arXiv subjects

Mikko Salo

Publications and source records attributed to Mikko Salo.

At least 19 recordsLinked to original sources

arXiv2026

Inverse Scattering for Single Photons in Quantum Optics

We study inverse problems for a time-harmonic one-photon model describing the interaction of a single photon with a medium of stationary two-level atoms. After time-harmonic reduction, the unknown compactly supported atomic density appears as a frequency-dependent potential in a scattering equation for the half Laplacian. We prove high-frequency uniqueness results for three types of intensity data: source-driven measurements, renormalized far-field intensity measurements, and phaseless far-field measurements obtained from coherent superpositions of incident plane waves. In each case, the corresponding data, given at all sufficiently large frequencies, determine the atomic density uniquely; the source-driven result requires a geometric visibility condition on the source and observation sets.

math.AP↗

arXiv2026

Improved stability of low Fourier modes in inverse problems for potentials

We prove Lipschitz, sub-Hölder and Hölder stability estimates for recovering the low Fourier modes of an unknown potential from the Dirichlet-to-Neumann (DN) map. We study three different cases, depending on the regularity of the difference $q_1-q_2$. First, we consider \[ (-Δ- λ^2 + q)u=0 \quad \text{in} \quad Ω\subset \mathbb{R}^n. \] We show that the difference $q_1-q_2$, assumed to be $M$-bandlimited, can be recovered in a Lipschitz stable way from the difference of the corresponding DN maps. This holds whenever $λ$ is sufficiently large relative to $M^{n/2}$. The proof involves real geometrical optics solutions. Secondly, we consider \[ (-Δ+ q)u=0 \quad \text{in} \quad (-π,π)^n. \] We show that the low Fourier coefficients of the difference $q_1-q_2$, assumed to be real-analytic and periodic, can be recovered in a sub-Hölder stable way from the difference of the corresponding DN maps. The number of recoverable Fourier modes grows as the DN maps become closer. Finally, we consider the case where the Fourier coefficients of the difference $q_1-q_2$ decay at a super-exponential rate $e^{-c|k|^{n/2}}$. We prove that the low Fourier modes can be recovered with Hölder stability, with the number of recoverable modes tending to infinity as the DN maps become closer. In all cases the $L^\infty$ potentials themselves do not need to satisfy additional assumptions or belong to a finite dimensional space. The constants in the stability estimates are uniform in the number of recovered Fourier modes.

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arXiv2026

Semiglobal uniqueness for the Lorentzian Calderón problem

We prove uniqueness in the Lorentzian Calderón problem in a semiglobal setting, comparing a potentially large perturbation of the Minkowski metric against a small one. Our proof uses a reduced amount of data, solving a formally determined version of the problem. In contrast to traditional approaches, it employs neither control-theoretic arguments nor a reduction to a geometric inverse problem via microlocal methods or high-frequency solutions. Instead, it relies on distorted plane waves and weighted $L^2$-estimates.

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arXiv2026

Fixed angle inverse scattering with non-constant velocity

In this article, we study formally determined inverse problems for wave equations in the presence of a variable sound speed. We prove that by measuring the boundary data of finitely many plane waves and their complementary solutions, one can uniquely recover the unknown coefficients of the highest order terms of a second order hyperbolic operator with time independent coefficients. This improves earlier rigidity results in [10], [11] which compared the wave operator generated by a Riemannian metric with the wave operator generated by the Euclidean metric. We compare two general second order hyperbolic operators with time independent coefficients, with the same lower order terms. However, we require the geometry associated with one of the operators to satisfy a pseudoconvexity condition, a no-caustics condition, and a spanning condition. In particular one of the operators could be a wave operator with the sound speed close to a constant and the other operator could be arbitrary. To prove the results, we introduce the notion of a complementary solution for a generalized plane wave solution generated by an incoming plane wave. The complementary solution extends smoothly, across an interface, the generalized plane wave. The unknown coefficients appear in a transport equation at the interface. We show that the unknown coefficients in the interior can be extracted from this transport equation, from the boundary data, via a sequence of Carleman estimates for the wave operator.

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arXiv2026

Regularity for the Geodesic X-Ray Transform in Nonsmooth Geometry

In this article we give regularity results for the geodesic X-ray transform on nonsmooth simple manifolds. As an application, we improve previous injectivity results for the geodesic X-ray transform acting on $L^p$ functions. The results are based on symbol smoothing arguments for the normal operator and mapping properties of pseudodifferential operators with low regularity symbols.

math.DG↗

arXiv2026

Inverse problems for semilinear elliptic PDE with a general nonlinearity $a(x,u)$

This article studies the inverse problem of recovering a nonlinearity in an elliptic equation $Δu + a(x,u) = 0$ from boundary measurements of solutions. Previous results based on first order linearization achieve this under a sign condition on $\partial_u a(x,u)$, and results based on higher order linearization recover the Taylor series of $a(x,u)$ with respect to $u$. We improve these results and show that a general nonlinearity, and not just its Taylor series, is uniquely determined up to gauge near a fixed solution. Our method is based on constructing a good solution map that locally parametrizes solutions of the nonlinear equation by solutions of the linearized equation.

math.AP↗

arXiv2026

Inverse problems for semilinear elliptic equations with low regularity

We show that a general nonlinearity $a(x,u)$ is uniquely determined, possibly up to a gauge, in a neighborhood of a fixed solution from boundary measurements of the corresponding semilinear equation. The main theorems are low regularity counterparts of the results in our recent paper (Johansson, Nurminen, Salo; ArXiv preprint 2312.12196).

math.AP↗

arXiv2026

Determination of an anisotropic perturbation in elastic inverse scattering

We consider a linearized inverse scattering problem for elastic waves. We prove that a fully anisotropic perturbation of the elastic parameters around an isotropic and homogeneous reference can be uniquely determined by (single-)scattered waves. We also give a quantitative stability estimate for an isotropic perturbation, and as a consequence a rigidity result is established.

math.AP↗

arXiv2026

Propagation of singularities and inverse problems for the viscoacoustic wave equation

We study an inverse problem for the viscoacoustic wave equation, an integro-differential model describing wave propagation in viscoacoustic media with memory in the leading order term. The medium is characterized by a spatially varying sound speed and a space-time dependent memory kernel. Assuming that waves are generated by sources supported outside the region of interest, we consider exterior measurements encoded by the source-to-solution map. To study this inverse problem, we construct solutions concentrating near fixed geodesics and establish a corresponding propagation of singularities result for the semiclassical wave front set. These results are valid without any restriction on the underlying sound speed. Then, under certain geometric conditions, we prove that the exterior data uniquely determine not just the sound speed inside the domain but also all time derivatives at zero of the memory kernel. This involves a reduction to the lens rigidity and geodesic ray transform inverse problems. As an application, we establish uniqueness for the recovery of variable parameters in the extended Maxwell model.

math.AP↗

arXiv2026

On Exponential Instability of an Inverse Problem for the Wave Equation

For a time-independent potential $q\in L^\infty$, consider the source-to-solution operator that maps a source $f$ to the solution $u=u(t,x)$ of $(\Box+q)u=f$ in Euclidean space with an obstacle, where we impose on $u$ vanishing Cauchy data at $t=0$ and vanishing Dirichlet data at the boundary of the obstacle. We study the inverse problem of recovering the potential $q$ from this source-to-solution map restricted to some measurement domain. By giving an example where measurements take place in some subset and the support of $q$ lies in the `shadow region' of the obstacle, we show that recovery of $q$ is exponentially unstable.

math.AP↗

arXiv2025

On scattering behavior of corner domains with anisotropic inhomogeneities: part II

We study the scattering behavior of an anisotropic inhomogeneous Lipschitz medium at a fixed wave number, continuing our previous work [SIAM J. Math. Anal., 56(4):4834-4853, 2024] and using free boundary techniques from [arXiv:2506.22328]. Our main results can be categorized into two distinct cases. In the first case, we show that in two dimensions, piecewise $C^{1}$ or convex penetrable obstacles with corners, and in higher dimensions, obstacles with edge points, always induce nontrivial scattering for any incoming wave. In the second case, we prove that piecewise $C^{1}$ obstacles with corners in two dimensions (and with edge points in higher dimensions) with angles $\notinπ\mathbb{Q}$ always produce nontrivial scattering for any incoming wave.

math.AP↗

arXiv2025

Generalized boundary rigidity and minimal surface transform

We study a generalized boundary rigidity problem, which investigates whether the areas of embedded minimal surfaces can uniquely determine a Riemannian manifold with boundary. We prove that for a conformal perturbation of an analytic metric in dimension $n+1$ ($n \geq 2$), the metric is determined by these volumes under an ampleness condition. Furthermore, we establish Hölder stability for this determination. This result extends earlier works in dimension $2+1$. Instead of relying on reductions to Calderón type problems and complex geometrical optics solutions, we study the linearized forward operator that gives rise to the minimal surface transform, a generalization of the X-ray/Radon transform. We demonstrate that this transform fits into the framework of double fibration transforms and satisfies the Bolker condition in the sense of Guillemin. Under certain assumptions, including a foliation condition, we prove invertibility of this transform on an analytic manifold as well as recovery of the analytic wave front set. The methods developed in this paper offer new tools for addressing the generalized boundary rigidity problem and expand the scope of applications of double fibration transforms. We anticipate that these techniques will also be applicable to other geometric inverse problems. Beyond mathematics, our results have implications for the AdS/CFT correspondence in physics.

math.AP↗

arXiv2025

A free boundary approach to non-scattering obstacles with vanishing contrast

Motivated by questions in inverse scattering theory, we develop free boundary methods in obstacle problems where both the solution and the right hand side of the equation may have varying sign. The key condition that prevents the appearance of corners is that the right hand side should be related to a harmonic polynomial. In this setting we prove new free boundary results not found in existing literature. Notably, our results imply that piecewise $C^1$ or convex penetrable obstacles in two dimensions and edge points in higher dimensions always cause nontrivial scattering of any incoming wave.

math.AP↗

arXiv2025

On instability mechanisms for inverse problems

In this article we present three robust instability mechanisms for linear and nonlinear inverse problems. All of these are based on strong compression properties (in the sense of singular value or entropy number bounds) which we deduce through either strong global smoothing, only weak global smoothing or microlocal smoothing for the corresponding forward operators, respectively. As applications we for instance present new instability arguments for unique continuation, for the backward heat equation and for linear and nonlinear Calderón type problems in general geometries, possibly in the presence of rough coefficients. Our instability mechanisms could also be of interest in the context of control theory, providing estimates on the cost of (approximate) controllability in rather general settings. This is a revised version of the article ``On instability mechanisms for inverse problems'' Ars Inveniendi Analytica (2021), Paper No. 7, 93 pp by the same authors.

math.AP↗

arXiv2025

Rigidity in fixed angle inverse scattering for Riemannian metrics

The fixed angle inverse scattering problem for a velocity consists in determining a sound speed, or a Riemannian metric up to diffeomorphism, from measurements obtained by probing the medium with a single plane wave. This is a formally determined inverse problem that is open in general. In this article we consider the rigidity question of distinguishing a sound speed or a Riemannian metric from the Euclidean metric. We prove that a general smooth metric that is Euclidean outside a ball can be distinguished from the Euclidean metric. The methods involve distorted plane waves and a combination of geometric, topological and unique continuation arguments.

math.AP↗

arXiv2024

Geometrical optics for the fractional Helmholtz equation and applications to inverse problems

In this paper we construct a parametrix for the fractional Helmholtz equation $((-Δ)^s - τ^{2s} r(x)^{2s} + q(x))u=0$ making use of geometrical optics solutions. We show that the associated eikonal equation is the same as in the classical case, while in the first transport equation the effect of nonlocality is only visible in the zero-th order term, which depends on $s$. Moreover, we show that the approximate geometrical optics solutions present different behaviors in the regimes $s\in(0,\frac 12)$ and $s\in [\frac 12,1)$. While the latter case is quite similar to the classical one, which corresponds to $s=1$, in the former case we find that the potential is a strong perturbation, which changes the propagation of singularities. As an application, we study the inverse problem consisting in recovering the potential $q$ from Cauchy data when the refraction index $r$ is fixed and simple. Using our parametrix based on the construction of approximate geometrical optics solutions, we prove that Hölder stability holds for this problem. This is a substantial improvement over the state of the art for fractional wave equations, for which the usual Runge approximation argument can provide only logarithmic stability. Besides its mathematical novelty, this study is motivated by envisioned applications in nonlocal elasticity models emerging from the geophysical sciences.

math.AP↗

arXiv2024

Rigidity in the Lorentzian Calderón problem with formally determined data

We study the Lorentzian Calderón problem, where the objective is to determine a globally hyperbolic Lorentzian metric up to a boundary fixing diffeomorphism from boundary measurements given by the hyperbolic Dirichlet-to-Neumann map. This problem is a wave equation analogue of the Calderón problem on Riemannian manifolds. We prove that if a globally hyperbolic metric agrees with the Minkowski metric outside a compact set and has the same hyperbolic Dirichlet-to-Neumann map as the Minkowski metric, then it must be the Minkowski metric up to diffeomorphism. In fact we prove the same result with a much smaller amount of measurements, thus solving a formally determined inverse problem. To prove these results we introduce a new method for geometric hyperbolic inverse problems. The method is based on distorted plane wave solutions and on a combination of geometric, topological and unique continuation arguments.

math.AP↗

arXiv2024

Coefficient Determination for Non-Linear Schrödinger Equations on manifolds

We consider an inverse problem of recovering the unknown coefficients $β(t,x)$ and $V(t,x)$ appearing in a time-dependent nonlinear Schrödinger equation $ (\mathrm{i} \partial_t +Δ+V)u + βu^2=0$ in $(0,T) \times M$, on Euclidean geometry as well as on Riemannian geometry. We consider measurements in $Ω\subset M$ that is a neighborhood of the boundary of $M$ and the source-to-solution map $ L_{β, V}$ that maps a source $f$ supported in $ Ω\times (0,T) $ to the restriction of the solution $u$ in $ Ω\times (0,T) $. We show that the map $L_{β, V}$ uniquely determines the time-dependent potential and the coefficient of the non-linearity, for the above non-linear Schrödinger equation and for the Gross-Pitaevskii equation, with a cubic non-linear term $β|u|^2 \, u$, that is encountered in quantum physics.

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