22 research outputs found
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Efficient nonparametric bayesian inference for X-ray transforms
We consider the statistical inverse problem of recovering a function , where is a smooth compact Riemannian manifold with
boundary, from measurements of general -ray transforms of ,
corrupted by additive Gaussian noise. For equal to the unit disk with
`flat' geometry and this reduces to the standard Radon transform, but our
general setting allows for anisotropic media and can further model local
`attenuation' effects -- both highly relevant in practical imaging problems
such as SPECT tomography. We propose a nonparametric Bayesian inference
approach based on standard Gaussian process priors for . The posterior
reconstruction of corresponds to a Tikhonov regulariser with a reproducing
kernel Hilbert space norm penalty that does not require the calculation of the
singular value decomposition of the forward operator . We prove
Bernstein-von Mises theorems that entail that posterior-based inferences such
as credible sets are valid and optimal from a frequentist point of view for a
large family of semi-parametric aspects of . In particular we derive the
asymptotic distribution of smooth linear functionals of the Tikhonov
regulariser, which is shown to attain the semi-parametric Cram\'er-Rao
information bound. The proofs rely on an invertibility result for the `Fisher
information' operator between suitable function spaces, a result of
independent interest that relies on techniques from microlocal analysis. We
illustrate the performance of the proposed method via simulations in various
settings
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Lens Rigidity for a Particle in a Yang–Mills Field
We consider the motion of a classical colored spinless particle under the
influence of an external Yang-Mills potential on a compact manifold with
boundary of dimension . We show that under suitable convexity
assumptions, we can recover the potential , up to gauge transformations,
from the lens data of the system, namely, scattering data plus travel times
between boundary points
Entropies, volumes, and Einstein metrics
We survey the definitions and some important properties of several asymptotic
invariants of smooth manifolds, and discuss some open questions related to
them. We prove that the (non-)vanishing of the minimal volume is a
differentiable property, which is not invariant under homeomorphisms. We also
formulate an obstruction to the existence of Einstein metrics on four-manifolds
involving the volume entropy. This generalizes both the Gromov--Hitchin--Thorpe
inequality and Sambusetti's obstruction.Comment: This is a substantial revision and expansion of the 2004 preprint,
which I prepared in spring of 2010 and which has since been published. The
version here is essentially the published one, minus the problems introduced
by Springer productio
The geodesic X-ray transform with matrix weights
Consider a compact Riemannian manifold of dimension with strictly
convex boundary, such that the manifold admits a strictly convex function. We
show that the attenuated ray transform in the presence of an arbitrary
connection and Higgs field is injective modulo the natural obstruction for
functions and one-forms. We also show that the connection and the Higgs field
are uniquely determined by the scattering relation modulo gauge
transformations. The proofs involve a reduction to a local result showing that
the geodesic X-ray transform with a matrix weight can be inverted locally near
a point of strict convexity at the boundary, and a detailed analysis of layer
stripping arguments based on strictly convex exhaustion functions. As a
somewhat striking corollary, we show that these integral geometry problems can
be solved on strictly convex manifolds of dimension having
non-negative sectional curvature (similar results were known earlier in
negative sectional curvature). We also apply our methods to solve some inverse
problems in quantum state tomography and polarization tomography
Invariant distributions and the geodesic ray transform
We establish an equivalence principle between the solenoidal injectivity of
the geodesic ray transform acting on symmetric -tensors and the existence of
invariant distributions or smooth first integrals with prescribed projection
over the set of solenoidal -tensors. We work with compact simple manifolds,
but several of our results apply to non-trapping manifolds with strictly convex
boundary.This is the final version of the article. It first appeared from Mathematical Science Publishers via https://doi.org/10.2140/apde.2016.9.190
Inverse problems for the connection Laplacian
We reconstruct a Riemannian manifold and a Hermitian vector bundle with compatible connection from the hyperbolic Dirichlet-to-Neumann operator associated with the wave equation of the connection Laplacian. The boundary data is local and the reconstruction is up to the natural gauge transformations of the problem. As a corollary we derive an elliptic analogue of the main result which solves a Calderón problem for connections on a cylinder
Symplectic and contact properties of the Mañé critical value of the universal cover
We discuss several symplectic aspects related to the MaA (c) critical value c (u) of the universal cover of a Tonelli Hamiltonian. In particular we show that the critical energy level is never of virtual contact type for manifolds of dimension greater than or equal to three. We also show the symplectic invariance of the finiteness of the Peierls barrier and the Aubry set of the universal cover. We also provide an example where c (u) coincides with the infimum of Mather's alpha function but the Aubry set of the universal cover is empty and the Peierls barrier is finite. A second example exhibits all the ergodic invariant minimizing measures with zero homotopy, showing, quite surprinsingly, that the union of their supports is not a graph, in contrast with Mather's celebrated graph theorem