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Recovery of initial displacement and velocity in anisotropic elastic systems by the time dimensional reduction method

T0 review · 4 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Boundary observations of elastic waves can determine both the initial displacement and the initial velocity, provided one selects the minimal-norm solution and approximates it with a Legendre-exponential time reduction.

desk verdict A genuinely new application of time-dimensional reduction to anisotropic elastodynamics, with honest numerics, but the main convergence theorem has a gap in (4.10) and the experiments do not sit inside the theorem's asymptotic regime. read the letter →

arxiv 2506.13000 v1 pith:CC2NIXCM submitted 2025-06-15 math.NA cs.NA

Trong D. Dang , Chanh V. Le , Khoa D. Luu , Loc H Nguyen This is my paper · ORCID
classification math.NAcs.NA MSC 35R3074J0565M3235B30
keywords inverseinitialdataproblemelasticwaveequationanisotropicelasticityLegendre-exponentialbasistime-dimensionalreductionquasi-reversibilitymethodminimal-normsolutionCauchyboundary
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 tries to establish that the inverse problem of recovering the initial displacement and velocity of an anisotropic elastic wave from boundary data becomes computationally tractable and provably convergent when the time variable is expanded in a specially chosen orthonormal basis. The basis, made of Legendre polynomials multiplied by an exponential weight, turns the space-time elastic system into a finite chain of elliptic systems for spatial Fourier coefficients. The paper proves that, as the truncation index grows and the noise and regularization parameters tend to zero in the relation $\delta^2 = o(\eta)$, the reconstructed field and its first time derivative converge in a weighted $L^2$\textendash Sobolev norm to the unique minimal-norm solution of the lateral Cauchy problem. A sympathetic reader would care because this offers a practical algorithm for seismology, nondestructive testing, and elastography, where only surface measurements are available and the medium may be inhomogeneous and anisotropic. The numerical experiments with 10% noise indicate that both the geometry and the amplitude of inclusion-like initial data can be recovered.

What carries the argument

The engine of the method is the orthonormal Legendre-exponential basis $\Psi_n(t) = e^t Q_n(t)$, where $Q_n$ are the normalized Legendre polynomials on $(0,T)$; this basis is orthonormal in the weighted space $L^2_{e^{-2t}}(0,T)$. Its second derivatives $\Psi_n''$ are never identically zero, so every Fourier mode contributes to the approximation of $\partial_{tt}u$, and the Sturm\textendash Liouville eigenvalue structure of the Legendre polynomials gives the decay estimate $\|\langle u,\Psi_n\rangle\|_{H^p} \le C n^{-k}$ together with $\|\Psi_n''\| \le C n^{7/2}$. The reduction replaces the wave equation by the coupled elliptic system (3.10) with coefficient matrix $s_{mn} = \int_0^T e^{-2t}\Psi_n''(t)\Psi_m(t)\,dt$, and the quasi-reversibility functional $J_{N,\eta,\delta}$ makes that system solvable with noisy data.

What would settle it

Take the piecewise-constant initial data used in Test 1, regard the corresponding elastic wave field as $u^*$, and compute the weighted sum $\sum_{n=0}^\infty n^{7/2}\|\langle u^*, \Psi_n\rangle\|_{H^p(\Omega)^d}$; if it diverges, then the convergence hypothesis of Theorem 4.1 is violated, and the numerical reconstructions in the paper cannot be explained by the theorem as stated. Alternatively, run the algorithm on a smooth Gaussian initial condition with known truth and check whether the error $\|S_N[U^{\,N,\eta,\delta}_{\min}] - u^*\|$ actually decreases to zero along a sequence satisfying $\delta^2 = o(\eta)$; failure to decrease would contradict the theorem's conclusion.

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Extended reading notes

Core claim

On its own terms, the paper's central claim is Theorem 4.1: for a maximal truncation level $N \ge N(\delta)$ and a regularization parameter $\eta(\delta)$ tending to zero with $\delta^2 = o(\eta)$, the regularized time-reduced reconstruction $S_N[U^{\,N,\eta,\delta}_{\min}]$ converges to the minimal-norm solution $u^*$ of the lateral Cauchy problem (3.1) in $L^2_{e^{-2t}}((0,T);H^p(\Omega)^d)$, together with its first time derivative. That means the approximate initial displacement $S_N[\cdot](\cdot,0)$ converges strongly in $H^p(\Omega)^d$ and the approximate initial velocity $\partial_t S_N[\cdot](\cdot,0)$ converges weakly in $H^p(\Omega)^d$. The proof combines the spectral properties of the Legendre-exponential basis, which justify replacing $\partial_{tt}u$ by the series $\sum_n u_n\Psi_n''$, with the quasi-reversibility functional that enforces the reduced elliptic system and the boundary data while penalizing the $H^{2+p}$ norm.

Load-bearing premise

The convergence proof rests on the assumption that the true minimal-norm solution is smooth enough in the time variable that a certain infinite series of its second derivatives converges; the numerical tests use sharp step-like inclusions that do not meet this smoothness condition.

Editorial extensions

If this is right

  • If Theorem 4.1 is correct, both the initial displacement and the initial velocity can be stably reconstructed from noisy boundary data without knowing either field in advance.
  • The method applies to a fully anisotropic and inhomogeneous elasticity tensor, with no isotropy or constant-coefficient restriction, which covers layered geological media and biological tissue.
  • Because the target is the unique minimal-norm solution, the algorithm returns a well-defined answer even when the lateral Cauchy problem has multiple solutions; if the problem is actually uniquely solvable, the reconstructed field converges to the true physical solution.
  • The convergence argument carries over when the measurements are restricted to a proper subset $\Gamma$ of the boundary, so partial sensor coverage does not destroy the guarantee.
  • At $t=0$, the reconstruction yields the initial displacement strongly and the initial velocity weakly in $H^p(\Omega)^d$, giving quantitative approximations of both fields.

Reading between the lines

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

  • A fair numerical test of the theorem would use smooth initial data, since the piecewise-constant inclusions in the paper's experiments fall outside the $H^k$, $k \ge 5$ time-regularity hypothesis; comparing convergence on Gaussian versus step profiles would separate the method's performance from the theorem's scope.
  • The same time-reduction construction should transfer to scalar wave, acoustic, or viscoelastic equations, because the argument only requires a second-order time derivative and the spectral basis; running the scheme on the acoustic wave equation with smooth sources is a direct testable extension.
  • The condition $\delta^2 = o(\eta)$ suggests a concrete parameter-selection rule for practice: choose $\eta$ to decay slower than the squared noise level, say $\eta \sim \delta^{3/2}$, and then increase $N$ until the projection error in (4.4) drops below $\delta^2$; this could be automated as a stopping criterion.
  • The minimal-norm selection implicitly biases the reconstruction toward smoother initial fields, a consequence relevant to seismic imaging where the true source may be rough; the paper does not quantify this bias.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The paper proposes a time-dimensional reduction method for recovering initial displacement and velocity in an anisotropic elastic wave equation from lateral Cauchy data. The time variable is expanded in a Legendre-polynomial-exponential basis, reducing the space-time inverse problem to a sequence of coupled elliptic systems in space, which are then solved by a Tikhonov-type quasi-reversibility functional. The paper proves existence and uniqueness of a minimal-norm solution to the lateral Cauchy problem, states a convergence theorem for the regularized reduced solution as the truncation level and regularization parameter are tuned relative to the noise level, and reports two-dimensional numerical reconstructions for isotropic, inhomogeneous, and anisotropic media with 10% noise.

Significance. If the convergence result in Theorem 4.1 is correct, the paper makes a useful methodological contribution: the Legendre-exponential basis is a genuinely new device in this context, and the reduction of a time-dependent ill-posed inverse problem to a sequence of elliptic systems is computationally attractive. The numerical experiments cover an anisotropic elasticity tensor, which goes beyond much of the existing scalar or isotropic literature, and the reconstructed inclusions are qualitatively reasonable. However, the theoretical statement as written rests on an extra regularity assumption that is not verified for the tested discontinuous data, the proof of the key boundedness step uses an invalid inequality, and the numerical parameters do not satisfy the scaling required by the theorem. The central idea is plausible and likely repairable, but the current manuscript oversells the agreement between theory and computation.

major comments (4)
  1. [Theorem 4.1 and Theorem 2.1(b), Eq. (4.4)] The convergence theorem assumes that the series Σ⟨u*,Ψ_n⟩Ψ''_n converges in L²((0,T);H^p(Ω)^d). By Lemma 2.2, ∥Ψ''_n∥ = O(n^{7/2}), so Theorem 2.1(b) guarantees this convergence only when u* has at least H^5 time regularity. The natural regularity supplied by Theorem 3.1 is only u* ∈ L²(0,T;H^{2+p}) ∩ H²(0,T;H^p), and the extra assumption is not verified anywhere. The numerical tests use piecewise-constant initial data, whose elastic wave solutions cannot be expected to possess H^5 time regularity; consequently the experiments are not covered by the convergence theorem as stated. The authors should either prove the series convergence under weaker assumptions, restrict the claim to high-regularity data and run numerical tests with such data, or clearly state that the theorem is not applicable to the experiments.
  2. [Proof of Theorem 4.1, inequality (4.10)] The display after (4.10) bounds ∥div(C:∇S) − ∂ttS∥_{L²H^p} by J_{N,η,δ}(U^{N,η,δ}_min), without a square root. Since J is defined as a sum of squared norms, the residual norm is bounded by √J, not by J. This invalidates the chain (4.10)–(4.12) as written; the boundedness of ∂ttS does not follow. The step appears repairable, because √J remains bounded under δ²=o(η), but the current proof is not correct at this point.
  3. [Section 5.1, Theorem 4.1 parameter regime] The theorem requires η→0, δ→0, and δ²=o(η), with N≥N(δ) chosen accordingly. The numerical experiments set η=10^{-6} and δ=10%, so δ²=10^{-2} is not o(η); moreover N=30 is fixed by trial-and-error for Test 1 rather than chosen as N≥N(δ). Thus the reported computations lie outside Θ, and the abstract's claim that the numerical experiments 'confirm the theory' is not supported. The authors should either run experiments satisfying the theorem's scaling or present the numerics as a heuristic demonstration independent of Theorem 4.1.
  4. [Remark 4.1(2)] The remark asserts that the convergence result remains valid when boundary data are available only on a proper subset Γ⊂∂Ω. The proof of Theorem 4.1 identifies the limit z by passing to the limit in boundary integrals over the full ∂Ω and concludes that z satisfies (3.1) with f* and g* on all Γ_T. For a proper subset, this identification is not available, and no alternative argument is supplied. The claim should be removed or substantiated with a separate proof.
minor comments (6)
  1. [Equation (4.1)] The norm subscript in the first term of J appears as [H^p(Ω)^d]^{N+1} inside the sum over m; it should presumably be H^p(Ω)^d, otherwise the expression is not dimensionally consistent.
  2. [Equation (4.1)] The second sum inside the first term runs over n=1 while earlier definitions, such as (3.7), sum over n=0; this inconsistency should be fixed.
  3. [Figures 1–3 captions] In all three figure captions, the bottom-row labels repeat pcomp_2 for panels (g) and (h); based on the text these should be qcomp_1 and qcomp_2.
  4. [Proof of Lemma 4.2] The line stating ∥w'_{n_k}∥ → ∥w' L²_{e^{-2t}}(0,T) is missing the norm symbol and argument; it should read ∥w'_{n_k}∥_{L²_{e^{-2t}}(0,T)} → ∥w'∥_{L²_{e^{-2t}}(0,T)}.
  5. [Introduction and Abstract] The abstract and introduction state that the method recovers 'the initial displacement and velocity fields' without always qualifying that, when uniqueness fails, the target is the minimal-norm solution rather than the true physical data. The qualification appears later in the paper but should be prominent from the outset.
  6. [Equation (4.11)] The display following (4.11) has a missing closing parenthesis in '≤ 2δ²/η + ∥u*∥_{L²...} + ∥u*∥_{L²...', making the inequality hard to parse; it should be rewritten.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the convergence theorem is conditional on an explicit high-regularity assumption, but the derivation does not fit the target result or rely on load-bearing self-citation.

full rationale

The central reconstruction is not obtained by fitting the target result from data. Theorem 3.1 defines u* independently as the unique minimizer of the L^2_{e^{-2t}}(0,T;H^{2+p}) norm over solutions of (3.1), using a standard weak-compactness/parallelogram argument. The quasi-reversibility functional (4.1) is a Tikhonov functional with residual and penalty terms; no parameter is fitted to the true initial data. The convergence proof in Theorem 4.1 is a standard consistency-stability-compactness chain: estimate (4.7) bounds the functional at the projected true solution by projection error plus data noise plus eta times the true norm, and the compactness argument shows any weak limit satisfies (3.1) and has norm no larger than that of u*, hence equals u* by minimality. This does not reduce to an input by construction. The only fragile point is the extra hypothesis in Theorem 4.1 that the series sum <u*,Psi_n> Psi''_n converges in L^2((0,T);H^p); by Theorem 2.1(b) this requires H^k time regularity for k>=5, which is not guaranteed by the natural H^2 time regularity and is likely violated by the piecewise-constant inclusions used in the numerical tests. That is a limitation and a mismatch between theory and experiments, not circularity. The inequality (4.10) also appears to bound a squared residual norm by the functional value rather than by its square root; this is a correctness issue, not a circular use of the conclusion. Self-citations (e.g., [22,30,34,40]) are contextual remarks about previous difficulties and are not used as the basis for any theorem here; the anisotropic elastic extension is derived from first principles in the paper. Therefore no circular step is present.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central theorem rests on standard spectral theory for Legendre polynomials, the Aubin-Lions compactness lemma, standard well-posedness assumptions on the elasticity tensor, and two paper-specific assumptions: high time regularity of the minimal-norm solution and the validity of the minimal-norm criterion as the physical target. The numerical method adds hyperparameters N and eta chosen by trial and error.

free parameters (2)
  • truncation level N = 30
    Selected by trial-and-error on Test 1, then fixed for all tests; numerical results depend on this hand-chosen hyperparameter.
  • regularization parameter eta = 1e-6
    Selected by trial-and-error on Test 1; violates the theorem's condition delta^2 = o(eta) because delta^2 = 0.01 with eta = 1e-6.
assumptions (6)
  • standard math Legendre polynomial spectral theory: the functions Q_n are eigenfunctions of the Sturm-Liouville operator K with eigenvalues n(n+1), used in Lemma 2.1.
    Used to establish coefficient decay estimates that underpin the convergence theorem.
  • standard math Aubin-Lions compactness lemma for bounded sequences in L^2((0,T);H^{2+p}) intersect H^2((0,T);H^p) to extract a strongly convergent subsequence in L^2((0,T);H^p).
    Used in the proof of Theorem 4.1(3) to pass to the limit and identify the limit with the minimal-norm solution.
  • domain assumption The elasticity tensor C is symmetric, coercive, and C^2-smooth on Omega, as stated in Section 1.
    Ensures the forward problem is well-posed and the lateral Cauchy problem has the unique continuation property.
  • domain assumption The boundary data f and g are generated by a true solution of the initial value problem, so the admissible set S is nonempty.
    The paper defines S as the set of solutions to (3.1); if no solution exists the method has nothing to reconstruct.
  • ad hoc to paper The series sum_n <u*, Psi_n> Psi''_n(t) converges in L^2((0,T);H^p(Omega)^d) for the minimal-norm solution u*, required in Theorem 4.1.
    This high time-regularity assumption, needing u* in H^k with k >= 5 by Theorem 2.1(b), is not derived from the natural regularity of the inverse problem and is violated by the discontinuous numerical test data.
  • ad hoc to paper The minimal-norm (Moore-Penrose) solution is the correct target, i.e., it coincides with the true physical initial data when uniqueness fails.
    Theorem 3.1 only proves uniqueness of the minimal-norm solution; Remark 4.1 concedes that uniqueness of (3.1) is only highly plausible, so the reconstructed limit may differ from the true p and q in non-unique cases.

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Pith. "Pith review of Recovery of initial displacement and velocity in anisotropic elastic systems by the time dimensional reduction method." pith.science (2026). https://pith.science/paper/CC2NIXCM

@misc{pith2026250613000,
  author       = {Pith},
  title        = {Pith review of: Recovery of initial displacement and velocity in anisotropic elastic systems by the time dimensional reduction method},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CC2NIXCM}},
  note         = {Machine review of arXiv:2506.13000}
}
read the original abstract

We introduce a time-dimensional reduction method for the inverse source problem in linear elasticity, where the goal is to reconstruct the initial displacement and velocity fields from partial boundary measurements of elastic wave propagation. The key idea is to employ a novel spectral representation in time, using an orthonormal basis composed of Legendre polynomials weighted by exponential functions. This Legendre polynomial-exponential basis enables a stable and accurate decomposition in the time variable, effectively reducing the original space-time inverse problem to a sequence of coupled spatial elasticity systems that no longer depend on time. These resulting systems are solved using the quasi-reversibility method. On the theoretical side, we establish a convergence theorem ensuring the stability and consistency of the regularized solution obtained by the quasi-reversibility method as the noise level tends to zero. On the computational side, two-dimensional numerical experiments confirm the theory and demonstrate the method's ability to accurately reconstruct both the geometry and amplitude of the initial data, even in the presence of substantial measurement noise. The results highlight the effectiveness of the proposed framework as a robust and computationally efficient strategy for inverse elastic source problems.

Figures

Figures reproduced from arXiv: 2506.13000 by the authors.

Figure 1
Figure 1. Comparison between the true and reconstructed initial displacement and velocity compo [PITH_FULL_IMAGE:figures/full_fig_p022_1.png] view at source ↗
Figure 2
Figure 2. Comparison between the true and reconstructed initial displacement and velocity compo [PITH_FULL_IMAGE:figures/full_fig_p024_2.png] view at source ↗
Figure 3
Figure 3. Comparison between the true and reconstructed initial displacement and velocity compo [PITH_FULL_IMAGE:figures/full_fig_p026_3.png] view at source ↗

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  1. Inverse initial data reconstruction for Maxwell's equations via time-dimensional reduction method

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Works this paper leans on

42 extracted references · 30 canonical work pages · cited by 1 Pith paper

  1. [25]

    M. V. Klibanov. Convexification of restricted Dirichlet to Neumann map. J. Inverse and Ill-Posed Problems, 25(5):669–685, 2017

    2017

  2. [1]

    Acosta and B

    S. Acosta and B. Palacios. Thermoacoustic tomography for an integro-differential wave equation modeling attenuation. J. Differential Equations , 5:1984–2010, 2018

    1984

  3. [2]

    Aki and P

    K. Aki and P. G. Richards. Quantitative Seismology. University Science Books, 2002

    work page 2002

  4. [3]

    Alves, N

    C. Alves, N. Martins, and N. Roberty. Full identification of acoustic sources with multiple frequencies and boundary measurements. Inverse Problems and Imaging , 3:275–294, 2009

    work page 2009

  5. [4]

    Ammari, E

    H. Ammari, E. Bretin, J. Garnier, H. Lee, and A. Wahab. Mathematical Methods in Elasticity Imaging. Princeton University Press, Princeton, NJ, 2015

    work page 2015

  6. [5]

    Ammari, E

    H. Ammari, E. Bretin, J. Garnier, and A. Wahab. Time reversal in viscoelastic media. preprint, 2011

    work page 2011

  7. [6]

    Ammari, E

    H. Ammari, E. Bretin, J. Garnier, and A. Wahab. Time reversal algorithms in viscoelastic media. European Journal of Applied Mathematics , 24:565–600, 2013

    work page 2013

  8. [7]

    Ammari, E

    H. Ammari, E. Bretin, J. Garnier, and V. Wahab. Time reversal in attenuating acoustic media. Contemp. Math. , 548:151–163, 2011

    2011

Show all 42 references
  1. [8]

    Ammari, E

    H. Ammari, E. Bretin, E. Jugnon, and V. Wahab. Photoacoustic imaging for attenuating acoustic media. In H. Ammari, editor, Mathematical Modeling in Biomedical Imaging II , pages 57–84. Springer, 2012

    2012

  2. [9]

    Baldassari, M

    L. Baldassari, M. V de Hoop, E. Francini, and S. Vessella. Early-warning inverse source problem for the elasto-gravitational equations. SIAM Journal on Applied Mathematics , 84(3):831–855, 2024

    2024

  3. [10]

    G. Bao, G. Hu, Y. Kian, and T. Yin. Inverse source problems in elastodynamics. Inverse Problems, 34:045009, 2018. 27

    2018

  4. [11]

    Bonnet and A

    M. Bonnet and A. Constantinescu. Inverse problems in elasticity. Inverse Problems, 21:R1–R50, 2005

    2005

  5. [12]

    Brevis, ´A

    I. Brevis, ´A. Rodr ´ ıguez-Rozas, J. H. Ortega, and D. Pardo. Source time reversal (str) method for linear elasticity. Computers & Mathematics with Applications , 77(5):1358–1375, 2019

    2019

  6. [13]

    Burgholzer, H

    P. Burgholzer, H. Gr¨ un, M. Haltmeier, R. Nuster, and G. Paltauf. Compensation of acous- tic attenuation for high-resolution photoa- coustic imaging with line detectors. Proc. SPIE, 6437:643724, 2007

    2007

  7. [14]

    Do and L

    N. Do and L. Kunyansky. Theoretically exact photoacoustic reconstruction from spatially and temporally reduced data. Inverse Problems, 34(9):094004, 2018

    2018

  8. [15]

    Duvaut and J

    G. Duvaut and J. L. Lions. Inequalities in Mechanics and Physics . Springer Verlag, Berlin Heidelberg New York, 1976

    1976

  9. [16]

    H. W. Engl, M. Hanke, and A. Neubauer. Regularization of Inverse Problems . Springer, 1996

    1996

  10. [17]

    M. Fink. Time reversed acoustics. Physics Today, 50(3):34–40, 1997

    1997

  11. [18]

    Fink and C

    M. Fink and C. Prada. Acoustic time-reversal mirrors. Inverse Problems, 17:R1–R38, 2001

    2001

  12. [19]

    M. A. Green, L. E. Bilston, and R. Sinkus. Ultrasound elastography for tissue characterization: current status and future directions. Ultrasound in Medicine & Biology , 45(4):715–736, 2019

    2019

  13. [20]

    Haltmeier

    M. Haltmeier. Inversion of circular means and the wave equation on convex planar domains. Comput. Math. Appl. , 65:1025–1036, 2013

    2013

  14. [21]

    P. C. Hansen. Discrete Inverse Problems: Insight and Algorithms . SIAM, 2010

    2010

  15. [22]

    D. N. H` ao, T. T. Le, and L. H. Nguyen. The Fourier-based dimensional reduction method for solving a nonlinear inverse heat conduction problem with limited boundary data. Communi- cations in Nonlinear Science and Numerical Simulation , 128:107679, 2024

    2024

  16. [23]

    V. Isakov. Inverse Problems for Partial Differential Equations . Springer, 2006

    2006

  17. [24]

    On the convergence of the time reversal method for thermoacoustic tomography in elastic media

    Vitaly Katsnelson and Linh V Nguyen. On the convergence of the time reversal method for thermoacoustic tomography in elastic media. Applied Mathematics Letters , 77:79–86, 2018

    2018

  18. [26]

    R. Kowar. On time reversal in photoacoustic tomography for tissue similar to water. SIAM J. Imaging Sci. , 7:509–527, 2014

    2014

  19. [27]

    Larmat, J

    C. Larmat, J. P. Montagner, M. Fink, Y. Capdeville, A. Tourin, and ´E. Cl´ ev´ ed´ e’. Time-reversal imaging of seismic sources and application to the great sumatra earthquake. Geophysical Research Letters, 33:L19312, 2006

    2006

  20. [28]

    Latt` es and J

    R. Latt` es and J. L. Lions.The Method of Quasireversibility: Applications to Partial Differential Equations. Elsevier, New York, 1969. 28

    1969

  21. [29]

    T. T. Le, L. H. Nguyen, T-P. Nguyen, and W. Powell. The quasi-reversibility method to numerically solve an inverse source problem for hyperbolic equations. Journal of Scientific Computing, 87:90, 2021

    2021

  22. [30]

    T. T. Le, L. V. Nguyen, L. H. Nguyen, and H. Park. The time dimensional reduction method to determine the initial conditions without the knowledge of damping coefficients. Computers and Mathematics with Applications , 166:77–90, 2024

    2024

  23. [31]

    Natterer

    F. Natterer. Photo-acoustic inversion in convex domains. Inverse Probl. Imaging , 6:315–320, 2012

    2012

  24. [32]

    L. H. Nguyen and M. V. Klibanov. Carleman estimates and the contraction principle for an inverse source problem for nonlinear hyperbolic equations. Inverse Problems, 38:035009, 2022

    2022

  25. [33]

    L. V. Nguyen. A family of inversion formulas in thermoacoustic tomography. Inverse Probl. Imaging, 3:649–675, 2009

    2009

  26. [34]

    P. M. Nguyen, L. H. Nguyen, and H. T. Vu. A robust approach with numerical demon- strations for the inverse scattering problem using a Carleman contraction map. preprint arXiv:2404.04145, 2024

    2024 arXiv

  27. [35]

    P. D. Norville and W. R. Scott. Time-reversal focusing of elastic surface waves. Journal of the Acoustical Society of America, 118:735–744, 2013

    2013

  28. [36]

    J. L. Rose. Ultrasonic Guided Waves in Solid Media . Cambridge University Press, 2014

    2014

  29. [37]

    Sarvazyan, T

    A. Sarvazyan, T. J. Hall, M. W. Urban, and M. Fatemi. Elasticity imaging: a new ultrasonic technology of medical diagnostics. Ultrasound in Medicine & Biology , 24(9):1419–1435, 1998

    1998

  30. [38]

    Tittelfitz

    J. Tittelfitz. Thermoacoustic tomography in elastic media. Inverse Problems , 28(5):055004, 2012

    2012

  31. [39]

    D. D. Trong, P. N. D. Alain, P. T. Nam, and T. T. Tuyen. Determination of the body force of a two-dimensional isotropic elastic body. Journal of computational and applied mathematics , 229(1):192–207, 2009

    2009

  32. [40]

    D. D. Trong, L. H. Nguyen, and H. T. Vu. Determining initial conditions for nonlinear hy- perbolic equations with time dimensional reduction and the Carleman contraction principle. Inverse Problems, 40:125021, 2024

    2024

  33. [41]

    Virieux and S

    J. Virieux and S. Operto. An overview of full-waveform inversion in exploration geophysics. Geophysics, 74(6):WCC1–WCC26, 2009

    2009

  34. [42]

    Yamamoto

    M. Yamamoto. Carleman estimates for parabolic equations and applications. Inverse Problems, 25(12):123013, 2009. 29

    2009

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