arXiv subjects
Publications and source records attributed to Jesse Railo.
Double fibration transforms integrate functions over families of submanifolds. We survey the recovery of singularities by both classical normal operator methods and direct analytic methods. We also discuss support theorems, injectivity and applications to geodesic ray transforms, seismic inversion and minimal surface data.
We show nonuniqueness for a discrete X-ray transform on $\mathbb Z^2$ under quadratic decay. We construct an explicit nonzero function in $\ell^1(\mathbb Z^2)$ with pointwise decay $O(|z|^{-2})$ whose sums over integer lines vanish. We also obtain an infinite-dimensional null space with the same decay assumption. Furthermore, we show a result which indicates that the decay rate $|z|^{-2}$ may be sharp and faster decay may imply uniqueness.
We present a finite element method for reconstructing a coefficient in a fractional semilinear equation from a single exterior measurement.
We study the joint recovery of attenuation and source coefficients from the attenuated X-ray transform, motivated by passive gamma emission tomography of spent nuclear fuel. Both coefficients are constant on known, disjoint disks of equal radius representing fuel rods. The surrounding moderator has known attenuation and zero source. We use tangential singularities of the data to give a sufficient condition on the geometry and coefficients for unique joint reconstruction from full data. For rods arranged on parallel rows separated by more than one rod diameter, with source coefficients uniformly bounded away from zero, we derive explicit reconstruction formulas and stability estimates for both coefficients. The estimates use a singular seminorm determined by two functionals per rod, computed from angular data near selected tangencies. Numerical experiments examine the approximation of these functionals from discretized data.
We develop a regularized finite element method for the fractional conductivity equation, recovering the conductivity on an exterior observation region from noisy Dirichlet-to-Neumann measurements and using this reconstruction to determine the interior conductivity. Mesh-localized inputs and mass-lumped Tikhonov regularization discretize the exterior determination principle of Covi--Railo--Zimmermann (2026, Calc. Var. PDE). We prove exterior \(L^2\)- and \(L^\infty\)-convergence with error estimates, including the logarithmic correction at the critical Hölder exponent. The exterior reconstruction enables the Liouville reduction to the Calderón problem for the fractional Schrödinger equation. Separated measurements recover the interior potential, after which overlapping measurements determine the exterior potential tail. Together with the recovered exterior conductivity, the tail supplies the exterior Cauchy data for a Tikhonov-regularized finite element least-squares continuation. We prove \(H^s\)-convergence of the transformed conductivity and \(L^2\)-convergence of the interior conductivity, together with logarithmic state-bias and conductivity error estimates for full observation spaces. The admissible reconstructions remain uniformly positive and bounded. We present one- and two-dimensional numerical experiments illustrating the accuracy of the exterior and interior reconstructions and their sensitivity to measurement noise and discretization.
Torus computed tomography (Torus CT) reconstructs the Fourier coefficients of an image from X-ray data along closed geodesics on the flat torus. We extend Torus CT to experimental X-ray tomographic data by converting the data to a parallel-beam geometry and mapping the measured projection angles to the geodesic directions on the torus. In addition, we introduce two extensions of the Torus CT framework: the Star TCT method, which recovers additional Fourier coefficients beyond the standard frequency box using the same data, and torus backprojection (TBP), which sums the data over the measured directions. We prove that filtered TBP converges to the Tikhonov-regularized solution for general data up to an explicit error term that vanishes for data which is invariant along fixed geodesics. We also show that the direction-wise averaging of the zero mode is an unbiased estimator and its mean squared error decays like $N^{-2}$. We evaluate the methods on simulated phantoms and experimental X-ray data of a walnut using an updated implementation. The results show that torus-based reconstruction methods are applicable to experimental data, avoid the streak artifacts of filtered backprojection (FBP), and yield competitive reconstruction errors on both simulated and experimental data.
This paper investigates the consistency of a posterior distribution in the single-measurement fractional Calderón problem with additive Gaussian noise. We consider a Bayesian framework with rescaled and Gaussian sieve priors, using a collection of noisy, discrete observations taken from a suitable exterior domain. Our main result shows that the posterior distribution concentrates around the true parameter as the number of measurements increases. Furthermore, we establish tight convergence rates for the reconstruction error of the posterior mean. A central technical challenge is to obtain refined stability estimates for both the forward and inverse problems. In particular, the required forward estimates are delicate to obtain because the fractional elliptic problems do not enjoy as strong regularity theory as their classical counterparts.
We show that the real-analytic matrix-weighted double fibration transform determines the analytic wavefront set of a vector-valued function. We apply this result to show that the matrix weighted ray transform is injective on a two-dimensional, non-trapping, real-analytic Riemannian manifold with strictly convex boundary. Additionally, we show that a real-analytic Higgs field can be uniquely determined from the nonabelian ray transform on real-analytic Riemannian manifolds of any dimension with a strictly convex boundary point.
We study a numerical reconstruction strategy for the potential in the fractional Calderón problem from a single partial exterior measurement. The forward model is the fractional Schrödinger equation in a bounded domain, with prescribed exterior Dirichlet datum and corresponding measurement of the exterior flux in an open observation set. Motivated by single-measurement uniqueness results based on unique continuation \cite{ghosh2020uniqueness}, we propose a decomposition strategy and a Galerkin--Tikhonov method to recover the potential by a stabilized least-squares quotient in a dedicated coefficient space. We prove the existence and uniqueness of the discrete reconstructor and establish conditional convergence under natural consistency and parameter choice assumptions. We further derive {\it a priori} error estimates for the reconstructed state and for the coefficient reconstruction, and combine the latter with logarithmic stability for the continuous inverse problem to obtain a total coefficient error bound. The framework cleanly separates the forward solver from the inverse reconstruction step and is compatible with practical truncation and quadrature schemes for the integral fractional Laplacian. Numerical experiments in one and two space dimensions illustrate stability with respect to noise and demonstrate reconstructions of both smooth and discontinuous potentials.
We consider the following inverse problem: Suppose a $(1+1)$-dimensional wave equation on $\mathbb{R}_+$ with zero initial conditions is excited with a Neumann boundary data modelled as a white noise process. Given also the Dirichlet data at the same point, determine the unknown first order coefficient function of the system. We first establish that direct problem is well-posed. The inverse problem is then solved by showing that correlations of the boundary data determine the Neumann-to-Dirichlet operator in the sense of distributions, which is known to uniquely identify the coefficient. This approach has applications in acoustic measurements of internal cross-sections of fluid pipes such as pressurised water supply pipes and vocal tract shape determination.
The classical Calderón problem with partial data is known to be log-log stable in some special cases, but even the uniqueness problem is open in general. We study the partial data stability of an analogous inverse fractional conductivity problem on bounded smooth domains. Using the fractional Liouville reduction, we obtain a log-log stability estimate when the conductivities a priori agree in the measurement set and their difference has compact support. In the case in which the conductivities are assumed to agree a priori in the whole exterior of the domain, we obtain a shaper logarithmic stability estimate.
We study the injectivity of the matrix attenuated and nonabelian ray transforms on compact surfaces with boundary for nontrapping $λ$-geodesic flows and the general linear group of invertible complex matrices. We generalize the loop group factorization argument of Paternain and Salo to reduce to the setting of the unitary group when $λ$ has the vertical Fourier degree at most $2$. This covers the magnetic and thermostatic flows as special cases. Our article settles the general injectivity question of the nonabelian ray transform for simple magnetic flows in combination with an earlier result by Ainsworth. We stress that the injectivity question in the unitary case for simple Gaussian thermostats remains open. Furthermore, we observe that the loop group argument does not apply when $λ$ has higher Fourier modes.
We survey recent developments in the theory and applications of the broken ray transforms. Furthermore, we discuss some open problems.
The article surveys inverse problems related to the twisted geodesic flows on Riemannian manifolds with boundary, focusing on the generalized ray transforms, tensor tomography, and rigidity problems. The twisted geodesic flow generalizes the usual geodesic flow, allowing to model richer dynamics.
We study Sobolev $H^s(\mathbb{R}^n)$, $s \in \mathbb{R}$, stability of the Fourier phase problem to recover $f$ from the knowledge of $|\hat{f}|$ with an additional Bessel potential $H^{t,p}(\mathbb{R}^n)$ a priori estimate when $t \in \mathbb{R}$ and $p \in [1,2]$. These estimates are related to the ones studied recently by Steinerberger in "On the stability of Fourier phase retrieval" J. Fourier Anal. Appl., 28(2):29, 2022. While our estimates in general are different, they share some comparable special cases and the main improvement given here is that we can remove an additional imaginary term and obtain sharper constants. We also consider these estimates for the quotient distances related to the non-uniqueness of the Fourier phase problem. Our arguments closely follow the Fourier analysis proof of the Sobolev embeddings for Bessel potential spaces with minor modifications.
We prove a uniqueness result for the broken ray transform acting on the sums of functions and $1$-forms on surfaces in the presence of an external force and a reflecting obstacle. We assume that the considered twisted geodesic flows have nonpositive curvature. The broken rays are generated from the twisted geodesic flows by the law of reflection on the boundary of a suitably convex obstacle. Our work generalizes recent results for the broken geodesic ray transform on surfaces to more general families of curves including the magnetic flows and Gaussian thermostats.
The geodesic ray transform, the mixed ray transform and the transverse ray transform of a tensor field on a manifold can all be seen as what we call mixing ray transforms, compositions of the geodesic ray transform and an invertible linear map on tensor fields. We show that the characterization of the kernel and the stability of a mixing ray transform can be reduced to the same properties of any other mixing ray transform. Our approach applies to various geometries and ray transforms, including the light ray transform. In particular, we extend studies in de Hoop--Saksala--Zhai (2019) from compact simple surfaces to orientable surfaces with solenoidally injective geodesic ray transform. Our proofs are based on algebraic arguments.
We investigate global uniqueness for an inverse problem for a nonlocal diffusion equation on domains that are bounded in one direction. The coefficients are assumed to be unknown and isotropic on the entire space. We first show that the partial exterior Dirichlet-to-Neumann map locally determines the diffusion coefficient in the exterior domain. In addition, we introduce a novel analysis of nonlocal Neumann derivatives to prove an interior determination result. Interior and exterior determination yield the desired global uniqueness theorem for the Calderón problem of nonlocal diffusion equations with time-dependent coefficients. This work extends recent studies from nonlocal elliptic equations with global coefficients to their parabolic counterparts. The results hold for any spatial dimension $n\geq 1$.