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Uniqueness and stability in determining the wave equation from a single passive boundary measurement

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Pith's one-line read From a single passive boundary recording, the paper proves that a piecewise constant wave speed and the initial source can be uniquely recovered in 3-D, with Hölder stability in the one-inclusion case.

desk verdict The low-frequency continuation idea is real and the equal-radius theorem looks right; but Theorem 2.3's proof assumes all inclusions are slower than background, which the theorem doesn't state, so the advertised generality for inclusions with holes isn't established. read the letter →

arxiv 2507.10012 v3 pith:WLJNRIXY submitted 2025-07-14 math.AP

Yavar Kian , Hongyu Liu This is my paper · ORCID
classification math.AP MSC 35R3035L05
keywords inverseboundaryproblemspassivemeasurementwaveequationphotoacoustictomographysimultaneousrecoveryuniquenessHölderstability
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 tackles the inverse problem of recovering two unknowns at once—the sound speed of a medium and the initial disturbance (source) that generates a wave—from a single recording of the wave on the boundary of a region in three dimensions. Its main uniqueness theorem says that if two admissible configurations, each consisting of a piecewise constant speed with disjoint ball inclusions of equal radius and an $H^1$ initial datum, produce exactly the same boundary trace, then the two speeds and the two sources are identical. A companion theorem for a single slower inclusion proves Hölder stability: small differences in the boundary measurement force small differences in the speed and source, in explicit powers. The authors emphasize that, unlike earlier work, no time-decay of the solution is required, so sources need not decay and piecewise constant speeds are allowed when they stay below an explicit multiple of the background speed. If correct, the results show that passive boundary listening alone is information-theoretically sufficient for simultaneous medium and source recovery in photoacoustic and thermoacoustic tomography.

What carries the argument

The load-bearing object is the analytic extension of the Laplace transform in time, $p \mapsto \hat{u}(p,\cdot)$, from the right half-plane to a fixed disk around $p = 0$. This is built from the cutoff resolvent $\chi_R(L + p^2)^{-1}\chi_{R_0}$ of the elliptic operator $L = -c^2\Delta$; under the contrast bound the perturbation series in powers of $(c^2/b_0^2 - 1)$ converges and the resolvent is boundedly invertible near zero. Coefficients $u^{(k)}$ of the resulting Taylor series satisfy a ladder of elliptic equations, with $u^{(1)}$ determined by $c^{-2}f$ and $u^{(2)}$ equal to the constant $-(2\pi b_0)^{-1}\int f/c^2\, dx$. Plugging harmonic test functions into the equation for $u^{(4)}$ and using the mean value theorem turns the boundary equality into equality of normal derivatives of transposition-solutions of an elliptic point-source problem; the Appendix's recovery theorem then matches the point sources one-to-one, giving $c_1 = c_2$, and the earlier wave-speed determination theorem gives $f_1 = f_2$. For stability, an observability inequality for wave equations with piecewise constant coefficients transfers control of the boundary measurement back to the initial data and to the low-frequency boundary coefficients.

What would settle it

Run a numerical experiment with two admissible configurations: same background speed, one ball inclusion of radius $r$ and speed $b$ at center $x$, another with radius $r$ and speed $b'$ at center $x' \neq x$, and choose $H^1$ sources $f$, $f'$ satisfying $\int f/c^2\, dx = \int f'/c'^2\, dx \neq 0$; if a search finds such a pair whose boundary traces match to solver precision, Theorem 2.1 would be contradicted. A more targeted check of the weakest assumption is to test whether the analytic continuation used in Theorem 3.1 still exists when $\operatorname{ess\,sup} c = \sqrt{2}\, b_0$ exactly, since the invertibility proof needs a strict inequality.

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Extended reading notes

Core claim

On the authors' own terms, the discovery is that the formally determined inverse problem of recovering $(c,f)$ from $u|_{\mathbb{R}_+ \times \partial\Omega}$ is solvable for piecewise constant $c$ with unknown ball inclusions. Theorem 2.1 proves that for admissible pairs whose inclusions are disjoint balls of the same radius, $u_1 = u_2$ on $\mathbb{R}_+ \times \partial\Omega$ implies $c_1 = c_2$ and $f_1 = f_2$; Corollary 2.1 extends this to unequal radii when the ball centres satisfy a partial order, Theorem 2.2 to inclusions that are unions of balls with holes, and Theorem 2.3 to spherical shells. Theorem 2.4 converts uniqueness into quantitative stability for one inclusion with $b_1 < b_0$: the $L^q$ difference of the wave speeds is bounded by the boundary-data difference to the first power, and the $H^1$ difference of the initial data by a boundary-measurement term plus a power $(3+2s)/6$ term. Admissibility means the non-degeneracy condition $\int_{\mathbb{R}^3} f/c^2\, dx \neq 0$ and the contrast bound $\|1 - c^2/b_0^2\|_{L^\infty} < 1$, equivalently $\operatorname{ess\,sup} c < \sqrt{2}\, b_0$. The whole construction works without any decay-in-time assumption on the solution, which is the main relaxation relative to prior simultaneous-recovery results.

Load-bearing premise

The argument breaks if the unknown inclusion is faster than roughly 1.414 times the known background speed, because the contrast bound that makes the resolvent invertible at frequency zero would fail; a second load-bearing assumption is that the inclusions can be matched one-to-one by equal radii or a partial order on their centers, which the authors themselves call a fundamental obstruction.

Editorial extensions

If this is right

  • A single boundary time trace, measured passively over all $t > 0$, determines both the piecewise constant wave speed and the initial source for the ball-inclusion classes covered by Theorems 2.1–2.3.
  • No local-energy decay is needed, so the result applies to sources that do not die out over time, as long as $\int f/c^2\, dx \neq 0$ and the contrast bound is met.
  • In the single-inclusion case with slower speed, the inverse map is Hölder continuous: boundary-data noise of size $\varepsilon$ leads to error at most $C\varepsilon^{1/q}$ in the $L^q$ speed difference and $C\varepsilon^{(3+2s)/6}$ in the $H^1$ source difference, up to an additional boundary-measurement term.
  • The stability estimate gives a rigorous starting point for regularized iterative reconstruction algorithms such as Tikhonov-type schemes, which the paper states are deferred to future work.
  • The geometric conditions (equal radii or partial order on centers) are, in the authors' view, not merely technical but reflect an obstruction: without a way to label inclusions one-to-one, the point-source matching step can fail.

Reading between the lines

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

  • The contrast ceiling $\operatorname{ess\,sup} c < \sqrt{2}\, b_0$ is visibly the price of avoiding decay: if an inclusion is faster than this, the Neumann-series argument for the resolvent at $p = 0$ diverges, and a natural testable question is whether uniqueness actually fails there or only this proof does.
  • The equal-radius or partial-order matching probably transfers to any configuration where the point-source recovery can separate inclusions—for example, by distinct amplitudes or by additional harmonic test functions—so the same low-frequency machinery may cover polyhedral or smoother inclusions with a similar combinatorial condition.
  • Because the stability estimate bounds $c$ in $L^q$ and $f$ in $H^1$ separately for one inclusion, a numerical study comparing convergence rates of the $(3+2s)/6$ exponent against actual reconstruction errors would reveal whether the estimate is sharp or merely sufficient.
  • The same analytic-extension device may extend to other passive-measurement problems, such as elastic or electromagnetic wave equations, whenever the contrast operator is bounded by less than one in the relevant norm.
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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

2 major / 5 minor

Summary. The paper studies the inverse problem of simultaneously recovering the wave speed c and the initial data f in the wave equation on R^3 from a single passive boundary measurement of u on R_+ × ∂Ω. The main results are: Theorem 2.1, uniqueness for piecewise constant speeds with equal-radius disjoint ball inclusions and unknown centers; Corollary 2.1, a variant with a partial order on centers and unequal radii; Theorems 2.2 and 2.3, uniqueness for more general inclusions formed by unions of balls with holes; and Theorem 2.4, Hölder stability for a single inclusion with slower speed. The proofs combine an analytic continuation of the cutoff resolvent near frequency zero, an expansion of the Laplace transform of the solution, and a point-source identification theorem for elliptic equations. The paper is a substantial attempt to extend simultaneous recovery results beyond the monotonicity and decay assumptions of prior work.

Significance. If the main results are correct, the paper makes a significant contribution: it shows that a single passive boundary trace can determine both a piecewise constant sound speed and the initial source for natural coefficient classes, without assuming local energy decay. The Section 3 analytic continuation mechanism is a genuine novelty; the algebra in (3.4) checks out, and the reduction to point-source identification is elegant. The equal-radius and single-slower-inclusion results, Theorem 2.1 and Theorem 2.4, appear internally consistent and are not affected by the concerns below. However, the advertised generality for unions of balls with holes depends on Theorem 2.3, and the proof of that theorem contains load-bearing gaps that must be repaired.

major comments (2)
  1. [6, Eq. (6.3)] The proof of Theorem 2.3 asserts that λ_j^k = ((b_j^k)^{-2} − b_0^{-2})|B_r| > 0 and λ_j^{N_j+ℓ} = −((b_j^k)^{-2} − b_0^{-2})|B_s| < 0 for every inclusion. This requires (b_j^k)^{-2} − b_0^{-2} > 0, i.e. b_j^k < b_0. But Definition 2.1(2) and the equivalent bound (2.1) only require b_j^k < √2 b_0, so faster inclusions with b_0 < b_j^k < √2 b_0 are admissible. For those inclusions both displayed signs reverse, and for a mixture of slower and faster inclusions the outer and hole point-source coefficients are not separated by sign. The subsequent deduction that the permutation σ from Theorem 8.1 maps outer centers to outer centers and holes to holes is therefore unjustified, and the equalities c1 = c2 and f1 = f2 do not follow for the generality stated in Theorem 2.3. The authors should either add a hypothesis ruling out mixed contrasts (for example, all inclusions slower than the background) or replace the sign-separation step with a different argument.
  2. [6, after Eq. (6.3)] Even when all contrasts have one common sign, the step 'Combining this with (6.3), we deduce that σ1 = σ2' is not justified by the displayed equalities. The equalities in (6.2) identify the multiset of signed point sources, but they do not by themselves fix which hole center y_k is associated with which outer center x_k in each configuration. An argument using geometric information, such as disjointness of the outer balls B(x_j^k, r) together with B(y_j^k, s) ⊂ B(x_j^k, r), or distinctness of the contrasts, is needed; neither assumption is stated in Theorem 2.3. As written, two configurations that differ only by permuting the association (x_k, y_k) while preserving the same signed point-source multiset are not distinguished by the proof. This is a second independent gap in the proof of Theorem 2.3.
minor comments (5)
  1. [2.2, Theorem 2.4] There are apparent typos in the definition of M and m: 'b2_2' should likely be 'b2_1', and the expression min(b1_1, b2_2, b0, r1) uses r1 without explanation.
  2. [6, Eq. (6.3)] The factors '2|B_r|' and '2|B_s|' in (6.3) do not match the definitions of λ_j^k and λ_j^{N_j+ℓ} given just before (6.2); the displayed formula should be corrected unless the factor 2 is intentional.
  3. [5, Theorem 2.2] The statement should make explicit that the contrast b1 is the same for c1 and c2. With that reading, the cancellation in (5.1) is legitimate; if the two contrasts were allowed to differ, the cancellation would be invalid.
  4. [8.2, Theorem 8.1] In the definition of O_r, the reference to B_R0 is confusing because the domain in the theorem is an arbitrary C^2 domain O; the notation should be cleaned up.
  5. [2.2, Eqs. (2.13)-(2.14)] The notation ∂^{2k}_p ∂^ℓ_ν \hat u_j(p, ·)|_{p=0} is used before the analytic extension is introduced in the main text; a forward reference to Lemma 3.1 would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the new analytic-continuation and point-source-recovery chain is self-contained.

full rationale

The central new step—analytic extension of the cutoff resolvent near frequency zero under the contrast bound ||1-c^2/b0^2||_L∞ < 1—is proved in Theorem 3.1 by an explicit Neumann-series inversion; the admissibility bound is an input assumption, not a disguised form of the theorem's conclusion. Lemma 3.1 and Lemma 3.2 convert boundary equality into the harmonic-moment identity (3.15) using only the Laplace coefficients and a Green's representation from [26], with the nonzero constant B supplied by the stated non-degeneracy condition (1.3). The point-source identification used to match inclusions is proved in the paper's own Appendix as Theorem 8.1, extending [10,11]; it is not an imported black box. Once c1=c2 is established, the recovery of f is delegated to [18, Theorem 8.1], a prior published theorem, not to the present argument, so no circular reduction occurs. The stability proof applies [19, Proposition 4.1] as an external observability input after an explicit time rescaling under the stated hypothesis b1_1 < b0. The only substantive concern found is a correctness gap, not a circularity: in the proof of Theorem 2.3, equation (6.3) asserts λj_k > 0 and λj_{N_j+ℓ} < 0, which requires b_j_k < b0, whereas Definition 2.1 only enforces c < sqrt(2) b0 and thereby admits faster inclusions; without that extra sign hypothesis the center-matching permutation argument can fail. This does not make the derivation circular and does not affect Theorem 2.1, Corollary 2.1, or Theorem 2.4.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

No free parameters are fitted to data: the constants delta, m, M, and T are chosen for estimates, and B=integral f/c^2 dx is an input assumed nonzero. No new physical entities are introduced. The central claim rests on standard PDE background, including self-adjoint wave operators, resolvent kernels, unique continuation, and observability, plus the paper's domain assumptions. The internal proof errors in Sections 5 and 6 are not additional assumptions and are recorded in the red flags.

assumptions (8)
  • standard math For f in H1(R3) and g in L2(R3), the initial value problem (8.1) has a unique solution in C1([0,infinity);L2) intersect C([0,infinity);H1), with H2 regularity for H2 data.
    Proved in Appendix 8.1 using the self-adjoint operator L=-c^2 Delta on L2(c^-2 dx) and Lions-Magenes semigroup results [22, Chapter 3].
  • standard math The cutoff free resolvent chi_R (L0+p^2)^-1 chi_R0 with L0=-b0^2 Delta extends analytically in p to a neighborhood of p=0 by the strong Huygens principle.
    Invoked in the proof of Theorem 3.1 with reference to [29, Section 2.1].
  • standard math Elliptic unique continuation: if v in H2(Ur), Av=0 in connected Ur, and v=partial_nu_a v=0 on a nonempty open subset S of the boundary, then v=0 in Ur.
    Used in the proof of Theorem 8.1 to identify point sources from conormal boundary data.
  • standard math Observability inequality of [19, Proposition 4.1]: for the wave equation with piecewise constant c, one slower inclusion b1<b0, and T sufficiently large, the initial energy is controlled by boundary Neumann data plus forcing.
    Used as a black box in Section 7 to prove the stability estimate (2.14).
  • standard math Once c1=c2, equality of boundary traces u1=u2 on R+ times the boundary implies f1=f2, per [18, Theorem 8.1].
    Cited in Theorems 2.1 to 2.3 to finish the source recovery argument.
  • domain assumption Non-degeneracy condition (1.3): integral of f/c^2 over R3 is nonzero.
    Definition 2.1(1); used to divide by the nonzero constant B in Lemma 3.2 and in Section 7.
  • domain assumption Contrast bound: ess sup c < sqrt(2) b0, equivalently ||1 - c^2/b0^2||_{L^infty} < 1.
    Definition 2.1(2); required for the invertibility argument in Theorem 3.1 and for the entire low-frequency extension.
  • domain assumption Geometric restrictions: equal radii or a partial order for ball centers in Theorems 2.1 and Corollary 2.1; disjoint balls and nested holes in Theorems 2.2 and 2.3; for stability, one inclusion with b1<b0, C2 domain, and T>4R0 b0/m.
    These are the hypotheses under which the uniqueness and stability proofs operate; the paper itself calls the partial-order condition a fundamental obstruction.

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Pith. "Pith review of Uniqueness and stability in determining the wave equation from a single passive boundary measurement." pith.science (2026). https://pith.science/paper/WLJNRIXY

@misc{pith2026250710012,
  author       = {Pith},
  title        = {Pith review of: Uniqueness and stability in determining the wave equation from a single passive boundary measurement},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WLJNRIXY}},
  note         = {Machine review of arXiv:2507.10012}
}
abstract

This article addresses the inverse problem of simultaneously recovering both the wave speed coefficient and an unknown initial condition (acting as the source) for the multidimensional wave equation from a single passive boundary measurement. Specifically, we establish uniqueness and H\"{o}lder stability estimates for determining these parameters in the wave equation on $\mathbb{R}^3$, where only a single boundary measurement of the solution--generated by the unknown source--is available. Our work connects to thermoacoustic and photoacoustic tomography (TAT/PAT) for the physically relevant case of piecewise constant sound speeds. We significantly relax the stringent conditions previously required for resolving this problem, extending results to general classes of piecewise constant sound speeds over inclusions with unknown locations. Moreover, we do not require decay properties in time of solutions to the wave equation, which enables our study to accommodate a much broader class of unknown sources. The approach combines low frequency-domain solution representations with distinctive properties of elliptic and hyperbolic equations.

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Forward citations

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