arXiv subjects

Hanming Zhou

Publications and source records attributed to Hanming Zhou.

At least 19 recordsLinked to original sources

arXiv2026

Conformal boundary rigidity for simple Finsler metrics

In this paper we prove that simple Finsler manifolds are conformally stable. Given a simple Finsler manifold and a class of conformal factors, we characterize the singularity of their induced boundary distance functions which allows us to define an appropiate $H^2$ norm. We then obtain a stability estimate with respect to this Sobolev norm and the $L^2$ norm on the class of the conformal factors. To prove the main theorems, we adapt the classical integration by parts technique employed by Mukhometov \cite{Muhometov} to the Finsler setting.

math.DG↗

arXiv2026

The BGK model with in-flow boundary condition: forward and inverse problems

We study the Bhatnagar-Gross-Krook (BGK) equation in a bounded domain with in-flow boundary condition. The BGK model is a simple relaxation of the Boltzmann equation through replacing the quadratic nonlinearity by the so-called local Maxwellian, which consists of the density, the velocity, and the temperature. The aim of this paper is twofold. First, we establish the Dirichlet boundary value problem for the BGK equation by showing a local existence result in the weighted $L^\infty$ norm if the initial data and boundary condition are close to the global Maxwellian. For the purpose of studying the inverse problem, we further derive an expansion of the BGK solution with respect to a small parameter. This allows us to decompose the nonlinear BGK into purely linear transport equations based on various order of the expansion. Second, we investigate an inverse problem of determining a general collision frequency, depending on both density and temperature, in the BGK equation from the boundary measurement operator, which maps from the in-flow boundary data to the out-flow data. We apply highly concentrated in-flow test data to the solution of the resulting linearized BGK equations to extract hidden information of the collision frequency.

math.AP↗

arXiv2026

The linearized minimal surfaces problem

We characterize the kernel of the linearization $R$ of the minimal surface problem about the Euclidean metric in a bounded smooth domain $\Omega\subset\mathbb{R}^n$, $n\ge2$, with the background minimal surfaces being the Euclidean planes. We show that, in the whole-space Euclidean decomposition, the kernel consists of potential fields and TT fields. For bounded domains, a similar phenomenon appears with additional boundary coupling conditions; in particular, the TT part may be coupled to a harmonic conformal component.

math.DG↗

arXiv2025

Stable Determination of Coefficients in Nonlinear Dynamical Schr\"odinger Equations by Carleman Estimates

We consider the inverse problem of recovering stationary coefficients in a class of dynamical Schr\"odinger equations with locally analytic nonlinear terms. Upon treating the well-posedness for small initial data and trivial boundary data, we proceed to establish stable and unique determination provided knowledge of the coefficients near the boundary and the measured Neumann data of the solution. We discuss both the case of measurement on a subset of the boundary large enough to satisfy a certain geometrical condition and, under stronger assumptions on the regularity and size of the coefficients, the case of measurement on arbitrary subsets of the boundary. Our argument relies on high-order linearization and Carleman estimates for the linear Schr\"odinger equation.

math.AP↗

arXiv2024

Inverse problems for time-dependent nonlinear transport equations

In this work, we investigate inverse problems of recovering the time-dependent coefficient in the nonlinear transport equation in both cases: two-dimensional Riemannian manifolds and Euclidean space $\mathbb{R}^n$, $n\geq 2$. Specifically, it is shown that its initial boundary value problem is well-posed for small initial and incoming data. Moreover, the time-dependent coefficient appearing in the nonlinear term can be uniquely determined from boundary measurements as well as initial and final data. To achieve this, the central techniques we utilize include the linearization technique and the construction of special geometrical optics solutions for the linear transport equation. This allows us to reduce the inverse coefficient problem to the inversion of certain weighted light ray transforms. Based on the developed methodology, the inverse source problem for the nonlinear transport equation in the scattering-free media is also studied.

math.AP↗

arXiv2023

Stability and Statistical Inversion of Travel time Tomography

In this paper, we consider the travel time tomography problem for conformal metrics on a bounded domain, which seeks to determine the conformal factor of the metric from the lengths of geodesics joining boundary points. We establish forward and inverse stability estimates for simple conformal metrics under some a priori conditions. We then apply the stability estimates to show the consistency of a Bayesian statistical inversion technique for travel time tomography with discrete, noisy measurements.

math.DG↗

arXiv2023

On the Identifiablility of Nonlocal Interaction Kernels in First-Order Systems of Interacting Particles on Riemannian Manifolds

In this paper, we tackle a critical issue in nonparametric inference for systems of interacting particles on Riemannian manifolds: the identifiability of the interaction functions. Specifically, we define the function spaces on which the interaction kernels can be identified given infinite i.i.d observational derivative data sampled from a distribution. Our methodology involves casting the learning problem as a linear statistical inverse problem using a operator theoretical framework. We prove the well-posedness of inverse problem by establishing the strict positivity of a related integral operator and our analysis allows us to refine the results on specific manifolds such as the sphere and Hyperbolic space. Our findings indicate that a numerically stable procedure exists to recover the interaction kernel from finite (noisy) data, and the estimator will be convergent to the ground truth. This also answers an open question in [MMQZ21] and demonstrate that least square estimators can be statistically optimal in certain scenarios. Finally, our theoretical analysis could be extended to the mean-field case, revealing that the corresponding nonparametric inverse problem is ill-posed in general and necessitates effective regularization techniques.

math.NA↗

arXiv2022

Recovery of coefficients in semilinear transport equations

We consider the inverse problem for time-dependent semilinear transport equations. We show that time-independent coefficients of both the linear (absorption or scattering coefficients) and nonlinear terms can be uniquely determined, in a stable way, from the boundary measurements by applying a linearization scheme and Carleman estimates for the linear transport equations. We establish results in both Euclidean and general geometry settings.

math.AP↗

arXiv2021

Single pixel X-ray transform and related inverse problems

In this paper, we analyze the nonlinear single pixel X-ray transform $K$ and study the reconstruction of $f$ from the measurement $Kf$. Different from the well-known X-ray transform, the transform $K$ is a nonlinear operator and uses a single detector that integrates all rays in the space. We derive stability estimates and an inversion formula of $K$. We also consider the case where we integrate along geodesics of a Riemannian metric. Moreover, we conduct several numerical experiments to corroborate the theoretical results.

math.AP↗

arXiv2021

Inverse source problems in transport equations with external forces

This paper is concerned with the inverse source problem for the transport equation with external force. We show that both direct and inverse problems are uniquely solvable for generic absorption and scattering coefficients. In particular, for inverse problems, generic injectivity and a stability estimate of the source are derived. The analysis employs the Fredholm theorem and the Santalo's formula.

math.AP↗

arXiv2020

Global determination for an inverse problem from the vortex dynamics

We consider the problem of reconstructing a background potential from the dynamical behavior of vortex dipole. We prove that under suitable conditions, one can uniquely reconstruct a real-analytic potential by measuring the entrance and exit positions as well as travel times between boundary points. In particular, the work removes the flatness assumption on the potential from the earlier result. A key step of our method is a constructional procedure of recovering the boundary jet of the potential.

math.AP↗

arXiv2020

Travel time tomography in stationary spacetimes

In this paper, we consider the boundary rigidity problem on a cylindrical domain in $\mathbb R^{1+n}$, $n\geq 2$, equipped with a stationary (time-invariant) Lorentzian metric. We show that the time separation function between pairs of points on the boundary of the cylindrical domain determines the stationary spacetime, up to some time-invariant diffeomorphism, assuming that the metric satisfies some a-priori conditions.

math.AP↗

arXiv2017

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 $A$ on a compact manifold with boundary of dimension $\geq 3$. We show that under suitable convexity assumptions, we can recover the potential $A$, up to gauge transformations, from the lens data of the system, namely, scattering data plus travel times between boundary points.

math.AP↗

arXiv2017

Reconstruction of a compact Riemannian manifold from the scattering data of internal sources

Given a smooth non-trapping compact manifold with strictly con- vex boundary, we consider an inverse problem of reconstructing the manifold from the scattering data initiated from internal sources. This data consist of the exit directions of geodesics that are emaneted from interior points of the manifold. We show that under certain generic assumption of the metric, one can reconstruct an isometric copy of the manifold from such scattering data measured on the boundary.

math.DG↗

arXiv2016

Generic injectivity and stability of inverse problems for connections

We consider the nonlinear problem of determining a connection and a Higgs field from the corresponding parallel transport along geodesics on a Riemannian manifold with boundary, in any dimension. The problem can be reduced to an integral geometry question of some attenuated geodesic ray transform through a pseudolinearization argument. We show injectivity (up to natural obstructions) and stability estimates for both the linear and nonlinear problems for generic simple metrics and generic connections and Higgs fields, including the real-analytic ones. We consider the problems on simple manifolds in order to make the exposition of the main ideas clear and concise, many results of this paper are still true under much weaker geometric assumptions, in particular conjugate points and trapped geodesics are allowed and the boundary is not necessarily convex.

math.AP↗

arXiv2016

The local magnetic ray transform of tensor fields

In this paper we study the local magnetic ray transform of symmetric tensor fields up to rank two on a Riemannian manifold of dimension $\geq 3$ with boundary. In particular, we consider the magnetic ray transform of the combinations of tensors of different orders due to the nature of magnetic flows. We show that such magnetic ray transforms can be stably inverted, up to natural obstructions, near a strictly convex (with respect to magnetic geodesics) boundary point. Moreover, a global invertibility result follows on a compact Riemannian manifold with strictly convex boundary assuming that some global foliation condition is satisfied.

math.DG↗

arXiv2016

The geodesic X-ray transform with matrix weights

Consider a compact Riemannian manifold of dimension $\geq 3$ 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 $\geq 3$ 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.

math.DG↗

arXiv2016

Lens rigidity with partial data in the presence of a magnetic field

In this paper we consider the lens rigidity problem with partial data for conformal metrics in the presence of a magnetic field on a compact manifold of dimension $\geq 3$ with boundary. We show that one can uniquely determine the conformal factor and the magnetic field near a strictly convex (with respect to the magnetic geodesics) boundary point where the lens data is accessible. We also prove a boundary rigidity result with partial data assuming the lengths of magnetic geodesics joining boundary points near a strictly convex boundary point are known. The local lens rigidity result also leads to a global rigidity result under some strictly convex foliation condition. A discussion of a weaker version of the lens rigidity problem with partial data for general smooth curves is given at the end of the paper.

math.DG↗