Stability, Partial Data, and Reconstruction Algorithms

Qualitative uniqueness answers whether exact infinite-dimensional data identify an unknown object modulo gauge. A usable reconstruction needs more: partial observations must see the target region, small data errors must cause controlled reconstruction errors, and a finite procedure must approximate the inverse. These are logically independent achievements. Nonlinear interaction methods have dramatically expanded uniqueness, while full metric stability and robust computation remain much less complete.

This assessment uses versions checked through 2026-08-12. It treats a withdrawal or proof correction as evidence, not as a footnote to be silently ignored.

Required background. Analysis for Mathematical Relativity supplies conditional stability and microlocal norms; PDEs and Green Functions for Gravitation supplies response operators; and Lorentzian metrics and signature supplies causal visibility and gauge.

Helpful background. The Einstein equation supplies constrained geometric unknowns, and Causality and Global Structure supplies accessible diamonds and trapped regions.

Three different upgrades

Let $\mathcal M(g)$ denote an observation operator and let $d_{\mathcal G}$ be a distance on metrics after minimizing over admissible boundary-fixing diffeomorphisms. The following statements are distinct:

$$\begin{aligned} \mathcal M(g_1)=\mathcal M(g_2) &\Longrightarrow d_{\mathcal G}(g_1,g_2)=0, &&\text{uniqueness},\\ d_{\mathcal G}(g_1,g_2) &\le C\,\lVert\mathcal M(g_1)-\mathcal M(g_2)\rVert^\alpha, &&\text{Hölder stability},\\ g_h&=\mathcal A_h(\mathcal M_\delta) \longrightarrow[g], &&\text{convergent reconstruction}. \end{aligned}$$

Here $C$, $\alpha$, the data norm, admissible class, noise model, discretization $h$, and gauge selection all matter. Logarithmic stability is weaker than every positive Hölder exponent; Lipschitz stability is stronger. An explicit proof of uniqueness may be “constructive” while giving neither a computable rate nor a noise-robust finite algorithm.

Reconstruction results through 2026-08-12

The formulation axes are the spacetime dimension and topology, asymptotic end, matter or scalar nonlinearity, source and receiver data class, regularity class, diffeomorphism or conformal gauge, boundary condition, observation norm, and exact conclusion.

Result family	Unknown and data	Conclusion through 2026-08-12	Status and limitation
Four-wave interaction in a local observation region	Conformal geometry in the causally reachable-and-returnable diamond; exact nonlinear source-to-solution map	Uniqueness modulo diffeomorphism and conformal gauge	Peer-reviewed theorem; exact operator and qualitative microlocal detection, not a noise estimate
Einstein–scalar active measurements	Smooth globally hyperbolic background; four independent scalar gradients near the source; conservation-compatible adaptive sources and source-dependent Fermi data	Conformal geometry of the diamond; full metric on a Ricci-flat target whose points connect internally to the apparatus	Peer-reviewed theorem; nondegeneracy, adaptive-source, conservation, and connectivity hypotheses; no general quantitative metric stability
Semilinear potential stability	Known $(n+1)$-dimensional globally hyperbolic metric, integer $m\ge4$, bounded $q\in C^{s+1}$ with $s+1>(n+1)/2$; full lateral DN map for small inputs	Hölder $L^\infty(W)$ recovery on compact $W\Subset I^-(\Sigma)\cap I^+(\Sigma)\cap([0,T]\times\Omega)$	Peer-reviewed 2025 theorem; coefficient on known geometry, not simultaneous metric recovery
Observation along one timelike curve	Conformal geometry; nonlinear source-to-solution response restricted to a curve	Qualitative conformal determination	Peer-reviewed 2025 theorem; severe partial receiver geometry but exact continuum data
Nonlinear scattering functionals	Semilinear equation on an open globally hyperbolic manifold with at least one asymptotically Minkowskian end	Topology, smooth structure, conformal type, and metric/nonlinearity up to multiplicative transformation are claimed	Alexakis–Isozaki–Lassas–Tyni arXiv:2411.09354v2; current preprint theorem claim, not a stability algorithm
Linear Lorentzian DN reconstruction proposed in 2023	Metric from the linear DN map	Claimed topological, smooth, and conformal recovery	arXiv:2303.08430 was withdrawn with “There is an error in the proof”; it is not positive theorem evidence

The Einstein–scalar entry uses small source–solution triples, not arbitrary stress-energy control. Its conformal conclusion is Theorem 1.1; the Ricci-flat metric upgrade and its internal connectivity condition belong to Corollary 1.2 Kurylev, Lassas, Oksanen, and Uhlmann 2022, Theorem 1.1, Corollary 1.2, and §§1.2–1.3, 5.

The Hölder result is equally precise. Lassas, Liimatainen, Potenciano-Machado, and Tyni work in dimension $n+1$ with $n\ge2$, take integer $m\ge4$ and uniformly bounded $q_j\in C^{s+1}$ with $s+1>(n+1)/2$, and compare the full lateral DN maps on a small Sobolev ball. They obtain an $L^\infty(W)$ Hölder estimate on a prescribed compact $W\Subset I^-(\Sigma)\cap I^+(\Sigma)\cap([0,T]\times\Omega)$, without assuming a convex boundary or that pairs of null geodesics meet only once Lassas et al. 2025, Theorem 1 and §§1.1, 4–6. The constants and exponent depend on the a priori class. This does not imply a Hölder estimate for an unknown Lorentzian metric.

At the partial-data extreme, Nursultanov, Oksanen, and Tzou show that nonlinear observations along one timelike curve determine conformal structure in the stated accessible region Nursultanov, Oksanen, and Tzou 2025, Theorem 1.1 and §§3–5. This is a uniqueness theorem from a thin receiver set, not a claim that finitely sampled noisy traces are well conditioned.

The current scattering preprint introduces nonlinear scattering functionals that remain defined even when some incoming waves lead to blowup; it claims recovery on asymptotically Minkowskian manifolds with nontrivial topology or several ends Alexakis et al. 2025, Theorems 1–3 and §§1.3–1.6, 3–5. Its formulation, equation, asymptotic end, multiplicative gauge, and preprint status must travel with the claim.

Partial-data visibility

Partial data determine only what controlled waves can reach and return from. For a source set $S$ and receiver set $R$, the basic candidate region is

$$\mathcal V(S,R)=J^+(S)\cap J^-(R),$$

refined by transversality, absence or management of caustics, and symbol nonvanishing. Points outside $\mathcal V$ are exactly invisible during the measurement interval. Inside it, trapping or missing angular directions can leave a microlocal nullspace.

Nonlinear interactions improve illumination: several visible incoming rays can meet at a point and create a new outgoing cone. They do not violate causal coverage. If the new cone never reaches $R$, the interaction is still unobserved. Nor do they automatically stabilize grazing intersections, nearly conjugate rays, or symbols close to cancellation.

A partial-data theorem should therefore state source and receiver subsets, time interval, accessible region, required ray configurations, coefficient or metric class, and boundary-fixing gauge. “Local data” alone is too vague to compare results.

From proofs to algorithms

A conceptual reconstruction pipeline would:

1. choose source packets approximating conormal or Gaussian-beam waves;
2. estimate mixed response derivatives by repeated amplitude experiments;
3. separate propagated and interaction singularities in phase space;
4. infer earliest light observation sets and assemble a recovered atlas;
5. reconstruct null cones, fix a gauge, and infer the conformal factor from amplitudes, time data, or field equations;
6. regularize and quantify uncertainty after discretization.

Several steps can be ill-conditioned. Finite differences amplify noise at high derivative order. Wave packets have nonzero width. Caustics mix branches. Atlas gluing accumulates error. Gauge optimization can introduce a nonconvex alignment problem. No general theorem currently composes these steps into an end-to-end, gauge-aware, convergent algorithm for an unknown Lorentzian metric from finite noisy nonlinear data.

For a concrete noise calculation, approximate one fourth mixed derivative by the $2^4=16$-term forward difference with source step $h$. If each response has error at most $\delta$ in the chosen data norm, the worst-case noise contribution is at most $16\delta/h^4$. With a bounded fifth source derivative, the truncation error is $O(h)$. Balancing these two errors suggests $h\sim\delta^{1/5}$ and an error of order $\delta^{1/5}$ for this particular derivative-estimation procedure. Decreasing $h$ indefinitely at fixed noise therefore makes the estimate worse. This elementary calculation is not a stability exponent for metric recovery; the subsequent geometric inversion can introduce further losses.

This does not make the uniqueness proofs nonconstructive or physically irrelevant. It identifies the open question accurately: prove stability for the geometric unknown in a useful quotient norm, then design a discretization whose convergence rate reflects that theorem. Claims of numerical success in special parametrized models would be a different evidence cell.

Why withdrawal tracking matters

The withdrawn linear-DN preprint is a discriminating counterexample to the practice of citing a title or v1 abstract without checking current status. The arXiv record now states that the paper is withdrawn because the proof has an error Enciso, Wrochna, and Uhlmann 2023, v2 status and comments. The proposed theorem may motivate future work, but as of the cutoff it cannot support a uniqueness claim. Establishing the proposed conclusion requires a corrected proof or an independent theorem with the same hypotheses.

Diagnostic: what does a stability estimate buy?

Suppose a theorem gives $\lVert q_1-q_2\rVert_{L^2}\le C\lVert\Lambda_{q_1}-\Lambda_{q_2}\rVert^\alpha$ for $0<\alpha<1$ on an a priori bounded coefficient class. Does it yield a stable metric-reconstruction algorithm?

Solution

No. It controls one coefficient on a known metric, in the theorem’s data norm and admissible class. Simultaneous metric recovery has diffeomorphism and often conformal gauges, so it needs a quotient metric and a separate estimate. Even for $q$, the inequality gives conditional continuity but not by itself a discretization, an optimizer, or a rule relating measurement noise to mesh size. It is an essential ingredient of an algorithmic theorem, not the whole theorem.

References

• Alexakis, Spyros, Hiroshi Isozaki, Matti Lassas, and Teemu Tyni. “Inverse Scattering Problems for Non-linear Wave Equations on Lorentzian Manifolds.” arXiv:2411.09354v2 (2025). arXiv.
• Enciso, Alberto, Michał Wrochna, and Gunther Uhlmann. “Reconstruction of a Lorentzian Manifold from Its Dirichlet-to-Neumann Map.” arXiv:2303.08430v2, withdrawn for a proof error (2023). arXiv record.
• Kurylev, Yaroslav, Matti Lassas, Lauri Oksanen, and Gunther Uhlmann. “Inverse Problem for Einstein–Scalar Field Equations.” Duke Mathematical Journal 171 (2022): 3215–3282. DOI.
• Lassas, Matti, Tony Liimatainen, Leyter Potenciano-Machado, and Teemu Tyni. “Stability Estimates for Inverse Problems for Semi-linear Wave Equations on Lorentzian Manifolds.” Analysis & PDE 18 (2025): 1065–1118. DOI.
• Nursultanov, Medet, Lauri Oksanen, and Leo Tzou. “Determining Lorentzian Manifold from Non-linear Wave Observation at a Single Point.” Journal of Differential Equations 440 (2025): 113465. arXiv.