem 2 states a nonconstructive Lipschitz stability estimate $\|\rho_1-\rho_2\|_{L^\infty(\Omega)} \le C\|\Lambda(\rho_1)-\Lambda(\rho_2)\|_\ast$ for densities in a finite-dimensional subspace with uniform positive bounds. Theorem 3 makes this constructive for piecewise constant densities: $\|\rho_1-\rho_2\|_\infty \le \alpha\|\Lambda(\rho_1)-\Lambda(\rho_2)\|_\ast$ with an explicit $\alpha$ computed from finitely many boundary loads. Theorem 5 extends the Lipschitz bound to simultaneous recovery, $\|(\lambda_1-\lambda_2,\mu_1-\mu_2,\rho_1-\rho_2)\|_\Delta \le C\|\Lambda_{\lambda_1,\mu_1,\rho_1}-\Lambda_{\lambda_2,\mu_2,\rho_2}\|_\ast$, provided the two parameter triples are ordered by coordinatewise inequalities. If true, these results convert the qualitative statement 'the problem is ill-posed' into a quantitative one: within the admissible class, boundary-measurement error controls parameter error linearly.

Load-bearing premise

The proof needs every admissible normalized parameter difference to be bounded away from zero with a fixed sign on some open subset of the domain, and it needs the complement of the localization sets to be connected—conditions that are not implied by the stated finite-dimensional-subspace hypothesis.

Editorial extensions

If this is right

  • Equal Neumann-to-Dirichlet maps force equal parameters inside the admissible class: uniqueness follows directly from the Lipschitz estimates, as the paper notes in Remark 1 and as the ordered-pair formulation of Theorem 5 implies.
  • Stability estimates of this form transfer to numerical inversion: the error in a recovered density or Lamé pair is controlled by the error in the measured boundary map, so iterative reconstruction algorithms inherit convergence rates from the stability constant.
  • The constructive estimate for piecewise constant densities supplies an explicit constant and a recipe for boundary loads from finitely many well-posed PDE solves, making the stability bound usable rather than purely existential.
  • Because the bound is Lipschitz, finite-dimensional approximations of the Neumann-to-Dirichlet operator can be plugged into the estimate, so errors made in approximating the operator by a finite amount of data propagate linearly into parameter errors.

Reading between the lines

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

  • The proof of Theorem 2 requires, although the statement does not say so, that every normalized parameter difference be bounded away from zero with a fixed sign on some open subset of the domain; this suggests the finite-dimensional theorem is really guaranteed for piecewise-constant or otherwise locally sign-definite parameter families. This is our inference from the proof strategy, not an explici
  • A natural testable strengthening would be to quantify the size of the sign-definite regions—their measure or diameter—and to track how the stability constant depends on it, potentially giving dimension-dependent constants for adaptive partitions.
  • The monotonicity-plus-localized-potentials template is not specific to isotropic linear elasticity; it should produce analogous Lipschitz stability estimates for other elliptic systems with monotone coefficient dependence, such as anisotropic elasticity or poroelastic models, whenever a Runge approximation and unique continuation are available.
  • The coordinatewise ordering condition in Theorem 5 means that simultaneous recovery is proved stable only along monotone parameter paths; outside those paths, instabilities of the kind already known for general Schrödinger-type inverse problems may persist.
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Referee Report

4 major / 6 minor

Summary. The paper studies the stable recovery of the Lamé parameters (λ, μ) and density ρ in the isotropic linear elasticity system (1) from the Neumann-to-Dirichlet map. After proving a monotonicity relation between the material parameters and the boundary map, the authors invoke localized-potential arguments to derive: (i) a Lipschitz stability estimate for ρ when λ, μ are known and ρ lies in a finite-dimensional subspace of L∞(Ω) with two-sided bounds (Theorem 2); (ii) a constructive version for piecewise-constant densities supported in a fixed partition (Theorem 3); and (iii) a simultaneous Lipschitz stability estimate for (λ, μ, ρ) under a definiteness/monotonicity condition and a finite-dimensional parametrization (Theorem 5). The proofs are built from variational identities and unique-continuation-based localized potentials, in the spirit of recent monotonicity methods for inverse coefficient problems.

Significance. The paper targets a genuinely ill-posed multiparameter inverse problem, and a quantitative Lipschitz estimate of the form ‖ρ1−ρ2‖∞ ≤ C‖Λ(ρ1)−Λ(ρ2)‖∗, or the simultaneous version (39), would be a useful addition to the elasticity imaging literature. The monotonicity identities (5) and (27) are derived carefully from the variational form, and the constructive character of Theorem 3, with an explicit stability constant in (24), is attractive for numerical applications. However, the main theorems as stated are not established: the positivity of the infimum — the load-bearing step in both stability proofs — rests on a hidden sign-definiteness-open-region property that is false for general finite-dimensional L∞ subspaces, and the localized-potential lemmas carry unstated connectivity and regularity assumptions. These are correctness issues rather than presentation issues, and they affect the central claims. The paper is therefore not acceptable in its current form, but the gaps appear repairable by adding explicit structural hypotheses.

major comments (4)
  1. [Section 3.1.3, proof of (18)] The proof of Theorem 2 asserts that for every ζ∈K there exists a non-empty open set D1⊂Ω and 0<β<1 such that either ζ|D1≥β or −ζ|D1≥β, and this assertion is the only mechanism producing the uniform lower bound in Lemma 4. The assertion is not implied by the standing assumption that E is a finite-dimensional subspace of L∞(Ω). For instance, if A⊂Ω is measurable with A and Ω\A both of positive measure and empty interior and E=span{1, 1_A−1_{Ω\A}}, then the normalized element f=1_A−1_{Ω\A} has no non-empty open set on which it is sign-definite with a positive lower bound. Theorem 2 is therefore unproved as stated; it needs an explicit assumption such as piecewise constancy on a fixed open partition, continuity, or an abstract open-sign-definiteness condition on every element of the unit sphere of E.
  2. [Lemmas 2 and 8 (Sections 3.1.2 and 4.1)] Both localized-potential results rely on hidden geometric hypotheses. In Lemma 2, D is only assumed to be a subset of Ω of positive measure, yet the proof uses traces on ∂D and concludes v|Ω\D=0 from unique continuation 'since Ω\D is connected'; neither the connectivity nor the regularity of D that makes the trace statement meaningful is stated. In Lemma 8, the assertions that R(Tj) and R(Zj) are dense and intersect only trivially are justified in one sentence by unique continuation, with no stated condition on the geometry of D1, D2 or Ω\(D1∪D2). Because Theorems 1 and 4 are direct consequences of these lemmas, the existence of the localized-potentials sequences used throughout the paper is not fully justified as written.
  3. [Section 4.2.1, proof of (44)] The proof of Theorem 5 repeats the same hidden sign-definiteness assumption in the multiparameter setting. For (ζ1,ζ2,ζ3)∈E+ it asserts the existence of a non-empty open D1 and δ>0 such that one component is bounded below by δ on D1 while the other two are nonnegative everywhere. This does not follow from ζi∈span(P), ζi≥0, and ‖(ζ1,ζ2,ζ3)‖Δ=1: a nonnegative L∞ function of unit norm may be supported only on a dense set with empty interior. Consequently the positivity of the infimum in Lemma 9, and with it the stability estimate (39), is not established without adding a structural hypothesis to P.
  4. [Section 3.1.4, Lemma 5 and Theorem 3] The constructive result also has missing justifications. Lemma 5(i) states that the boundary data g(j,k) exist by Theorem 1, but Theorem 1 requires D1,D2 open, disjoint, with Ω\(D1∪D2) connected and meeting Γ_N; for D1=Sj and D2=S\Sj these conditions are not verified from the stated assumptions on S and Sj. In part (ii), inequality (21) is stated only for δ∈L∞_+(S), but it is applied to a sign-changing δ (positive on Sj, negative on S\Sj); the proof chain also refers to the undefined solution u_{η(j,k)}. In part (iii), the displayed convergence uses (5b/(3a)−3/2) while the target (20) has (5b/(2a)−3/2). These points affect the validity of the explicit stability constant α in (24).
minor comments (6)
  1. [Abstract and Section 2] There are numerous typos: 'constrcutive' and 'simultameousely' in the abstract, 'dispslacement' and 'symetric' in Section 2, and inconsistent hyphenation throughout; a careful proofreading pass is needed.
  2. [Lemma 5, inequality (21)] Inequality (21) omits the squares inside the integrals: it should read ∫_S δ|u_ρ|² dx ≥ ∫_S δ|u_{ρ+δ}|² dx, and the restriction δ∈L∞_+(S) is unnecessarily strong; Lemma 1 gives the inequality for any δ with ρ+δ∈L∞_+(S).
  3. [Lemma 3] In the sentence following (17), 'The second argument of the function h' refers to a function h that has not been defined; the intended object is J.
  4. [Lemma 9 and proof of Theorem 5] In the line after (43), Φ is called with arguments (g, Θ1, Θ2, Θ3, (λ1,μ1,ρ1), (λ2,μ2,ρ2)), which does not match the definition of Φ in (40).
  5. [Section 4.2] The sentence introducing the proof of Theorem 5 says 'monotonicity relations in Lemma 1' but the relevant statement is Lemma 6.
  6. [References] References [33] and [37] are the same work (Eberle, Harrach, Meftahi, Rezgui, Lipschitz stability estimate and reconstruction of Lamé parameters in linear elasticity), listed with different publication years and page ranges; one duplicate should be removed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the stability estimates are derived from monotonicity and localized potentials, with self-citations used only for technique.

full rationale

The derivation chain is self-contained against the stated benchmarks. Lemma 1 and Lemma 6 prove the monotonicity relations (5) and (27) directly by subtracting the variational formulations, so the central inequalities are not assumed. The positivity of the infimum that yields the stability constant is not imported from the authors' earlier papers: it is proved by constructing localized potentials, with the Runge approximation in Lemma 2 justified from the external unique continuation result [42], and the range non-inclusions in Lemma 8 justified by [43]; the compactness/lower-semicontinuity arguments are reproduced in Lemma 4 and in the proof of Theorem 5. The self-citations [9,30,32,33] are employed as methodological pointers (e.g., 'Following the ideas of [8,9]' and 'we use a similar approach as in the construction of localized potentials in [9]'), but the required estimates are re-derived in the manuscript rather than assumed as black boxes. No parameter is fitted to boundary data and later renamed a prediction: the constants C and alpha are either existential via compactness or explicit in terms of finitely many constructed boundary data g(j,k). The reader's identified gap, namely the assumption that every normalized finite-dimensional difference has an open sign-definite region, is a correctness/regularity issue in the proof, not a circular reduction; it does not affect the circularity score.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The arguments rest on standard functional analysis (Lax-Milgram, CGNE convergence, lower semicontinuity) plus two domain-specific tools: the unique continuation theorem of [42] and the finite-dimensional parameter class. The load-bearing ad hoc ingredients are the hidden regularity of the parameter subspace (open sign-definite regions) and the geometric connectivity of the complement of the localization sets; both are used without being stated as assumptions on the parameter class.

assumptions (6)
  • domain assumption Hypothesis 1: μ ∈ C^{0,1}(Ω), λ, ρ ∈ L∞_+(Ω) with uniform lower bound δ0 on μ and λ+2μ, and uniform upper bound M0.
    Ensures well-posedness of the forward problem and the applicability of the quantitative unique continuation property from [42], used in Lemmas 2 and 8.
  • standard math Quantitative unique continuation for the Lamé system with C^{0,1} coefficients (Theorem in [42]).
    Used in the proofs of Lemma 2 (Runge approximation injectivity) and Lemma 8 (dense ranges, disjoint ranges) to propagate zero Cauchy data through connected gaps.
  • domain assumption The finite-dimensional subspace E (resp. P) and the bounds a, b (resp. a, b, c, d, e, f) are known a priori.
    The stability constants C and α depend on these; without this, the problem is known to lack Lipschitz stability [31].
  • ad hoc to paper Every ζ ∈ K has an open region where it is sign-definite and bounded away from zero.
    Required in the proof of Theorem 2 (Section 3.1.3) but not implied by E ⊂ L∞; the paper does not flag this loss of generality.
  • ad hoc to paper Geometric connectivity: Ω ∖ (D1 ∪ D2) is connected and meets Γ_N for the chosen D1, D2 (used in Theorem 1 and the proof of (18)).
    Needed for the unique continuation step in the localized potentials construction; not verified for the D1 arising from arbitrary ζ in Theorem 2.
  • domain assumption Monotonicity condition in Theorem 5: parameters are assumed ordered (λ1 ≤ λ2, μ1 ≤ μ2, ρ1 ≤ ρ2 or reversed).
    This is an explicit hypothesis of simultaneous recovery; without it, the Lipschitz estimate is not claimed.

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Pith. "Pith review of Stability analysis of an inverse coefficients problem in a system of partial differential equations." pith.science (2026). https://pith.science/paper/QT6KPJHY

@misc{pith2026250505116,
  author       = {Pith},
  title        = {Pith review of: Stability analysis of an inverse coefficients problem in a system of partial differential equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QT6KPJHY}},
  note         = {Machine review of arXiv:2505.05116}
}
abstract

In this study, we address the inverse problem of recovering the Lam\'e parameters ($\lambda, \mu$) and the density $\rho$ of a medium from the Neumann-to-Dirichlet map for any dimension $d\geq 2$. This inverse problem finds its motivation in the reconstruction of mechanical properties of tissues in medical diagnostics. We first assume that the Lam\'e parameters ($\lambda, \mu$) are know and we look for the inverse problem of recovering the density $\rho$. In this context, we derive a constrcutive Lipschitz stability estimate in terms of the Neumann to Dirichlet map in the case of piecewise constant parameters. Then, we look for the inverse problem of recovering $\lambda$, $\mu$ and $\rho$ simultameousely. We establish Lipschitz stability estimate, provided that the parameters $\lambda$, $\mu$ and $\rho$ have upper and lower bounds and belong to a known finite-dimensional subspace. The proofs hinge on monotonicity relations between the parameters and the Neumann-to-Dirichlet operator, coupled with the techniques of localized potentials.

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Reference graph

Works this paper leans on

43 extracted references · 43 canonical work pages · cited by 1 Pith paper

  1. [1]

    Mechanical models of pat tern and form in bio- logical tissues: The role of stress–strain constitutive equations

    Villa C, Chaplain MA, Gerisch A, Lorenzi T. Mechanical models of pat tern and form in bio- logical tissues: The role of stress–strain constitutive equations. Bulletin of Mathematical Biology. 2021;83(7):80

    work page 2021

  2. [2]

    Quantitative Seismology: Theory and Method s; 1980

    Aki K, Richards PG. Quantitative Seismology: Theory and Method s; 1980. Available from: https:// api.semanticscholar.org/CorpusID:58794764

    work page 1980

  3. [3]

    Ultrasound elastogra phy: principles and techniques

    Gennisson JL, Deffieux T, Fink M, Tanter M. Ultrasound elastogra phy: principles and techniques. Diagnostic and interventional imaging. 2013;94(5):487–495

    work page 2013

  4. [4]

    Imaging the elastic properties of tissue: the 20 year perspective

    Parker KJ, Doyley MM, Rubens DJ. Imaging the elastic properties of tissue: the 20 year perspective. Physics in medicine & biology. 2010;56(1):R1

    work page 2010

  5. [5]

    Model-based elastography: a survey of approaches to the inverse elasticity problem

    Doyley MM. Model-based elastography: a survey of approaches to the inverse elasticity problem. Physics in Medicine & Biology. 2012;57(3):R35

    work page 2012

  6. [6]

    Application of topological sensitivity t oward tissue elasticity imaging using magnetic resonance data

    Yuan H, Guzina BB, Sinkus R. Application of topological sensitivity t oward tissue elasticity imaging using magnetic resonance data. Journal of Engineering Mechanics . 2014;140(3):443–453

    work page 2014

  7. [7]

    Stable determination of conductivity by boundar y measurements

    Alessandrini G. Stable determination of conductivity by boundar y measurements. Applicable Analysis. 1988;27(1-3):153–172

    work page 1988

  8. [8]

    Uniqueness and Lipschitz stability in electrical impedan ce tomography with finitely many electrodes

    Harrach B. Uniqueness and Lipschitz stability in electrical impedan ce tomography with finitely many electrodes. Inverse problems. 2019;35(2):024005

    work page 2019

Show all 43 references
  1. [9]

    Global uniqueness and Lipschitz stability f or the inverse Robin transmission problem

    Harrach B, Meftahi H. Global uniqueness and Lipschitz stability f or the inverse Robin transmission problem. SIAM Journal on Applied Mathematics. 2019;79(2):525–55 0

    2019

  2. [10]

    Inversion formulas for the linearized problem for an inverse boundary value problem in elastic prospection

    Ikehata M. Inversion formulas for the linearized problem for an inverse boundary value problem in elastic prospection. SIAM Journal on Applied Mathematics. 1990;50 (6):1635–1644

    1990

  3. [11]

    Global uniqueness in inverse bounda ry value problems for the Navier– Stokes equations and Lam´ e system in two dimensions

    Imanuvilov OY, Yamamoto M. Global uniqueness in inverse bounda ry value problems for the Navier– Stokes equations and Lam´ e system in two dimensions. Inverse Pro blems. 2015;31(3):035004

    2015

  4. [12]

    Global uniqueness for an inverse bound ary problem arising in elasticity

    Nakamura G, Uhlmann G. Global uniqueness for an inverse bound ary problem arising in elasticity. Inventiones mathematicae. 1994;118(1):457–474

    1994

  5. [13]

    Uniqueness and Lipschitz stability for the identification of Lam´ e parameters from boundary measurements

    Elena Beretta SV Elisa Francini. Uniqueness and Lipschitz stability for the identification of Lam´ e parameters from boundary measurements. Inverse Problems & I maging. 2014;8(3):611–644

    2014

  6. [14]

    Identification of Lam´ e coefficients from boundary observations

    Akamatsu M, Nakamura G, Steinberg S. Identification of Lam´ e coefficients from boundary observations. Inverse Problems. 1991;7(3):335

    1991

  7. [15]

    Inverse problems at the boundary for an elastic medium

    Nakamura G, Uhlmann G. Inverse problems at the boundary for an elastic medium. SIAM journal on mathematical analysis. 1995;26(2):263–279. 21

    1995

  8. [16]

    On the inverse boundary value problem for line ar isotropic elasticity

    Eskin G, Ralston J. On the inverse boundary value problem for line ar isotropic elasticity. Inverse Problems. 2002;18(3):907

    2002

  9. [17]

    Identification of Lam´ e parameters by boundary measurements

    Nakamura G, Uhlmann G. Identification of Lam´ e parameters by boundary measurements. American Journal of Mathematics. 1993;p. 1161–1187

    1993

  10. [18]

    Global uniqueness for an inverse bound ary value problem arising in elasticity

    Nakamura G, Uhlmann G. Global uniqueness for an inverse bound ary value problem arising in elasticity. Inventiones mathematicae. 2003;152(1):205–207

    2003

  11. [19]

    Lipschitz stability for the electrical impe dance tomography problem: the complex case

    Beretta E, Francini E. Lipschitz stability for the electrical impe dance tomography problem: the complex case. Communications in Partial Differential Equations. 201 1;36(10):1723–1749

  12. [20]

    Lipschitz stability for the inverse con ductivity problem

    Alessandrini G, Vessella S. Lipschitz stability for the inverse con ductivity problem. Advances in Applied Mathematics. 2005;35(2):207–241

    2005

  13. [21]

    Lipschitz stability for the inverse conduct ivity problem for a conformal class of anisotropic conductivities

    Gaburro R, Sincich E. Lipschitz stability for the inverse conduct ivity problem for a conformal class of anisotropic conductivities. Inverse Problems. 2015;31(1):0150 08

    2015

  14. [22]

    Lipschitz stab ility for a piecewise linear Schr¨ odinger potential from local Cauchy data

    Alessandrini G, de Hoop MV, Gaburro R, Sincich E. Lipschitz stab ility for a piecewise linear Schr¨ odinger potential from local Cauchy data. Asymptotic Analy sis. 2018;108(3):115–149

    2018

  15. [23]

    Lipschitz stability of an inverse bou ndary value problem for a Schr¨ odinger-type equation

    Beretta E, De Hoop MV, Qiu L. Lipschitz stability of an inverse bou ndary value problem for a Schr¨ odinger-type equation. SIAM Journal on Mathematical Ana lysis. 2013;45(2):679–699

    2013

  16. [24]

    Calder´ on’s inverse problem with a fi nite number of measurements

    Alberti GS, Santacesaria M. Calder´ on’s inverse problem with a fi nite number of measurements. In: Forum of Mathematics, Sigma. vol. 7. Cambridge University Press; 2 019

  17. [25]

    A direct linear inversion for discontinuous elas- tic parameters recovery from internal displacement information o nly

    Ammari H, Bretin E, Millien P, Seppecher L. A direct linear inversion for discontinuous elas- tic parameters recovery from internal displacement information o nly. Numerische Mathematik. 2021;147:189–226

    2021

  18. [26]

    Mathema tical methods in elasticity imaging

    Ammari H, Bretin E, Garnier J, Kang H, Lee H, Wahab A. Mathema tical methods in elasticity imaging. Princeton University Press; 2015

    2015

  19. [27]

    A method of biological tissu es elasticity reconstruction using magnetic resonance elastography measurements

    Ammari H, Garapon P, Kang H, Lee H. A method of biological tissu es elasticity reconstruction using magnetic resonance elastography measurements. Quarterly of A pplied Mathematics. 2008;66(1):139– 175

    2008

  20. [28]

    Nonuniqueness in diffusion-based opt ical tomography

    Arridge SR, Lionheart WR. Nonuniqueness in diffusion-based opt ical tomography. Optics letters. 1998;23(11):882–884

    1998

  21. [29]

    Simultaneous determination of the diffusion and abso rption coefficient from boundary data

    Harrach B. Simultaneous determination of the diffusion and abso rption coefficient from boundary data. Inverse Problems & Imaging. 2012;6(4):663

    2012

  22. [30]

    Uniqueness, Lipschitz Stability, and Reconstructio n for the Inverse Optical Tomography Problem

    Meftahi H. Uniqueness, Lipschitz Stability, and Reconstructio n for the Inverse Optical Tomography Problem. SIAM Journal on Mathematical Analysis. 2021;53(6):6326 –6354. 22

    2021

  23. [31]

    Exponential instability in an inverse problem for the Schr¨ odinger equation

    Mandache N. Exponential instability in an inverse problem for the Schr¨ odinger equation. Inverse Problems. 2001;17(5):1435

    2001

  24. [32]

    Elastic shear modulus and density profi les inversion: Lipschitz stability results

    Meftahi H, Potschka A. Elastic shear modulus and density profi les inversion: Lipschitz stability results. Applicable Analysis. 2023;p. 1–16

    2023

  25. [33]

    Lipschitz stability es timate and reconstruction of Lam´ e parameters in linear elasticity

    Eberle S, Harrach B, Meftahi H, Rezgui T. Lipschitz stability es timate and reconstruction of Lam´ e parameters in linear elasticity. Inverse Problems in Science and Engin eering. 2021;29(3):396–417

    2021

  26. [34]

    On localizing and concentrating electrom agnetic fields

    Harrach B, Lin YH, Liu H. On localizing and concentrating electrom agnetic fields. SIAM Journal on Applied Mathematics. 2018;78(5):2558–2574

    2018

  27. [35]

    Exact shape-reconstruction by one-ste p linearization in electrical impedance tomography

    Harrach B, Seo JK. Exact shape-reconstruction by one-ste p linearization in electrical impedance tomography. SIAM Journal on Mathematical Analysis. 2010;42(4) :1505–1518

    2010

  28. [36]

    Local uniqueness for an inverse boundary value problem with partial data

    Harrach B, Ullrich M. Local uniqueness for an inverse boundary value problem with partial data. Proceedings of the American Mathematical Society. 2017;145(3):1 087–1095

    2017

  29. [37]

    Lipschitz stability es timate and reconstruction of Lam´ e parameters in linear elasticity

    Eberle S, Harrach B, Meftahi H, Rezgui T. Lipschitz stability es timate and reconstruction of Lam´ e parameters in linear elasticity. Inverse Problems in Science and Engin eering. 2020;p. 1–22

    2020

  30. [38]

    Uniquene ss and Lipschitz stabil- ity of an inverse boundary value problem for time-harmonic elastic wa ves

    Beretta E, de Hoop MV, Francini E, Vessella S, Zhai J. Uniquene ss and Lipschitz stabil- ity of an inverse boundary value problem for time-harmonic elastic wa ves. Inverse Problems. 2017;33(3):035013

    2017

  31. [39]

    Local analysis of inverse problem s: H¨ older stability and iterative reconstruction

    De Hoop MV, Qiu L, Scherzer O. Local analysis of inverse problem s: H¨ older stability and iterative reconstruction. Inverse Problems. 2012;28(4):045001

    2012

  32. [40]

    An analysis of a multi-level projec ted steepest descent itera- tion for nonlinear inverse problems in Banach spaces subject to sta bility constraints

    Maarten V, Qiu L, Scherzer O. An analysis of a multi-level projec ted steepest descent itera- tion for nonlinear inverse problems in Banach spaces subject to sta bility constraints. Numerische Mathematik. 2015;129(1):127–148

    2015

  33. [41]

    Infinite-dimensional inverse prob lems with finite measurements

    Alberti GS, Santacesaria M. Infinite-dimensional inverse prob lems with finite measurements. arXiv preprint arXiv:190610028. 2019

    2019

  34. [42]

    Quantitative strong un ique continuation for the Lam´ e system with less regular coefficients

    Uhlmann G, Lin CL, Nakamura G, Wang JN. Quantitative strong un ique continuation for the Lam´ e system with less regular coefficients. Methods and Applications of An alysis. 2011;18(1):085–092

    2011

  35. [43]

    Localized potentials in electrical impedance tomogra phy

    Gebauer B. Localized potentials in electrical impedance tomogra phy. Inverse Problems & Imaging. 2008;2(2):251–269. 23

    2008

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