Pith. sign in

REVIEW 3 major objections 4 minor 1 cited by

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

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read The paper proves that projection onto a Legendre polynomial-exponential basis converts Maxwell initial-field recovery into stable spatial solves converging to a minimum-norm solution.

desk verdict The convergence theorem has a genuine n=0 gap, and even if repaired, it proves convergence to a minimum-norm solution rather than to the physical initial field; the 3D numerics look good but outrun the theory. read the letter →

arxiv 2506.20777 v1 pith:MSNHK6YN submitted 2025-06-25 math.NA cs.NA

Thuy T. Le , Cong B. Van , Trong D. Dang , Loc H. Nguyen This is my paper · ORCID
classification math.NAcs.NA MSC 35R3035L5035Q6165M3278A25
keywords inverseproblemMaxwellequationsinitialelectricfieldrecoverytime-dimensionalreductionLegendrepolynomial-exponentialbasisminimum-normsolutionquasi-reversibilitymethodboundarymeasurements
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 claims that the initial electric field of a time-dependent Maxwell system can be recovered from boundary measurements of the field and its normal derivative over a finite time interval, even when the initial magnetic field and charge density are unknown. The central move is to expand the electric field in a Legendre polynomial-exponential basis in time, turning the space-time inverse problem into a sequence of three-dimensional elliptic systems for time-Fourier coefficients. A quasi-reversibility method with $H^3$-regularization and a minimum-norm selection rule solves these systems. The authors prove that as the truncation order grows and the noise and regularization parameters vanish, with $\delta^2 = o(\epsilon)$, the regularized reconstruction converges to the unique minimum-norm solution of the lateral boundary value problem. Numerical experiments in 3D with 10% noise show that the selected solution matches prescribed test fields within roughly 8-16% relative error in peak values.

What carries the argument

The central object is the Legendre polynomial-exponential basis $\Psi_n(t) = e^t Q_n(t)$, where $\{Q_n\}$ is the rescaled orthonormal Legendre basis of $L^2(0,T)$; the basis is orthonormal in the weighted space $L^2_{e^{-2t}}(0,T)$ and, crucially, no $\Psi_n''$ vanishes identically, so the projection does not annihilate the second time derivative. The machinery applies the time-projection operator $P_N$ to the Maxwell equation, producing the coupled spatial system (4.9) for the Fourier coefficient modes $e_m(x)$, with coupling matrix $s_{mn} = \int_0^T e^{-2t}\Psi_n''(t)\Psi_m(t)\,dt$. The numerical solver minimizes the Tikhonov-type functional $J_{N,\delta,\epsilon}$ that penalizes the equation residual, mismatch of both Cauchy boundary data, and $\epsilon\|\cdot\|^2_{H^3(\Omega)^3}$; the minimizer is expanded back in time to give the reconstructed field and its value at $t=0$.

What would settle it

Take a smooth initial field that satisfies given lateral boundary data but is not the $H^3$-norm minimizer among all compatible fields, generate exact boundary data from it, and run the algorithm with vanishing noise and regularization. If the reconstruction converges to that true field instead of to the independently computed minimum-norm field, the theorem's identification of the convergence target is wrong; if it converges to the minimizer, the method is provably recovering only the selected solution, not the physical field.

Watch

Extended reading notes

Core claim

The paper's central discovery is a stable reconstruction route for an underdetermined electromagnetic inverse problem: by projecting the space-time Maxwell equation onto a Legendre polynomial-exponential basis, the time variable is removed, and the resulting spatial systems can be solved by minimizing a regularized least-squares functional. The rigorously stated payoff is Theorem 5.1: when the noisy boundary data satisfy the stated noise bound and the regularization parameter obeys $\delta^2 = o(\epsilon)$, the reconstruction $S_N[V^{\min}_{N,\delta,\epsilon}]$ converges as $(N,\delta,\epsilon) \to (\infty,0^+,0^+)$ in $L^2((0,T);H^2(\Omega)^3)$ to $E^*$, the unique minimum-$H^3$-norm solution of the lateral boundary value problem (3.1). The paper frames this convergence as showing that time-dimensional reduction, quasi-reversibility, and minimum-norm selection together turn the underdetermined inverse initial-value problem into a well-posed reconstruction. The numerical experiments demonstrate that the method preserves support and amplitude of discontinuous test fields under 10% multiplicative noise.

Load-bearing premise

The load-bearing premise is that the true physical initial field coincides with the minimum-norm solution selected by the boundary value problem; the paper proves convergence to that selected field, and nothing in the argument links the minimizer to the physical $E_0$.

Editorial extensions

If this is right

  • If Theorem 5.1 holds, the algorithm provably recovers the $H^3$-minimum-norm electric field compatible with the lateral boundary data, without needing the initial magnetic field, the initial time derivative, or the charge density.
  • The time-dimensional reduction replaces the $(3+1)$-dimensional inverse problem by a sequence of spatial elliptic solves, so the computational cost scales with the number $N$ of temporal modes rather than with a full space-time inversion.
  • The convergence result gives a parameter recipe: choose $N$ large enough to make the time-projection residual small, and choose $\epsilon$ so that $\delta^2/\epsilon \to 0$.
  • The convergence of $S_N[V^{\min}_{N,\delta,\epsilon}](\cdot,0)$ to $E^*(\cdot,0)$ in $L^2(\Omega)^3$ is the precise sense in which the method solves the initial-data recovery problem.
  • The method also controls the first time derivative in (5.7), so it gives a stable approximation to the full space-time field, not only its initial value.

Reading between the lines

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

  • Editorial inference: because the convergence target is the minimum-norm solution of the boundary value problem, the algorithm's output is best interpreted as the smoothest field consistent with the measurements; the physical initial field is recovered only if it happens to be that selected field.
  • Editorial inference: the numerical tests use discontinuous true fields that do not satisfy the $H^3$ regularity assumed in Theorem 5.1, so the observed accuracy is evidence of numerical behavior beyond the theorem's stated hypotheses.
  • Editorial inference: the same time-projection plus quasi-reversibility construction may transfer to other hyperbolic systems with unknown initial velocity, such as elastic or acoustic wave equations, whenever lateral Cauchy data are available.
  • Editorial inference: a quantitative version of Theorem 5.1, giving explicit rates in $N$, $\delta$, and $\epsilon$, would let practitioners set the truncation order and regularization parameter directly from the measured noise level; the paper leaves that open.
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.

Referee Report

3 major / 4 minor

Summary. The paper studies the recovery of the initial electric field E0 in a bounded domain from boundary measurements of the electric field and its normal derivative over (0,T), for the time-domain Maxwell system in an inhomogeneous anisotropic medium. The authors formulate an underdetermined lateral boundary value problem (3.1), select a minimum-H^3-norm solution E*, project the field in time onto a Legendre polynomial-exponential basis, reduce the problem to a sequence of coupled spatial systems (4.9), and solve these systems by minimizing a Tikhonov-type functional (5.3). Theorem 5.1 claims convergence of the reconstructed space-time field S_N[V_min] to E* as N tends to infinity, the noise level delta tends to 0, and the regularization parameter epsilon tends to 0 with delta^2 = o(epsilon). Numerical experiments with 10% multiplicative noise on three-dimensional discontinuous test fields show reconstructions of the initial field.

Significance. If Theorem 5.1 were valid as stated, the contribution would be valuable: the time-dimensional reduction converts a (3+1)-dimensional inverse Maxwell problem into a sequence of 3D elliptic systems, with an explicit quasi-reversibility regularization and a convergence theorem. The manuscript also contains a detailed Algorithm 1, three fully three-dimensional numerical tests, and reconstructions that remain accurate at 10% noise. However, as written the central convergence proof contains a concrete missing-mode error and a regularity gap, and the theorem's target is the minimum-norm solution of (3.1), not the physical initial field. The numerical tests use discontinuous true fields that do not satisfy the hypotheses of the theorem. These issues must be resolved before the convergence claim can be accepted. With repairs and a corrected framing, the method itself remains plausible and potentially useful.

major comments (3)
  1. [§5, Theorem 5.1, Eq. (5.4)] Equation (5.4) uses the truncation sum from n=1 to N, while Lemma 4.1, which is invoked to prove it, provides the approximation with the sum from n=0 to N. The omitted n=0 mode is not negligible. Since Psi_0(t)=e^t/sqrt(T), one has Psi_0'' = Psi_0 and s_m0 = delta_m0, so the missing term is epsilon(x) e*_0(x) in the m=0 component. Consequently, as N tends to infinity, the left-hand side of (5.4) tends to ||epsilon e*_0||^2_{L^2(Omega)^3}, which need not vanish. A concrete counterexample is E*(x,t)=phi(x)Psi_0(t) with nonzero smooth phi satisfying the elliptic equation in (3.1); then e*_0=phi and e*_n=0 for n>=1, so the left-hand side of (5.4) equals ||epsilon phi||^2_{L^2(Omega)^3} for every N. The same omission appears in (5.10) when evaluating J_{N,delta,epsilon}(F_N[E*]). Because (5.13) and all subsequent compactness and minimum-norm steps rely on the bound (5.4), Theorem 5.1 is not proved as stated. Replacing n=1 by n=0 in (5.4) and (5.10) appears to repair this specific point, provided the regularity issue in the next comment is also resolved.
  2. [§4, Lemma 4.1 and §5, Theorem 5.1] Lemma 4.1 is stated for u in H^2((0,T);L^2(Omega)), but its proof invokes Proposition 2.1, which requires u in H^k((0,T);L^2(Omega)) for some k>=5. The convergence of the differentiated projection sum_{n=0}^N <u,Psi_n>Psi_n'' to u_tt requires additional temporal regularity and compatibility conditions; for a general H^2 function it is not established by the cited result. The field E* in Theorem 5.1 belongs to S with only H^2((0,T);L^2(Omega)^3), so the hypothesis of Lemma 4.1 as used in the proof of (5.4) is not satisfied. This is a second load-bearing gap in the convergence proof. The authors should either strengthen the assumptions on E* (for example, H^5 temporal regularity) or provide a direct proof of Lemma 4.1 under the stated H^2 assumption.
  3. [§3, Theorem 3.1 and §5, Theorem 5.1] The convergence statements (5.6)-(5.8) are only to E*, the unique minimizer of the H^3 norm over the admissible set S of solutions to (3.1). Nothing in Theorem 3.1 or Section 5 shows that the physical initial electric field E0 satisfies this minimum-norm selection; indeed any solution of (3.1) is admissible, and the true field need not be the norm minimizer. Thus the abstract and Section 1 claims that the method 'recovers the initial electric field' or 'converges to the true solution' are stronger than what is proved. Moreover, in all three numerical tests the true initial fields are discontinuous indicator functions, which are not in H^3(Omega)^3, and the data are generated with partial_t E(x,0)=0; these settings lie outside the hypotheses of Theorem 5.1. The numerical validation should be presented as heuristic, and the theorem should be stated as convergence to the selected minimum-norm solution, or an argument relating E* to the physical E0 should be supplied.
minor comments (4)
  1. [§6, data generation] The boundary data G* are generated as (curl E*) x nu, whereas the inverse problem in (1.2) uses G = partial_nu E; this discrepancy should be clarified or the two settings should be aligned.
  2. [§3, Theorem 3.1] Theorem 3.1 states that E* belongs to L^2((0,T);H^3(Omega)^3), but the minimization is performed in the weighted norm L^2_{e^{-2t}}((0,T);H^3(Omega)^3); the norms are equivalent on the finite interval, but the notation should be made consistent.
  3. [§6, error reporting] The relative errors reported in the numerical section appear to be computed from maximum values in target regions rather than from a global L^2 error; the precise quantity being reported should be defined.
  4. [§2 and §4] Propositions 2.1, 2.2, and Lemma 2.1 are imported from the authors' preprint [37]; since that reference is not yet published, the statements and proofs should be included or a published source should be cited to make the paper self-contained.

Circularity Check

2 steps flagged · score 6.0 of 10

Theorem 5.1 converges to the method's own minimum-norm target E*, not to the physical E0, and the key spectral lemmas are imported from the authors' unpublished preprint [37].

  1. self definitional [Section 3, Theorem 3.1; Section 5, Theorem 5.1 proof Step 4 and Eq. (5.3)]
    "Then, problem (3.1) admits a unique solution E∗ ∈ L2((0, T); H3(Ω)3) ∩ H2((0, T); L2(Ω)3) satisfying ∥E∗∥L2_{e−2t}((0,T);H3(Ω)3) = min{...}. ... Since z belongs to the admissible set and achieves the minimum norm, the uniqueness of the minimizer implies that z = E∗."

    Theorem 3.1 defines E* as the unique H3-norm minimizer over the admissible set S. The regularized functional (5.3) contains precisely the same H3 penalty, ϵ∥v_m∥²_{H3(Ω)³}, and Step 4 of Theorem 5.1 identifies the limit z with E* solely by invoking uniqueness of the H3 minimizer. Hence the claimed convergence is to the field the method was built to select; the physical initial field E0 is never shown to be that minimizer. The abstract calls this 'the true solution,' but the theorem's target is E* by definition. This is a self-definitional reduction: the output of the minimization is renamed as the prediction.

  2. self citation load bearing [Section 2, Propositions 2.1–2.2 and Lemma 2.1; Section 5, Theorem 5.1 proof, use of Lemma 4.1 for Eq. (5.4)]
    ""For details of the proofs of Propositions 2.1 and 2.2, see [37]." "The following lemma plays an essential role in establishing our convergence result. Its proof can be found in [37, Lemma 4.2]." "The existence of N (δ) and the estimate in (5.4) follow from Lemma 4.1 directly.""

    Lemma 4.1—and behind it Proposition 2.1 and Lemma 2.1—supplies the spectral approximation used at the single most load-bearing point of Theorem 5.1, the estimate (5.4), which enters the bound (5.13) and all later compactness and minimum-norm steps. The proofs are not reproduced; the paper imports them from reference [37], an unpublished preprint by overlapping authors (D. D. Trong, C. V. Le, K. D. Luu, and L. H. Nguyen). The self-citation is therefore not a peripheral pointer; it carries the foundation of the convergence proof.

full rationale

The central advertised claim is that the quasi-reversibility/time-reduction method reconstructs the initial electric field E0 from lateral data. What Theorem 5.1 actually proves, if estimate (5.4) were valid, is that the regularized minimizers converge to E*, the H3-minimum-norm solution of the underdetermined system (3.1). Because E* is defined by minimizing exactly the H3 norm that the regularizer penalizes in (5.3), the theorem is largely a consistency statement for the selection rule; it never shows E0 = E* or that the true field is the H3-minimizer. This is the self-definitional core of the paper. In addition, the key spectral-approximation tools (Propositions 2.1–2.2, Lemma 2.1, and Lemma 4.1 behind (5.4)) are not proved here but deferred to the authors' own unpublished preprint [37]; this is load-bearing self-citation. These two features justify a partial circularity score of 6. I am not counting the apparent omission of the n = 0 mode in (5.4) as circularity: if that gap is real, it means Theorem 5.1 as stated is not proved, which is a correctness defect rather than a circularity. Likewise, the numerical tests use discontinuous characteristic-function fields outside H3(Ω), so the theorem's hypotheses do not cover them; this is a limitation, not a circular step.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The central claim rests on the existence of a smooth minimum-norm solution in H^3, on external spectral approximation lemmas from the authors' companion preprint, and on exact knowledge of the medium. The two tuned numerical parameters (ϵ and N) are fitted to make the tests succeed, and the theory requires a noise-to-regularization scaling that the experiments violate.

free parameters (2)
  • regularization parameter ϵ = 1e-6
    Chosen by trial-and-error on Test 1 and fixed for all experiments; the numerical noise level δ=0.1 gives δ^2=0.01, which is not o(ϵ) as required by Theorem 5.1.
  • truncation order N = 15
    Chosen by trial-and-error on Test 1 and fixed for all experiments; convergence theory requires N > N(δ) with N depending on the tolerance, but no systematic selection is given.
assumptions (3)
  • domain assumption The boundary value problem (3.1) admits a nonempty admissible set S of H^3-regular solutions when data are smooth.
    Used in Theorem 3.1 to guarantee existence of the minimum-norm solution; the forward problem is assumed to give such smooth solutions, but the numerical tests use discontinuous initial data.
  • domain assumption The Legendre polynomial-exponential basis approximation results (Propositions 2.1, 2.2, Lemma 2.1) from [37] hold.
    These external lemmas drive the spectral convergence and the bound (5.4); they are not proved in this paper and come from a companion preprint.
  • domain assumption The medium parameters µ and ε are smooth, strictly positive definite, and known exactly.
    Assumed in Problem 1.1 and Theorem 3.1; the forward data generation uses a smooth µ with a radial bump.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Inverse initial data reconstruction for Maxwell's equations via time-dimensional reduction method." pith.science (2026). https://pith.science/paper/MSNHK6YN

@misc{pith2026250620777,
  author       = {Pith},
  title        = {Pith review of: Inverse initial data reconstruction for Maxwell's equations via time-dimensional reduction method},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MSNHK6YN}},
  note         = {Machine review of arXiv:2506.20777}
}
abstract

We study an inverse problem for the time-dependent Maxwell system in an inhomogeneous and anisotropic medium. The objective is to recover the initial electric field $\mathbf{E}_0$ in a bounded domain $\Omega \subset \mathbb{R}^3$, using boundary measurements of the electric field and its normal derivative over a finite time interval. Informed by practical constraints, we adopt an under-determined formulation of Maxwell's equations that avoids the need for initial magnetic field data and charge density information. To address this inverse problem, we develop a time-dimension reduction approach by projecting the electric field onto a finite-dimensional Legendre polynomial-exponential basis in time. This reformulates the original space-time problem into a sequence of spatial systems for the projection coefficients. The reconstruction is carried out using the quasi-reversibility method within a minimum-norm framework, which accommodates the inherent non-uniqueness of the under-determined setting. We prove a convergence theorem that ensures the quasi-reversibility solution approximates the true solution as the noise and regularization parameters vanish. Numerical experiments in a fully three-dimensional setting validate the method's performance. The reconstructed initial electric field remains accurate even with $10\%$ noise in the data, demonstrating the robustness and applicability of the proposed approach to realistic inverse electromagnetic problems.

Figures

Figures reproduced from arXiv: 2506.20777 by the authors.

Figure 1
Figure 1. Visualization of the true and reconstructed components of the initial electric field [PITH_FULL_IMAGE:figures/full_fig_p019_1.png] view at source ↗
Figure 2
Figure 2. Visualization of the true and reconstructed components of the initial electric field [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗
Figure 3
Figure 3. Visualization of the true and reconstructed components of the initial electric field [PITH_FULL_IMAGE:figures/full_fig_p023_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Carleman--Picard and time-dimensional reduction for inverse initial-data problems in nonlinear transport with memory

    math.NA 2026-08 conditional novelty 5.0 of 10

    A Carleman-Picard iteration with Legendre-exponential time reduction globally converges, within a truncated reduced model, for reconstructing initial data of quasilinear transport with memory from outflow measurements.

Reference graph

Works this paper leans on

37 extracted references · 36 canonical work pages · cited by 1 Pith paper

  1. [37]

    D. D. Trong, C. V. Le, K. D. Luu, and L. H. Nguyen. Recovery of initial displacement and velocity in anisotropic elastic systems by the time dimensional reduction method. Preprint arXiv:2506.13000, 2025. 27

    work page Pith review arXiv 2025

  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

    work page 1984

  3. [2]

    Albanese and P

    R. Albanese and P. B. Monk. The inverse source problem for maxwell’s equations. Inverse Problems, 22(3):1023–1035, 2006

    work page 2006

  4. [3]

    Ammari, G

    H. Ammari, G. Bao, and J. L. Fleming. An inverse source problem for maxwell’s equations in magnetoencephalography. SIAM Journal on Applied Mathematics , 62(4):1369–1382, 2002

    work page 2002

  5. [4]

    Ammari, E

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

    work page 2011

  6. [5]

    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

    work page 2012

  7. [6]

    G. Bao, P. Li, and Y. Zhao. Stability for the inverse source problems in elastic and electro- magnetic waves. Journal de Math´ ematiques Pures et Appliqu´ ees, 134:122–178, 2020

    work page 2020

  8. [7]

    Beilina and M

    L. Beilina and M. V. Klibanov. Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems. Springer, New York, 2012

    work page 2012

Show all 37 references
  1. [8]

    Benoit, C

    J. Benoit, C. Chauvi` ere, and P. Bonnet. Time-dependent current source identification for numerical simulations of maxwell’s equations. Journal of Computational Physics , 289:116–128, 2015

    2015

  2. [9]

    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

  3. [10]

    Clason and M

    C. Clason and M. V. Klibanov. The quasi-reversibility method for thermoacoustic tomography in a heterogeneous medium. SIAM J. Sci. Comput. , 30:1–23, 2007

    2007

  4. [11]

    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

  5. [12]

    Griesmaier, R

    R. Griesmaier, R. K. Mishra, and C. Schmiedecke. Inverse source problems for maxwell’s equations and the windowed fourier transform. SIAM Journal on Scientific Computing , 40(3):A1204–A1223, 2018

    2018

  6. [13]

    Haltmeier

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

    2013

  7. [14]

    Haltmeier and L

    M. Haltmeier and L. V. Nguyen. Reconstruction algorithms for photoacoustic tomography in heterogeneous damping media. Journal of Mathematical Imaging and Vision , 61:1007–1021, 2019

    2019

  8. [15]

    Harris, T

    I. Harris, T. Le, and D-L Nguyen. A direct reconstruction method for radiating sources in Maxwell’s equations with single frequency data. Inverse Problems., 41:015003, 2024. 25

    2024

  9. [16]

    A. Homan. Multi-wave imaging in attenuating media. Inverse Probl. Imaging , 7:1235–1250, 2013

    2013

  10. [17]

    Hristova

    Y. Hristova. Time reversal in thermoacoustic tomography–an error estimate. Inverse Problems, 25:055008, 2009

    2009

  11. [18]

    Hristova, P

    Y. Hristova, P. Kuchment, and L. V. Nguyen. Reconstruction and time reversal in thermoa- coustic tomography in acoustically homogeneous and inhomogeneous media. Inverse Problems, 24:055006, 2008

    2008

  12. [19]

    Huang, K

    C. Huang, K. Wang, L. Nie, L. V. Wang, and M. A. Anastasio. Full-wave iterative image reconstruction in photoacoustic tomography with acoustically inhomogeneous media. IEEE Trans. Med. Imaging, 32:1097–1110, 2013

    2013

  13. [20]

    Isakov and J-N

    V. Isakov and J-N. Wang. Uniqueness and increasing stability in electromagnetic inverse source problems. Journal of Differential Equations , 283:110–135, 2021

    2021

  14. [21]

    T. Kang, K. Van Bockstal, and R. Wang. The reconstruction of a time-dependent source from a surface measurement for full maxwell’s equations by means of the potential field method. Computers & Mathematics with Applications , 75:764–786, 2018

    2018

  15. [22]

    Katsnelson and L

    V. Katsnelson and L. V. Nguyen. On the convergence of time reversal method for thermoa- coustic tomography in elastic media. Applied Mathematics Letters , 77:79–86, 2018

    2018

  16. [23]

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

    2017

  17. [24]

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

    2014

  18. [25]

    Kowar and O

    R. Kowar and O. Scherzer. Photoacoustic imaging taking into account attenuation. In H. Am- mari, editor, Mathematics and Algorithms in Tomography II, Lecture Notes in Mathematics , pages 85–130. Springer, 2012

    2012

  19. [26]

    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

  20. [27]

    Li and M

    S. Li and M. Yamamoto. An inverse source problem for maxwell’s equations in anisotropic media. Applicable Analysis, 84:1051–1067, 2005

    2005

  21. [28]

    A. I. Nachman, J. F. Smith III, and R.C. Waag. An equation for acoustic propagation in inhomogeneous media with relaxation losses. J. Acoust. Soc. Am. , 88:1584–1595, 1990

    1990

  22. [29]

    Natterer

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

    2012

  23. [30]

    L. V. Nguyen. A family of inversion formulas in thermoacoustic tomography. Inverse Probl. Imaging, 3:649–675, 2009. 26

    2009

  24. [31]

    Paltauf, R

    G. Paltauf, R. Nuster, M. Haltmeier, and P. Burgholzer. Experimental evaluation of recon- struction algorithms for limited view photoacoustic tomography with line detectors. Inverse Problems, 23:S81–S94, 2007

    2007

  25. [32]

    Paltauf, J

    G. Paltauf, J. A. Viator, S. A. Prahl, and S. L. Jacques. Iterative reconstruction algorithm for optoacoustic imaging. J. Opt. Soc. Am. , 112:1536–1544, 2002

    2002

  26. [33]

    V. G. Romanov and S. I. Kabanikhin. Inverse Problems for Maxwell’s Equations . Inverse and Ill-Posed Problems Series. VSP, Utrecht, 1994

    1994

  27. [34]

    Slodicka and M

    M. Slodicka and M. Galba. Recovery of a time dependent source from a surface measurement in maxwell’s equations. Computers & Mathematics with Applications , 71:368–380, 2016

    2016

  28. [35]

    Stefanov and G

    P. Stefanov and G. Uhlmann. Thermoacoustic tomography with variable sound speed. Inverse Problems, 25:075011, 2009

    2009

  29. [36]

    Stefanov and G

    P. Stefanov and G. Uhlmann. Thermoacoustic tomography arising in brain imaging. Inverse Problems, 27:045004, 2011

    2011

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.