arXiv research

We consider inverse boundary value problems for general real principal type differential operators. The first results state that the Cauchy data set uniquely determines the scattering relation of the operator and bicharacteristic ray transforms of lower order coefficients. We also give two different boundary determinat…

Study on uniquely determining thermal properties from boundary temperature and heat flux measurements.

Stable solution found for manifold topology from boundary data.

Study inverse boundary value problem for Monge-Ampère equation on convex domains.

New boundary condition for weak inverse mean curvature flow in bounded domains.

Method constructs spirals with given tangents and curvatures.

This study recovers electromagnetic parameters on boundaries from impedance and admittance data.

Paper develops a new method for solving IBVPs on star-shaped domains.

We prove uniqueness results for a Calderon type inverse problem for the Hodge Laplacian acting on graded forms on certain manifolds in three dimensions. In particular, we show that partial measurements of the relative-to-absolute or absolute-to-relative boundary value maps uniquely determine a zeroth order potential. T…

Unified framework for forward and inverse PDE problems in multiphase media.

Study shows stability of travel time data reconstruction from closed subsets.

Geometric Hydrodynamics tackles open problems in fluid dynamics.

We consider a Dirac-type operator $D_P$ on a vector bundle $V$ over a compact Riemannian manifold $(M,g)$ with a nonempty boundary. The operator $D_P$ is specified by a boundary condition $P(u|_{\p M})=0$ where $P$ is a projector which may be a non-local, i.e. a pseudodifferential operator. We assume the existence of a…

We survey recent results on inverse problems for geodesic X-ray transforms and other linear and non-linear geometric inverse problems for Riemannian metrics, connections and Higgs fields defined on manifolds with boundary.

Proposes neural networks for solving complex free boundary problems.

We show that the travel time difference functions, measured on the boundary, determine a compact Riemannian manifold with smooth boundary up to Riemannian isometry, if boundary satisfies a certain visibility condition. This corresponds with the inverse microseismicity problem. The novelty of our paper is a new type of …

Study tackles inverse problems on low-dimensional manifolds, proving stability and proposing a reconstruction algorithm.

Study inverse problems for twisted geodesic flows on manifolds.

Study the geometry of gas giant planets to infer their internal structure.

Researchers recover Riemannian manifolds and lower order terms from travel time data.

This paper explores and ties together three themes. The first is to establish regularity of a metric tensor, on a manifold with boundary, on which there are given Ricci curvature bounds, on the manifold and its boundary, and a Lipschitz bound on the mean curvature of the boundary. The second is to establish geometric c…

We consider the boundary rigidity problem for asymptotically hyperbolic manifolds. We show injectivity of the X-ray transform in several cases and consider the non-linear inverse problem which consists of recovering a metric from boundary measurements for the geodesic flow.

Study determines a minimal surface in a Riemannian manifold from boundary data.

Researchers solve a formally determined inverse problem in Lorentzian geometry.

Inverse mean curvature flow converges to a disk in hyperbolic space.

We consider integral geometry inverse problems for unitary connections and skew-Hermitian Higgs fields on manifolds with negative sectional curvature. The results apply to manifolds in any dimension, with or without boundary, and also in the presence of trapped geodesics. In the boundary case, we show injectivity of th…

Researchers reconstruct simple Riemannian manifolds from boundary wave arrival times.

Summary of tensor tomography proofs on manifolds with boundaries.

We define a class of boundary value problems on manifolds with fibered boundary. This class is in a certain sense a deformation between the classical boundary value problems and the Atiyah-Patodi-Singer problems in subspaces. The boundary conditions in this theory are taken as elements of the C^*-algebra generated by p…

Study determines minimal surfaces from boundary data, proving topological and conformal recoverability.

We prove new pinching estimate for the inverse curvature flow of strictly convex hypersurfaces in the space form $N$ of constant sectional curvature $K_N$ with speed given by $F^{-α}$, where $α\in (0,1]$ for $K_N=0,-1$ and $α=1$ for $K_N=1$, $F$ is a smooth, symmetric homogeneous of degree one function which is inverse…

Study magnetic potentials on Anosov manifolds using spectral data.

Consider a broken geodesics $α([0,l])$ on a compact Riemannian manifold $(M,g)$ with boundary of dimension $n\geq 3$. The broken geodesics are unions of two geodesics with the property that they have a common end point. Assume that for every broken geodesic $α([0,l])$ starting at and ending to the boundary $\partial M$…

Study on heat flow across two half-lines with special boundary conditions.

In this paper, we consider the variational regularization of manifold-valued data in the inverse problems setting. In particular, we consider TV and TGV regularization for manifold-valued data with indirect measurement operators. We provide results on the well-posedness and present algorithms for a numerical realizatio…

Study solves Gel'fand's inverse problem in non-smooth spaces with Ricci curvature bounds.

Paper addresses travel time tomography stability and statistical inversion.

Boundary value problems for operators of Dirac type arise naturally in connection with the conformal geometry of surfaces immersed in Euclidean 3--space. Recently such boundary value problems have been successfully applied to a variety of problems from computer graphics. Here we investigate under which conditions these…

Inverse scattering result on AH manifolds determines metric up to diffeo and conformal factor.

In this paper we continue our study of bifurcations of solutions of boundary-value problems for symplectic maps arising as Hamiltonian diffeomorphisms. These have been shown to be connected to catastrophe theory via generating functions and ordinary and reversal phase space symmetries have been considered. Here we pres…

Taking an elementary and straightforward approach, we develop the concept of a regular value for a smooth map f: O -> P between smooth orbifolds O and P. We show that Sard's theorem holds and that the inverse image of a regular value is a smooth full suborbifold of O. We also study some constraints that the existence o…

In this paper, we discuss the uniqueness in an integral geometry problem in a strongly convex domain. Our problem is related to the problem of finding a Riemannian metric by the distances between all pairs of the boundary points. For the proof, the problem is reduced to an inverse source problem for a kinetic equation …

Proves the Hodge conjecture for complex projective manifolds.

We outline an approach to the inverse problem of Calderón that highlights the role of microlocal normal forms and propagation of singularities and extends a number of earlier results also in the anisotropic case. The main result states that from the boundary measurements it is possible to recover integrals of the unkno…

In this paper we consider two inverse problems on a closed connected Riemannian manifold $(M,g)$. The first one is a direct analog of the Gel'fand inverse boundary spectral problem. To formulate it, assume that $M$ is divided by a hypersurface $Σ$ into two components and we know the eigenvalues $λ_j$ of the Laplace ope…

The paper solves the Steklov spectral inverse problem for conformal metrics.

A guide for solving first-order elliptic boundary value problems.

We describe the range of a restricted spherical mean transform, which sends a function supported inside a closed ball in a hyperbolic space to its mean values on the geodesics spheres centered at the boundary of the ball. The description resembles that of the same transform on the Euclidean spaces obtained by Mark Agra…