arXiv subjects

Jesse Railo

Publications and source records attributed to Jesse Railo.

At least 19 recordsLinked to original sources

arXiv2026

Torus computed tomography for experimental data

We implement the torus-based X-ray tomography method introduced by Ilmavirta, Koskela, and Railo in "Torus computed tomography", SIAM J. Appl. Math., 80(4):1947--1976, 2020, for experimental X-ray tomographic data. The numerical implementation is extended to accommodate fan-beam measurements by converting the data to a parallel-beam format and mapping the projection angles to the closed-geodesic directions on the torus. In addition, we consider two extensions of the original framework: the Star TCT method which extends the frequency coverage of the reconstruction, and a numerical implementation of torus backprojection developed by Railo in "Fourier analysis of periodic Radon transforms", J. Fourier Anal. Appl., 26(4):64, 2020, for which we also derive a corresponding regularized formulation. We demonstrate the methods on experimental X-ray data of a walnut and compare them with filtered backprojection. We also introduce a pointwise positivity constraint as a post-processing step, which substantially improves the reconstruction accuracy. The simulated data experiments are revisited using an updated implementation. The results indicate that the proposed extensions improve reconstruction quality and support the applicability of torus-based reconstruction methods to experimental data.

math.NA↗

arXiv2026

A Galerkin Finite Element Method for the Fractional Calder\'on Problem

We study a numerical reconstruction strategy for the potential in the fractional Calder\'on problem from a single partial exterior measurement. The forward model is the fractional Schr\"odinger 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.

math.NA↗

arXiv2025

Bayesian inference for the fractional Calder\'on problem with a single measurement

This paper investigates the consistency of a posterior distribution in the single-measurement fractional Calder\'on 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.

math.ST↗

arXiv2025

The matrix weighted real-analytic double fibration transforms

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.

math.AP↗

arXiv2025

Partial data stability for the inverse fractional conductivity problem

The classical Calder\'on 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.

math.AP↗

arXiv2025

Recovering a (1+1)-dimensional wave equation from a single white noise boundary measurement

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.

math.AP↗

arXiv2024

Survey on Broken Ray Transforms

We survey recent developments in the theory and applications of the broken ray transforms. Furthermore, we discuss some open problems.

math.DG↗

arXiv2024

Inverse Problems for Twisted Geodesic Flows

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.

math.DG↗

arXiv2024

A note on the Fourier magnitude data and Sobolev embeddings

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.

math.FA↗

arXiv2023

Loop group factorization method for the magnetic and thermostatic nonabelian ray transforms

We study the injectivity of the matrix attenuated and nonabelian ray transforms on compact surfaces with boundary for nontrapping $\lambda$-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 $\lambda$ 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 $\lambda$ has higher Fourier modes.

math.DG↗

arXiv2023

Broken ray transform for twisted geodesics on surfaces with a reflecting obstacle

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.

math.DG↗

arXiv2022

The Calderón problem for a nonlocal diffusion equation with time-dependent coefficients

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 $n\geq 1$.

math.AP↗

arXiv2022

Stability estimates for the inverse fractional conductivity problem

We study the stability of an inverse problem for the fractional conductivity equation on bounded smooth domains. We obtain a logarithmic stability estimate for the inverse problem under suitable a priori bounds on the globally defined conductivities. The argument has three main ingredients: 1. the logarithmic stability of the related inverse problem for the fractional Schr\"odinger equation by R\"uland and Salo; 2. the Lipschitz stability of the exterior determination problem; 3. utilizing and identifying nonlocal analogies of Alessandrini's work on the stability of the classical Calder\'on problem. The main contribution of the article is the resolution of the technical difficulties related to the last mentioned step. Furthermore, we show the optimality of the logarithmic stability estimates, following the earlier works by Mandache on the instability of the inverse conductivity problem, and by R\"uland and Salo on the analogous problem for the fractional Schr\"odinger equation.

math.AP↗

arXiv2022

Low regularity theory for the inverse fractional conductivity problem

We characterize partial data uniqueness for the inverse fractional conductivity problem with $H^{s,n/s}$ regularity assumptions in all dimensions. This extends the earlier results for $H^{2s,\frac{n}{2s}}\cap H^s$ conductivities by Covi and the authors. We construct counterexamples to uniqueness on domains bounded in one direction whenever measurements are performed in disjoint open sets having positive distance to the domain. In particular, we provide counterexamples in the special cases $s \in (n/4,1)$, $n=2,3$, missing in the literature due to the earlier regularity conditions. We also give a new proof of the uniqueness result which is not based on the Runge approximation property. Our work can be seen as a fractional counterpart of Haberman's uniqueness theorem for the classical Calder\'on problem with $W^{1,n}$ conductivities when $n=3,4$. One motivation of this work is Brown's conjecture that uniqueness for the classical Calder\'on problem holds for $W^{1,n}$ conductivities also in dimensions $n \geq 5$.

math.AP↗

arXiv2022

The fractional $p\,$-biharmonic systems: optimal Poincar\'e constants, unique continuation and inverse problems

This article investigates nonlocal, fully nonlinear generalizations of the classical biharmonic operator $(-\Delta)^2$. These fractional $p$-biharmonic operators appear naturally in the variational characterization of the optimal fractional Poincar\'e constants in Bessel potential spaces. We study the following basic questions for anisotropic fractional $p$-biharmonic systems: existence and uniqueness of weak solutions to the associated interior source and exterior value problems, unique continuation properties (UCP), monotonicity relations, and inverse problems for the exterior Dirichlet-to-Neumann maps. Furthermore, we show the UCP for the fractional Laplacian in all Bessel potential spaces $H^{t,p}$ for any $t\in \mathbb{R}$, $1 \leq p < \infty$ and $s \in \mathbb{R}_+ \setminus \mathbb{N}$: If $u\in H^{t,p}(\mathbb{R}^n)$ satisfies $(-\Delta)^su=u=0$ in a nonempty open set $V$, then $u\equiv 0$ in $\mathbb{R}^n$. This property of the fractional Laplacian is then used to obtain a UCP for the fractional $p$-biharmonic systems and plays a central role in the analysis of the associated inverse problems. Our proofs use variational methods and the Caffarelli-Silvestre extension.

math.AP↗

arXiv2022

The global inverse fractional conductivity problem

We prove \emph{global} uniqueness for an inverse problem for the fractional conductivity equation on domains that are bounded in one direction. The conductivities are assumed to be isotropic and nontrivial in the exterior of the domain, while the data is given in the form of partial Dirichlet-to-Neumann (DN) maps measured in nondisjoint open subsets of the exterior. This can be seen as the fractional counterpart of the classical inverse conductivity problem. The proof is based on a unique continuation property (UCP) for the DN maps and an exterior determination method from the partial exterior DN maps. This is analogous to the classical boundary determination method by Kohn and Vogelius. The most important technical novelty is the construction of sequences of special solutions to the fractional conductivity equation whose Dirichlet energies in the limit can be concentrated at any given point in the exterior. This is achieved independently of the UCP and despite the nonlocality of the equation. Due to the recent counterexamples by the last two authors, our results almost completely characterize uniqueness for the inverse fractional conductivity problem with partial data for isotropic global conductivities.

math.AP↗

arXiv2022

Counterexamples to uniqueness in the inverse fractional conductivity problem with partial data

We construct counterexamples for the partial data inverse problem for the fractional conductivity equation in all dimensions on general bounded open sets. In particular, we show that for any bounded domain $\Omega \subset \mathbb{R}^n$ and any disjoint open sets $W_1,W_2 \Subset \mathbb{R}^n \setminus \bar{\Omega}$ there always exist two positive, bounded, smooth, conductivities $\gamma_1,\gamma_2$, $\gamma_1 \neq \gamma_2$, with equal partial exterior Dirichlet-to-Neumann maps $\Lambda_{\gamma_1}f|_{W_2} = \Lambda_{\gamma_2}f|_{W_2}$ for all $f \in C_c^\infty(W_1)$. The proof uses the characterization of equal exterior data from another work of the authors in combination with the maximum principle of fractional Laplacians. The main technical difficulty arises from the requirement that the conductivities should be strictly positive and have a special regularity property $\gamma_i^{1/2}-1 \in H^{2s,\frac{n}{2s}}(\mathbb{R}^n)$ for $i=1,2$. We also provide counterexamples on domains that are bounded in one direction when $n \geq 4$ or $s \in (0,n/4]$ when $n=2,3$ using a modification of the argument on bounded domains.

math.AP↗

arXiv2022

Fractional Calder\'on problems and Poincar\'e inequalities on unbounded domains

We generalize many recent uniqueness results on the fractional Calder\'on problem to cover the cases of all domains with nonempty exterior. The highlight of our work is the characterization of uniqueness and nonuniqueness of partial data inverse problems for the fractional conductivity equation on domains that are bounded in one direction for conductivities supported in the whole Euclidean space and decaying to a constant background conductivity at infinity. We generalize the uniqueness proof for the fractional Calder\'on problem by Ghosh, Salo and Uhlmann to a general abstract setting in order to use the full strength of their argument. This allows us to observe that there are also uniqueness results for many inverse problems for higher order local perturbations of a lower order fractional Laplacian. We give concrete example models to illustrate these curious situations and prove Poincar\'e inequalities for the fractional Laplacians of any order on domains that are bounded in one direction. We establish Runge approximation results in these general settings, improve regularity assumptions also in the cases of bounded sets and prove general exterior determination results. Counterexamples to uniqueness in the inverse fractional conductivity problem with partial data are constructed in another companion work.

math.AP↗