Researchers find counterexamples to inverse problems for wave equations.
problem Inverse problems for wave equations on domains and Lorentzian manifolds.
method Constructing non-isometric Lorentzian metrics leading to same partial data measurements.
result Non-isometric Lorentzian metrics can produce identical partial data measurements.
Study fixed angle inverse scattering with Riemannian metrics, proving uniqueness and stability.
problem Inverse scattering problem with Riemannian metrics.
method Direct problem via progressing wave expansion; inverse problem using symmetry assumptions.
result Uniqueness and stability results for inverse scattering with Riemannian metrics.
Researchers solve a formally determined inverse problem in Lorentzian geometry.
problem Determining a Lorentzian metric from boundary measurements of the Dirichlet-to-Neumann map.
method New method using distorted plane wave solutions and geometric, topological, and unique continuation arguments.
result A globally hyperbolic metric agreeing with the Minkowski metric outside a compact set and having the same Dirichlet-to-Neumann map must be the Minkowski metric up to diffeomorphism.
Study uses renormalized area to determine metric expansion from minimal surfaces.
problem Recovering the expansion of asymptotically hyperbolic metrics from minimal surfaces.
method Uses renormalized area functional on minimal submanifolds to recover metric expansion.
result Proves rigidity for log-analytic metrics and determines obstruction tensor.
Paper addresses travel time tomography stability and statistical inversion.
problem Determining conformal factors of metrics from geodesic lengths.
method Established forward and inverse stability estimates; applied to Bayesian statistical inversion.
result Consistency of statistical inversion technique for travel time tomography.
Study inverse boundary value problem for Monge-Ampère equation on convex domains.
problem Determine a positive source function from the Dirichlet-to-Neumann map for Monge-Ampère equation.
method Recover Hessian as Riemannian metric, prove DN map uniqueness, develop asymptotic expansions, solve nonlocal ∂-equation. result DN map uniquely determines positive source function in convex Euclidean plane domains.
Inverse scattering result on AH manifolds determines metric up to diffeo and conformal factor.
problem Determining the metric on an AH manifold from scattering data.
method Relating eigenvalue problem to Conformal Laplacian and using Guillarmou--Guillopé and Chang--González results.
result Scattering matrix at energy 2n+1 determines jet of the metric on the boundary up to diffeomorphism and conformal factor. Fractional Laplacian inverse problem solved for connection Laplacians.
problem Determining structures from metric, bundle, and map knowledge.
method Local knowledge of metric, bundle, and map determines global structures.
result Global structures determined from local knowledge of metric, bundle, and map.
Paper shows stability of metric reconstruction for orbifolds from spectral data.
problem Determining the metric structure of collapsing orbifolds from spectral data.
method Improved quantitative unique continuation for wave operator on Riemannian manifolds.
result Quantitative stability of inverse problem for Riemannian orbifolds.
Paper examines stability of Bayesian posterior measures using integral probability metrics.
problem Stability of Bayesian inference in large-scale inverse problems.
method New families of integral probability metrics for likelihood and prior perturbations.
result Constructs new stability results for Bayesian posterior measures.
The well-known Funk metric F(x,y) is projectively flat with constant flag curvature K=-1/4 and the Hilbert metric H(x,y):=(F(x,y)+F(x,-y))/2 is projectively flat with constant curvature K=-1. These metrics are the special solutions to Hilbert's Fourth Problem. In this paper, we construct a non-trivial R-flat spray usin…
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.
We study the inverse resonance problem for conformally compact manifolds which are hyperbolic outside a compact set. Our results include compactness of isoresonant metrics in dimension two and of isophasal negatively curved metrics in dimension three. In dimensions four or higher we prove topological finiteness theorem…
Paper explores stability, regularization, and gradient flows for stochastic inverse problems.
problem Recovering random probability distributions from measurements.
method Direct inversion, variational formulation with regularization, and optimization via gradient flows.
result The choice of metric impacts stability and properties of the optimizer.
Study shows stability of travel time data reconstruction from closed subsets.
problem Reconstruction of length spaces from travel time data on a closed subset.
method Lipschitz stability proof for certain types of length spaces.
result Reconstruction of length spaces is Lipschitz stable from travel time data on a closed subset.
Stable solution found for manifold topology from boundary data.
problem Determining manifold properties from boundary data and eigenvalues.
method Quantitative stability estimates and unique continuation for the wave operator.
result Eigenvalues and boundary values determine a metric space close to the manifold.
In this article we consider the anisotropic Calderon problem and related inverse problems. The approach is based on limiting Carleman weights, introduced in Kenig-Sjoestrand-Uhlmann (Ann. of Math. 2007) in the Euclidean case. We characterize those Riemannian manifolds which admit limiting Carleman weights, and give a c…
This paper is devoted to an inverse Steklov problem for a particular class of n-dimensional manifolds having the topology of a hollow sphere and equipped with a warped product metric. We prove that the knowledge of the Steklov spectrum determines uniquely the associated warping function up to a natural invariance.
Study solves Gel'fand's inverse problem in non-smooth spaces with Ricci curvature bounds.
problem Determining a Riemannian manifold from heat kernel on subsets.
method Analyzes mRCD(K,N) spaces with synthetic Ricci curvature bounds. result Unique solvability of Gel'fand's inverse problem for compact mRCD(K,N) spaces. Proposes a method to learn both constraints and objective functions from data.
problem Data-driven inverse optimization for mixed-integer linear programs (MILPs).
method Two-stage approach: first learns constraints, then estimates objective-function weights conditioned on learned constraints.
result Proposes and validates a method for learning both objective functions and constraints from data.
This work characterizes, analytically and numerically, two major effects of the quadratic Wasserstein (W2) distance as the measure of data discrepancy in computational solutions of inverse problems. First, we show, in the infinite-dimensional setup, that the W2 distance has a smoothing effect on the inversion pro…
We present a numerical implementation of the geodesic ray transform and its inversion over functions and solenoidal vector fields on two-dimensional Riemannian manifolds. For each problem, inversion formulas previously derived in \cite{Pestov2004,Krishnan2010} are implemented in the case of simple and some non-simple m…
Abstract: Generalizes Milnor-Schwarz lemma to inverse monoids.
problem Applying Milnor-Schwarz lemma to inverse monoids.
method Two proofs provided: elementary and using Vietoris-Rips complex.
result Generalization of Milnor-Schwarz lemma to inverse monoids.
Computational method approximates homology groups of compact metric spaces.
problem Computing homology groups of compact metric spaces.
method Inversely sequence of finite topological spaces, inverse limit, homeomorphic copy, strong deformation retract.
result Approximates homology groups of compact metric spaces.
Deep learning models for inverse problems are evaluated over time.
problem Solving inverse problems for natural systems from measurements.
method Comparing deep learning approaches on benchmark tasks and proposing neural-adjoint method.
result Neural-adjoint method achieves best performance in many scenarios.
Inversive distance circle packing metric was introduced by P Bowers and K Stephenson \cite{BS} as a generalization of Thurston's circle packing metric \cite{T1}. They conjectured that the inversive distance circle packings are rigid. For nonnegative inversive distance, Guo \cite{Guo} proved the infinitesimal rigidity a…
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 …
3D metrics get scalar curvature bounds via IMCF.
problem Bounding scalar curvature for C0 metrics. method Inverse Mean Curvature Flow (IMCF) and Hawking mass monotonicity.
result Stability theorem for nonnegative scalar curvature.
The paper studies Möbius inversion on surfaces in Minkowski 3-space.
problem Understanding transformations of surfaces in Minkowski space.
method Definition and properties of Möbius inversion on surfaces in Minkowski 3-space.
result Möbius inversion preserves lines of principal curvature and degenerate metric points but not parabolic sets.
The paper solves the Steklov spectral inverse problem for conformal metrics.
problem Recovering a metric from its Steklov spectrum in dimension n≥3.
method Combines wave trace formula techniques with geodesic X-ray transform.
result Steklov isospectral metrics must coincide under real-analyticity assumption.
Researchers reconstruct simple Riemannian manifolds from boundary wave arrival times.
problem Reconstructing Riemannian manifolds from unknown interior sources and arrival times.
method Discrete metric approximation using labeled Gromov--Hausdorff distance.
result Finite-time approximations converge to the true Riemannian manifold.
Given a bounded domain M in Rn with a conformally Euclidean metric g=ρdx2, in this paper we consider the inverse problem of recovering a semigeodesic neighborhood of a domain Γ⊂∂M and the conformal factor ρ in the neighborhood from the travel time data (defined below) and the Carte…
Paired autoencoders solve inverse problems using latent space projections.
problem Solving inverse problems in scientific computing.
method Paired autoencoder framework that projects data and quantity of interest into a latent space.
result Paired autoencoders generate multiple reconstruction metrics and enable latent-space refinement for accurate data fitting.
Paper proposes a robust method for inferring parameters in multiobjective optimization.
problem Uncertainty in hypothetical decision-making problem, data quality, and parameter space.
method Wasserstein distributionally robust approach for inverse multiobjective optimization.
result WRO-IMOP minimizes worst-case expected loss over a Wasserstein ball of distributions.
Study shows stability of Schrödinger operator spectral data on a manifold.
problem Determining a manifold and potential function from spectral data.
method Approximation of spectral data on a subset to determine manifold and potential.
result Quantitative stability estimate for Schrödinger operator inverse problem.
The paper uses polyhedral expansions to capture the shape of compact metric spaces.
problem Capturing the shape of compact metric spaces using finite approximations.
method Inverse sequences of polyhedra based on finite approximations of a compact metric space.
result Proves the General Principle and computes inverse persistent homology groups.
We find coordinates, the metric tensor, the inverse metric tensor and the Laplace-Beltrami operator for the orbit space of Hamiltonian SU(2) gauge theory on a finite, rectangular lattice. This is done using a complete axial gauge fixing. The Gribov problem can be completely solved, with no remaining gauge ambiguities.
A new algorithm improves posterior sampling for linear inverse problems.
problem Efficiently sampling from posterior distributions in noisy linear inverse problems.
method Proposes \pddim, a DDIM-type sampler that separately samples along singular directions of the measurement operator.
result The method converges to the Bayesian posterior conditioned on the measurements.
Improved inverse problem solving with data consistency in diffusion models.
problem Speed and data consistency issues in diffusion model-based inverse problems.
method Data Consistent Direct Diffusion Bridges (CDDB) that ensures data consistency without fine-tuning.
result CDDB outperforms inconsistent DDB in perception and distortion metrics.
Finite approximations help reconstruct countable metric and ultrametric spaces.
problem Reconstructing countable metric and ultrametric spaces.
method Topological reconstruction using inverse limits of finite T0 spaces. result Countable metric and ultrametric spaces can be reconstructed as finite approximations.
New method finds metrics on surfaces with prescribed curvatures using circle packings and surgery.
problem Finding piecewise Euclidean metrics on surfaces with prescribed combinatorial curvatures.
method Combinatorial curvature flows with surgery for inversive distance circle packings.
result Longtime existence and global convergence of combinatorial curvature flows with surgery.
The article resolves complex structures in transport twistor spaces, proving a Newlander-Nirenberg theorem.
problem Degenerate complex structures in transport twistor spaces.
method Holomorphic blow-down structure maps to resolve degeneracy and gain insight into complex geometry.
result Global and local β-maps for various metrics, proving a Newlander-Nirenberg theorem for degenerate complex structures.
Inverse problem solved for relativistic Boltzmann equation on spacetime.
problem Determining spacetime from causal measurements.
method Using the nonlinearity of the Boltzmann equation to uniquely determine the spacetime.
result The spacetime is uniquely determined up to isometry in the causal set I+(x−)∩I−(x+). Exact posterior score estimation for solving linear inverse problems
problem Solving linear inverse problems
method Derive the exact posterior score and use it as a denoising training objective
result EPS outperforms training-free and training-based baselines on various metrics
In this article, we study the properties of the geodesic X-ray transform for asymptotically Euclidean or conic Riemannian metrics and show injectivity under non-trapping and no conjugate point assumptions. We also define a notion of lens data for such metrics and study the associated inverse problem.
The flow converges without Kähler-Einstein and develops ideal sheaves.
problem Analyzing convergence of inverse Monge-Ampere flow without Kähler-Einstein metrics.
method Generalizing the flow and providing conditions for convergence and ideal sheaves development.
result The flow converges without Kähler-Einstein metrics and develops Nadel multiplier ideal sheaves.
A solution of Hilberts fourth problem lead to integral equation of the type generalized cosine transform. The present paper considers the solution that integral equation by integral geometry methods and propose an inversion formula for reconstruction of Crofton measures from projective smooth Finsler metrics in R3.
Extending a previous paper, we present a generalization in dimension 3 of the traditional Szebehely-type inverse problem. In that traditional setting, the data are curves determined as the intersection of two families of surfaces, and the problem is to find a potential V such that the Lagrangian L = T - V, where T is t…