Simultaneous Recovery of the Initial Source and Sound Speed for the Wave Equation under a Constitutive Constraint

Amir Moradifam

REVIEW 1 major objections 4 minor 30 references

Simultaneous Recovery of the Initial Source and Sound Speed for the Wave Equation under a Constitutive Constraint

T0 review · 1 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read A known material link between source and sound speed makes simultaneous recovery of both from one boundary measurement stable.

desk verdict A clean reduction of a coupled source-speed problem to an inverse source problem under a constitutive constraint, with the main unresolved step being a uniformity claim in the full-boundary stability estimate. read the letter →

arxiv 2607.19799 v1 pith:YC3CAFA5 submitted 2026-07-22 math.AP

Amir Moradifam This is my paper · ORCID

classification math.AP MSC 35R3035L05

keywords inversewaveproblemsimultaneousrecoveryinitialsourcesoundspeedconstitutiveconstraintLipschitzstabilitysingleboundarymeasurementphotoacoustictomography

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 if the initial source and the sound speed are coupled by a known constitutive relation f(x)=F(x,c(x)) whose derivative with respect to speed is bounded away from zero, then one boundary measurement of the wave field determines both unknowns simultaneously, with Lipschitz stability. This matters because recovering either quantity alone is classical, while simultaneous recovery was previously open in general and known to be unstable at the linearized level. The mechanism is a reduction: the two-unknown problem is rewritten as a single inverse source problem for the squared-speed difference, with a time-dependent coefficient that is uniformly nonzero at t=0. Under strictly convex foliation and geodesic visibility conditions, global uniqueness and stability follow; a local version recovers both quantities in the swept visible region from partial boundary data.

What carries the argument

The core device is the second time primitive of the wave difference, V(t,x)=∫_0^t (t−s)(u_{c1}−u_{c2})(s,x) ds. Lemma 2.2 shows that V satisfies P_{c1}V = α(x)A(t,x) with α = c1^2 − c2^2, where A(t,x)=B(x)+∫_0^t (t−s)Δu_{c2}(s,x) ds and B is the divided difference (F(x,c1)−F(x,c2))/(c1^2−c2^2). Nondegeneracy forces |B(x)| ≥ m0/(2c+), so A(0,x) is uniformly nonzero. Together with the trace identity ∂_t^2V|_{∂Ω} = Λ_{f1,c1} − Λ_{f2,c2}, this converts the two-unknown comparison into the one-unknown inverse source problem that microlocal and Carleman source-recovery methods already solve.

What would settle it

Run a numerical experiment in a domain satisfying the stated convex-foliation and visibility conditions: generate two admissible pairs (c1,f1=F(·,c1)) and (c2,f2=F(·,c2)) with |∂_rF| ≥ m0 > 0 over the admissible speed range, and compute the ratio (‖c1−c2‖_{L2(K)}+‖f1−f2‖_{L2(K)}) / ‖Λ_{f1,c1}−Λ_{f2,c2}‖_{L2((0,T)×∂Ω)} over a bounded smooth admissible class. If the ratio is unbounded, the Lipschitz claim is false. Alternatively, test necessity: choose F with ∂_rF = 0 on an open set, search for two distinct positive speeds in the admissible range with equal initial sources and equal boundary tra

Watch

Extended reading notes

Core claim

The paper establishes that the simultaneous inverse problem becomes well-posed when f and c are linked by a prescribed smooth constitutive law F with |∂_rF(x,r)| ≥ m0 > 0 on the recovery set. In that case, the difference of two candidate wave fields, after one time integration, satisfies a single-source equation whose spatial factor is the squared-speed difference α = c1^2 − c2^2 and whose coefficient A(t,x) satisfies |A(0,x)| ≥ m0/(2c+) > 0. Equality of full boundary measurements then implies c1 = c2 and f1 = f2 under a convex-foliation condition, and the estimate ‖c1−c2‖_{L2(K)} + ‖f1−f2‖_{L2(K)} ≤ C‖Λ_{f1,c1} − Λ_{f2,c2}‖_{L2((0,T)×∂Ω)} holds under an added visibility condition. The same

Load-bearing premise

The whole argument rests on the material law f=F(x,c) being exactly known, smooth, and having a speed-derivative bounded away from zero in the region where recovery is sought; if the law is misspecified or its derivative vanishes locally, the reduction to a single inverse source problem fails and the known instability of simultaneous recovery returns.

Editorial extensions

If this is right

When a nondegenerate constitutive relation holds, simultaneous recovery of the initial source and sound speed from a single boundary measurement is unique and Lipschitz stable in the full-data geometry.
The stability estimate uses the measured boundary trace difference itself, with no differentiation of the data lost.
The partial-data version shows that the constitutive law needs to be nondegenerate only in a neighborhood of the swept visible region, not in the whole medium.
In the thermoelastic calibration example f(x)=ρ(x)ε_T(x)c(x)^2, nondegeneracy holds wherever ρ ε_T ≥ q0 > 0 and c ≥ c− > 0, so the theorem applies to that concrete material class.
Local partial-data uniqueness and stability hold in compact visible regions satisfying the stated convex foliation, cone, and microlocal visibility conditions.

Reading between the lines

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

The reduction suggests a design principle for imaging modalities: calibrate source amplitude and wave speed to a common scalar material parameter, such as cure degree or thermoelastic stress; the nondegeneracy condition is then an experimentally testable slope condition on the calibrated response curves.
The method converts a known instability into a stable problem by restricting to a submanifold of admissible pairs; the same strategy may apply to other coupled parameter pairs, such as density and source or attenuation and speed, whenever a similar monotone calibration is available.
The divided-difference lower bound |B| ≥ m0/(2c+) implies that stability degrades as m0 approaches zero; numerical experiments sweeping a family of constitutive laws toward degeneracy could test this predicted blow-up of the constant.
The local theorem implies that a mis-calibrated constitutive law in one region does not spoil recovery elsewhere, as long as nondegeneracy holds along the propagation path from the observed boundary to the target—relevant for multi-material samples.

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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,

This paper does something genuinely new: it shows that if the initial source and the sound speed are tied by a prescribed constitutive law f(x) = F(x, c(x)) with |∂_r F| ≥ m0, then the simultaneous recovery problem collapses to a single inverse source problem. The reduction is elegant. The time-primitive lemma (Lemma 2.2) is the key trick, and it works: the difference of two solutions is transformed into a source problem with a coefficient A(t,x) that is bounded below at t=0. Lemmas 2.1–2.3 check out algebraically. The uniqueness theorems (2.4, 2.5) then follow directly from Stefanov–Uhlmann, and the geometric hypotheses are the standard foliation and visibility conditions. The partial-data version (Theorem 3.1) is a nice local formulation: the constitutive law only needs to be non-degenerate along the swept recoverable region, not globally. The motivating examples (cure monitoring, thermoelastic generation) are plausible and make the strong structural assumption less artificial.

The main soft spot is the stability estimate, Theorem 2.6. The reduced equation is P_{c1} V = α A_{c2}, where A depends on c2 through u_{c2}. The proof asserts that the constant in the cited source-recovery theorem [26, Thm 3.4] is uniform over the family A_{c2}, but it does not demonstrate this. The class C(E,T,b,M) is defined precisely to capture the needed uniform bounds, so the claim is plausible, but the verification is missing. Notice that the partial-data analogue (Proposition 3.3) contains exactly the compactness and continuity argument needed to make the constant uniform, but that argument is not carried out for the full-boundary case. This is a gap in the written proof, not a fatal flaw. A referee should ask the author to fill it in or point to a version of [26] that already proves uniformity over such a class. A second, minor issue is that Proposition 3.3 is fairly sketched, leaning heavily on [24] and [26]; that is acceptable for a specialist journal but not fully self-contained.

Overall, the central mechanism is correct and the paper is a real advance for a class of material-constrained inverse problems. The constitutive assumption is strong but explicitly stated, and the paper does not oversell its practical reach. It deserves a serious referee; with the stability uniformity argument patched, it would be a solid publication.

Recommendation: engage with it. Send it to a referee who knows the Stefanov–Uhlmann machinery and ask specifically about the uniformity step.

Referee Report

1 major / 4 minor

Summary. The paper considers simultaneous recovery of the sound speed c and the initial source f for the scalar wave equation from one boundary measurement, under the constitutive constraint f(x)=F(x,c(x)). The main idea is to reduce the coupled problem to a single inverse source problem: subtracting the two wave equations and taking one time primitive gives a source equation P_{c1}V = α(x)A(t,x) with V initially zero, where α=c_1^2-c_2^2 and A(0,x) satisfies the lower bound |A(0,x)|≥m_0/(2c_+) under the nondegeneracy condition |∂_r F|≥m_0. The authors then apply Stefanov--Uhlmann source-recovery theorems to obtain global uniqueness (Theorem 2.5) and Lipschitz stability (Theorem 2.6) under convex-foliation and visibility conditions, as well as partial-data analogues (Theorems 3.1 and 3.4). The paper is clearly written and the reduction in Lemmas 2.1--2.3 is internally consistent.

Significance. If the results hold, they give a positive structural answer to a simultaneous recovery problem that is known to be unstable in general (Stefanov--Uhlmann [27]). The constitutive-reduction idea is elegant and potentially transferable to other two-parameter inverse problems. The paper explicitly identifies the nondegeneracy condition as the mechanism restoring well-posedness, and carefully separates the geometric assumptions inherited from the source-recovery framework from the new constitutive assumption. The partial-data localization in Section 3 is a useful contribution, as it allows the constitutive law to degenerate outside the recoverable region. However, the central stability claim in Theorem 2.6 relies on an unverified uniformity assertion, which currently prevents the main theorem from being fully established as written.

major comments (1)

[Theorem 2.6, Eq. (2.16)] The Lipschitz stability estimate requires the constant in the source-recovery theorem [26, Thm 3.4] to be uniform over the family A_{c2}(t,x)=B(x)+∫_0^t(t-s)Δu_{c2}(s,x)ds as c2 varies in the admissible class. The proof asserts this uniformity in one sentence ('where C is uniform as c2 varies in the admissible class') without verifying that the constant in [26] depends only on the C^2 bounds of A and the lower bound |A(0)|≥b0. Since A_{c2} contains Δu_{c2}, the dependence is nontrivial and not covered by the class C(K,T,b,M) defined in Section 2.1, which controls only time regularity and |a(0)|. If the constant in [26] depends on finer data of A, Theorem 2.6 does not follow from the cited result as written. The authors should quote the exact uniformity statement in [26] or prove it by adapting the finite-covering/C^2-precompactness argument used in Proposition 3.3. This issue is load-bea

minor comments (4)

[Section 2.1, Lemma 2.2] The definition of the class C(E,T,b,M) uses C^2([0,T]; C(Ω)); it may be clearer to state explicitly the spatial regularity of the coefficient a, since A(t,x) contains Δu_{c2} and the source stability estimates in [26] require spatial smoothness assumptions. The current formulation only specifies time regularity.
[Throughout] There are several typographical artifacts, e.g., 'w(s, x), ds' and 'Z t 0' in Lemmas 2.2 and 2.3; these should be cleaned up.
[Theorem 3.1 and Definition (3.1)] The notation DΓ versus D_Γ is inconsistent; also the definition of τ(y) and the cone condition could be restated more explicitly to avoid ambiguity about the initial time slice.
[References] Reference [24] is cited for both the source-stability argument and the exterior cone condition; the precise statements used in Proposition 3.3 would be easier to verify if the citation included theorem numbers.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the constitutive reduction is a structural assumption and the inverse-source theorems are external.

full rationale

The paper's central reduction is algebraic and non-circular. Lemma 2.1 factors the difference of the constitutive sources as F(x,c1)-F(x,c2) = (c1^2-c2^2) B(x), where B is the divided difference of F; the lower bound |B| >= m0/(2c+) is exactly the explicit non-degeneracy hypothesis (1.3), not a fitted or hidden input. Lemma 2.2 then derives an inverse-source equation P_{c1} V = α A for the time-primitive V, with α = c1^2-c2^2 and A(0)=B, and Lemma 2.3 connects the boundary data to ∂^2_t V. These steps are identities following from the assumed constitutive law; they do not presuppose c1=c2 or f1=f2. The uniqueness and stability theorems (Theorems 2.4-2.6, 3.1, 3.4) invoke the external, independent source-recovery results of Stefanov and Uhlmann [26] (and [24] in Proposition 3.3), whose hypotheses do not include the target equality. The only self-citation, [15], appears in the introduction as background context and is not load-bearing in any proof. The uniformity of the stability constant over the c2-dependent coefficient A is asserted briefly in Theorem 2.6; this is a potential proof gap but not a circularity, because the verification given uses only a priori admissible-class bounds, smoothness, A_t(0)=0, and |A(0)| >= b0, none of which encode the desired conclusion. No fitted parameter is renamed as a prediction and no load-bearing claim reduces to its own input.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new physical entities. Its main load-bearing assumptions are the constitutive relation and the inherited S-U geometric and regularity hypotheses; no constants are fitted to data.

assumptions (4)

domain assumption The constitutive law f=F(x,c) is known a priori and C∞, and |∂_r F| ≥ m0 > 0 on K×[c−,c+].
Introduced in (1.2)-(1.3); it is the central modeling premise. If F is unknown or degenerate, the reduction fails and instability (Stefanov-Uhlmann [27]) returns.
domain assumption Speeds lie in a C∞ admissible class with c−≤c≤c+, c≡1 outside Ω, and are C^N bounded; supp(c1−c2)⊂K.
Section 2.1; needed for uniform constants and exterior normalization.
domain assumption Geometric hypotheses: strictly convex foliations (full or partial), geodesic visibility, and sufficiently large T; for partial data, the cone condition.
Required by Theorems 2.3/3.2/3.4 of Stefanov-Uhlmann [26] used as black boxes.
standard math Carleman estimates and microlocal parametrix results of [24] and [26] hold as stated.
The paper does not reprove these; they are external published theorems.

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Cite this review

Pith. "Pith review of Simultaneous Recovery of the Initial Source and Sound Speed for the Wave Equation under a Constitutive Constraint." pith.science (2026). https://pith.science/paper/YC3CAFA5

@misc{pith2026260719799,
  author       = {Pith},
  title        = {Pith review of: Simultaneous Recovery of the Initial Source and Sound Speed for the Wave Equation under a Constitutive Constraint},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YC3CAFA5}},
  note         = {Machine review of arXiv:2607.19799}
}

read the original abstract

We study the simultaneous recovery of the initial source and sound speed for the scalar wave equation from a single boundary measurement. Although the recovery of either parameter separately is well understood under suitable geometric hypotheses, simultaneous recovery remains open in general, and stable recovery is further obstructed by the inherent instability of the linearized problem. We show that both uniqueness and stability can be obtained when the two unknowns are coupled through a prescribed constitutive relation arising from a common underlying material. Under a quantitative nondegeneracy condition, motivated by calibrated material regimes, the coupled inverse problem reduces to a single inverse source problem. Applying the microlocal and Carleman framework of Stefanov and Uhlmann to this reduced problem, we establish global uniqueness and Lipschitz stability under geometric conditions involving strictly convex foliations and geodesic visibility. We also obtain partial-data results, proving local uniqueness and Lipschitz stability in compact visible regions where the constitutive nondegeneracy condition holds.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [26]

Stefanov and G

P. Stefanov and G. Uhlmann, Recovery of a source term or a speed with one measurement and applications,Transactions of the American Mathematical Society, 365 (2013), no. 11, 5737–5758

2013

[2] [24]

Stefanov and G

P. Stefanov and G. Uhlmann, Thermoacoustic tomography with variable sound speed,Inverse Problems, 25 (2009), no. 7, 075011

2009

[3] [27]

Stefanov and G

P. Stefanov and G. Uhlmann, Instability of the linearized problem in multiwave tomography of recovery both the source and the speed,Inverse Problems and Imaging, 7 (2013), no. 4, 1367–1377

2013

[4] [1]

J. D. Achenbach, The thermoelasticity of laser-based ultrasonics,Journal of Thermal Stresses, 28 (2005), no. 6–7, pp. 713–727

2005

[5] [2]

Agranovsky, P

M. Agranovsky, P. Kuchment, and L. Kunyansky, On reconstruction formulas and algorithms for the thermoacoustic tomography, inPhotoacoustic Imaging and Spectroscopy, L. V. Wang, ed., CRC Press, Boca Raton, FL, 2009, pp. 89–102

2009

[6] [3]

Ammari, E

H. Ammari, E. Bossy, V. Jugnon, and H. Kang, Mathematical modeling in photoacoustic imaging of small absorbers,SIAM Review, 52 (2010), no. 4, pp. 677–695. 16

2010

[7] [4]

A. Elia, P. M. Lugar` a, C. Di Franco, and V. Spagnolo, Photoacoustic techniques for trace gas sensing based on semiconductor laser sources,Sensors, 9 (2009), no. 12, 9616–9628

2009

[8] [5]

Finch and K

D. Finch and K. S. Hickmann, Transmission eigenvalues and thermoacoustic tomography, Inverse Problems, 29 (2013), no. 10, 104016

2013