arXiv subjects

Joonas Ilmavirta

Publications and source records attributed to Joonas Ilmavirta.

At least 19 recordsLinked to original sources

arXiv2026

Determination of Wave Speed from interior sources

We consider the wave equation with variable wave speed in Euclidean space, with point sources modeled by Dirac delta initial displacement data. There are three special subsets of the whole space: (1) the source set where the delta initial conditions are supported, (2) the unknown set where the wave speed is not known a priori, and (3) the receiver set where the waves are measured. The inverse problem is to reconstruct the wave speed uniquely in the unknown set from an unlabeled collection of waves generated by point sources and measured in the receiver set. We use propagation of singularities and sharp finite speed of propagation to reduce this data to geometric travel-time data, whose form depends on how the three sets lie in relation to each other. We give three scenarios where this procedure leads to unique determination of the wave speed.

math.AP↗

arXiv2026

Generic Recovery of Permittivity and Permeability in Anisotropic Maxwell Systems

We study the inverse problem of recovering the constitutive tensors of a homogeneous anisotropic electromagnetic medium without magnetoelectric coupling (non-chiral) from its Fresnel surface, the characteristic variety of Maxwell's equations governing electromagnetic wave propagation. For known isotropic permeability, normalized to $μ= I$, we prove that the Fresnel surface uniquely determines the permittivity tensor $\varepsilon$, and that the associated Fresnel polynomial is reducible precisely when $\varepsilon$ has a repeated eigenvalue. For general, positive-definite symmetric tensors $(\varepsilon,μ)$, we prove that the Fresnel polynomial is generically irreducible over $\mathbb{C}$ and we identify the natural gauge symmetry under which it is invariant. Using geometric invariant theory, a powerful tool of modern algebraic geometry, we construct an affine quotient of the parameter space by this gauge action and prove that the induced Fresnel-polynomial map is birational onto its image. We deduce that, outside a proper real algebraic exceptional set, the real Fresnel surface determines $(\varepsilon,μ)$ up to gauge. This establishes generic uniqueness for the inverse problem and, to our knowledge, provides a new application of affine geometric invariant theory to gauge freedom in a PDE inverse problem.

math.AP↗

arXiv2026

Classification of singularities of planar slowness surfaces

Slowness surfaces are algebraic varieties arising from propagation of elastic waves. In dimensions $2$, we completely classify the types of singularities slowness surfaces can have. The two types of possible singularities are a transversal self-intersection and a tangential singularity produced by a concentric circle and ellipse that are tangent to each other. To interpret these results analytically, in the case that the slowness surface has transversal self-intersections, we show that the principal symbol of the elastic wave operator is locally smoothly diagonalizable.

math.AG↗

arXiv2026

Stable recovery of a simple irreversible Finsler geometry from travel time data

We show that a simple irreversible Finsler geometry can be recovered uniquely and Lipschitz-stably from its travel time data. We introduce and use a version of Gromov--Hausdorff distance adapted to irreversible metric spaces. In contrast to reversible (e.g. Riemannian) geometry, even the question of stability becomes ill-defined without simplicity.

math.MG↗

arXiv2026

Rigidity of homogeneous Lamé systems

In this short paper, we show that any Lamé system whose Dirichlet-to-Neumann map for the elastic wave equation agrees with the one arising from the homogeneous Lamé system must actually be homogeneous. We do not need to impose any assumptions for the Lamé coefficients that we aim to recover. We use the fact that the homogeneous system gives rise to a geometry that is both simple and admits a strictly convex foliation.

math.AP↗

arXiv2026

Tomography of 1-forms on a gas giant

We show that on gas giant manifolds the geodesic X-ray transform is solenoidally injective on one-forms that are smooth up to the boundary in an appropriate smooth structure. A gas giant manifold is a conformally blown up Riemannian manifold whose boundary singularity is milder than asymptotically hyperbolic. The proof is based on a Pestov identity and asymptotic analysis of short geodesics.

math.DG↗

arXiv2025

Horizontal and Vertical Regularity of Elastic Wave Geometry

The elastic properties of a material are encoded in a stiffness tensor field and the propagation of elastic waves is modeled by the elastic wave equation. We characterize analytic and algebraic properties a general anisotropic stiffness tensor field has to satisfy in order for Finsler-geometric methods to be applicable in studying inverse problems related to imaging with elastic waves.

math.DG↗

arXiv2025

Analysis and X-ray tomography

These are lecture notes for the course "Analysis and X-ray tomography". The course is a broad overview of various tools in analysis that can be used to study X-ray tomography. The focus is on tools and ideas, not so much on technical details and minimal assumptions. Only very basic functional analysis is assumed as background. Exercise problems are included.

math.FA↗

arXiv2025

The elastic ray transform

We introduce and study a new family of tensor tomography problems. At rank 2 it corresponds to linearization of travel time of elastic waves, measured for all polarizations. We provide a kernel characterization for ranks up to 2. The kernels consist of potential tensors, but in an unusual sense: the associated differential operators have degree 2 instead of the familiar 1. The proofs are based on Fourier analysis, Helmholtz decompositions, and cohomology.

math.FA↗

arXiv2025

Reconstruction of anisotropic stiffness tensors from partial data around one polarization

We study inverse problems in anisotropic elasticity using tools from algebraic geometry. The singularities of solutions to the elastic wave equation in dimension $n$ with an anisotropic stiffness tensor have propagation kinematics captured by so-called slowness surfaces, which are hypersurfaces in the cotangent bundle of $\mathbb{R}^n$ that turn out to be algebraic varieties. Leveraging the algebraic geometry of families of slowness surfaces we show that, for tensors in a dense open subset in a space of anisotropic two-dimensional stiffness tensors, a small amount of data around one polarization in an individual slowness surface uniquely determines the entire slowness surface and its stiffness tensor. In three dimensions, for generic orthorhombic and monoclinic stiffness tensors, a small number of anomalous companions give rise to the same slowness surface; nevertheless, we conjecture that in the most anisotropic setting (triclinic) the tensor is unique, as in two dimensions. The partial data needed to determine a tensor arises naturally from seismological measurements or geometrized versions of seismic inverse problems. Additionally, we explain how the reconstruction of the stiffness tensor can be carried out effectively, using Gröbner bases. Our uniqueness or finiteness results fail for symmetric materials (e.g., fully isotropic), evidencing the counterintuitive claim that inverse problems in elasticity can become more tractable with increasing asymmetry.

math.DG↗

arXiv2025

Principal spectral rigidity implies subprincipal spectral rigidity

We study the inverse spectral problem of jointly recovering a radially symmetric Riemannian metric and an additional coefficient from the Dirichlet spectrum of a perturbed Laplace-Beltrami operator on a bounded domain. Specifically, we consider the elliptic operator \[ L_{a,b} := e^{a-b} \nabla \cdot e^b \nabla \] on the unit ball $ B \subset \mathbb{R}^3 $, where the scalar functions $ a = a(|x|) $ and $ b = b(|x|) $ are spherically symmetric and satisfy certain geometric conditions. While the function $ a $ influences the principal symbol of $ L $, the function $ b $ appears in its first-order terms. We investigate the extent to which the Dirichlet eigenvalues of $ L_{a,b} $ uniquely determine the pair $ (a, b) $ and establish spectral rigidity results under suitable assumptions.

math.AP↗

arXiv2025

Reconstruction along a geodesic from sphere data in Finsler geometry and anisotropic elasticity

Dix formulated the inverse problem of recovering an elastic body from the measurements of wave fronts of point sources. We geometrize this problem in the context of seismology, leading to the geometrical inverse problem of recovering a Finsler manifold from certain sphere data in a given open subset of the manifold. We solve this problem locally along any geodesic through the measurement set.

math.DG↗

arXiv2025

Unique continuation for the momentum ray transform

The present article focuses on a unique continuation result for certain weighted ray transforms, utilizing the unique continuation property (UCP) of the fractional Laplace operator. Specifically, we demonstrate a conservative property for momentum ray transforms acting on tensors, as well as the antilocality property for both weighted ray and cone transforms acting on functions.

math.AP↗

arXiv2024

Lipschitz Stability of Travel Time Data

We prove that the reconstruction of a certain type of length spaces from their travel time data on a closed subset is Lipschitz stable. The travel time data is the set of distance functions from the entire space, measured on the chosen closed subset. The case of a Riemannian manifold with boundary with the boundary as the measurement set appears is a classical geometric inverse problem arising from Gel'fand's inverse boundary spectral problem. Examples of spaces satisfying our assumptions include some non-simple Riemannian manifolds, Euclidean domains with non-trivial topology, and metric trees.

math.MG↗

arXiv2024

Gauge freedoms in the anisotropic elastic Dirichlet-to-Neumann map

We address the inverse problem of recovering the stiffness tensor and density of mass from the Dirichlet-to-Neumann map. We study the invariance of the Euclidean and Riemannian elastic wave equation under coordinate transformations. Furthermore, we present gauge freedoms between the parameters that leave the elastic wave equations invariant. We use these results to present gauge freedoms in the Dirichlet-to-Neumann map associated to the Riemannian elastic wave equation.

math.AP↗

arXiv2024

Geometric inverse problems on gas giants

On gas giant planets the speed of sound is isotropic and goes to zero at the surface. Geometrically, this corresponds to a Riemannian manifold whose metric tensor has a conformal blow-up near the boundary. The blow-up is tamer than in asymptotically hyperbolic geometry: the boundary is at a finite distance. We study the differential geometry of such manifolds, especially the asymptotic behavior of geodesics near the boundary. We relate the geometry to the propagation of singularities of a hydrodynamic PDE and we give the basic properties of the Laplace--Beltrami operator. We solve two inverse problems, showing that the interior structure of a gas giant is uniquely determined by different types of boundary data.

math.DG↗

arXiv2024

Tensor tomography on negatively curved manifolds of low regularity

We prove solenoidal injectivity for the geodesic X-ray transform of tensor fields on simple Riemannian manifolds with $C^{1,1}$ metrics and non-positive sectional curvature. The proof of the result rests on Pestov energy estimates for a transport equation on the non-smooth unit sphere bundle of the manifold. Our low regularity setting requires keeping track of regularity and making use of many functions on the sphere bundle having more vertical than horizontal regularity. Some of the methods, such as boundary determination up to gauge and regularity estimates for the integral function, have to be changed substantially from the smooth proof. The natural differential operators such as covariant derivatives are not smooth.

math.DG↗

arXiv2024

Quantum computing algorithms for inverse problems on graphs and an NP-complete inverse problem

We consider an inverse problem for a finite graph $(X,E)$ where we are given a subset of vertices $B\subset X$ and the distances $d_{(X,E)}(b_1,b_2)$ of all vertices $b_1,b_2\in B$. The distance of points $x_1,x_2\in X$ is defined as the minimal number of edges needed to connect two vertices, so all edges have length 1. The inverse problem is a discrete version of the boundary rigidity problem in Riemannian geometry or the inverse travel time problem in geophysics. We will show that this problem has unique solution under certain conditions and develop quantum computing methods to solve it. We prove the following uniqueness result: when $(X,E)$ is a tree and $B$ is the set of leaves of the tree, the graph $(X,E)$ can be uniquely determined in the class of all graphs having a fixed number of vertices. We present a quantum computing algorithm which produces a graph $(X,E)$, or one of those, which has a given number of vertices and the required distances between vertices in $B$. To this end we develop an algorithm that takes in a qubit representation of a graph and combine it with Grover's search algorithm. The algorithm can be implemented using only $O(|X|^2)$ qubits, the same order as the number of elements in the adjacency matrix of $(X,E)$. It also has a quadratic improvement in computational cost compared to standard classical algorithms. Finally, we consider applications in theory of computation, and show that a slight modification of the above inverse problem is NP-complete: all NP-problems can be reduced to a discrete inverse problem we consider.

math.CO↗