arXiv research

Shape manifold and elastic energy regularization help reconstruct complex obstacles from scattering data.

Regularizes 3D inverse scattering with tangent-point energy for better solutions.

We prove that if two non-trapping obstacles in $\mathbb{R}^n$ satisfy some rather weak non-degeneracy conditions and the scattering rays in their exteriors have (almost) the same travelling times or (almost) the same scattering length spectrum, then they coincide.

Billiard trajectories (broken generalised geodesics) are considered in the exterior of an obstacle $K$ with smooth boundary on an arbitrary Riemannian manifold. We prove a generalisation of the well-known Santalo's formula. As a consequence, it is established that if the set of trapped points has positive measure, then…

The paper deals with some problems related to recovering information about an obstacle in an Euclidean space from certain measurements of lengths of generalized geodesics in the exterior of the obstacle. The main result is that if two obstacles satisfy some generic regularity conditions and have (almost) the same trave…

Study fixed angle inverse scattering with Riemannian metrics, proving uniqueness and stability.

Study time-dependent scattering on manifolds, resolving conjectures.

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

Deep learning solves wave-based inverse problems, including super-resolution imaging.

Billiard trajectories in curved spaces have predictable travel times.

WideBNet learns inverse scattering from wide-band data efficiently and stably.

We consider scattering by an abstract compactly supported perturbation in R^n. To include the traditional cases of potential, obstacle and metric scattering without going into their particular nature we adopt the "black box" formalism developed jointly with Sjostrand [23]. It is quite likely that one could extend the r…

New method learns soliton dynamics from scattering data without assuming known equations.

We study inverse scattering for $Δ_g+V$ on $(X,g)$ a conformally compact manifold with metric $g,$ with variable sectional curvature $-\alf^2(y)$ at the boundary and $V\in C^\infty(X)$ not vanishing at the boundary. We prove that the scattering matrix at a fixed energies $(λ_1,$ $λ_2)$ in a suitable subset of $\mc$, de…

It was proved in \cite{NS1} that obstacles $K$ in $\R^d$ that are finite disjoint unions of strictly convex domains with $C^3$ boundaries are uniquely determined by the travelling times of billiard trajectories in their exteriors and also by their so called scattering length spectra. However the case $d = 2$ is not pro…

Study applies inverse scattering to BKM systems, linking spectra and integrable systems.

Based on our previous study [IS2] we develop fully the stationary scattering theory for the Schrodinger operator on a manifold possessing an escape function. A particular class of examples are manifolds with Euclidean and/or hyperbolic ends, possibly with unbounded and non-smooth obstacles. We develop the theory largel…

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

This paper is an expository account of the development of soliton mathematics, from its inception in famous numerical experiments of Fermi-Pasta-Ulam and Zabusky-Kruskal to the recent synthesis of Terng-Uhlenbeck (dg-ga/9707004) that explains hidden symmetries of soliton equations in terms of loop-groups acting by dres…

Inverse problem solved for rotationally symmetric manifolds using eigenvalues and resonances.

Paper develops formulas for shape derivatives in wave scattering.

Novel method uses Gaussian process to estimate particle sizes from scattering data.

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…

New method designs multilayer nanoparticles using AI.

Transfer learning improves lensless imaging through scattering media with fewer samples.

We discuss positivity properties of `distinguished propagators', i.e. distinguished inverses of operators that frequently occur in scattering theory and wave propagation. We relate this to the work of Duistermaat and Hörmander on distinguished parametrices (approximate inverses), which has played a major role in quantu…

We analyze the inverse problem, originally formulated by Dix in geophysics, of reconstructing the wave speed inside a domain from boundary measurements associated with the single scattering of seismic waves. We consider a domain $\tilde M$ with a varying and possibly anisotropic wave speed which we model as a Riemannia…

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…

Generative Adversarial Nets (GANs) and Variational Auto-Encoders (VAEs) provide impressive image generations from Gaussian white noise, but the underlying mathematics are not well understood. We compute deep convolutional network generators by inverting a fixed embedding operator. Therefore, they do not require to be o…

In this paper, we are concerned with the 2D and 3D geometric shape generation by prescribing a set of characteristic values of a specific geometric body. One of the major motivations of our study is the 3D human body generation in various applications. We develop a novel method that can generate the desired body with c…

We consider an inverse problem associated with some 2-dimensional non-compact surfaces with conical singularities, cusps and regular ends. Our motivating example is a Riemann surface $\mathcal M = Γ\backslash{\bf H}^2$ associated with a Fuchsian group of the 1st kind $Γ$ containing parabolic elements. $\mathcal M$ is t…

Study solves inverse problems for real principal type operators using unique data sets and ray transforms.

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 $\partial M$…

A CNN-based method improves DTI of the human heart, compensating for motion.

The space-time monopole equation is obtained from a dimension reduction of the anti-self dual Yang-Mills equation on $\R^{2,2}$. A family of Ward equations is obtained by gauge fixing from the monopole equation. In this paper, we give an introduction and a survey of the space-time monopole equation. Included are altern…

The paper proves existence and growth estimates for inverse mean curvature flow and related $p$-Laplacian Green kernel decay.

Special class of surfaces in five-dimensional sphere in $C^3$ is considered. Immersion equations for minimal tori of that class are shown to be reducible to the equation $u_{z\bar z}=e^u-e^{-2u}$ which is integrable by means of inverse scattering method. Finite-gap minimal tori are constructed.

There is a general method for constructing a soliton hierarchy from a splitting of a loop group as a positive and a negative sub-groups together with a commuting linearly independent sequence in the positive Lie subalgebra. Many known soliton hierarchies can be constructed this way. The formal inverse scattering associ…

Given a triangulated surface $M$, we use Ge-Xu's $α$-flow \cite{Ge-Xu1} to deform any initial inversive distance circle packing metric to a metric with constant $α$-curvature. More precisely, we prove that the inversive distance circle packing with constant $α$-curvature is unique if $αχ(M)\leq 0$, which generalize And…

We show that the Dirichlet-to-Neumann operator of the Laplacian on an open subset of the boundary of a connected compact Einstein manifold with boundary determines the manifold up to isometries. Similarly, for connected conformally compact Einstein manifolds of even dimension $n+1$, we prove that the scattering matrix …

We discuss $SU(2)$ Bogomolny monopoles of arbitrary charge $k$ invariant under various symmetry groups. The analysis is largely in terms of the spectral curves, the rational maps, and the Nahm equations associated with monopoles. We consider monopoles invariant under inversion in a plane, monopoles with cyclic symmetry…

For a given smooth compact manifold $M$, we introduce an open class $\mathcal G(M)$ of Riemannian metrics, which we call \emph{metrics of the gradient type}. For such metrics $g$, the geodesic flow $v^g$ on the spherical tangent bundle $SM \to M$ admits a Lyapunov function (so the $v^g$-flow is traversing). It turns ou…

Formula removes geometric patterns from random hyperbolic surfaces.

On a fixed smooth compact Riemann surface with boundary $(M_0,g)$, we show that the Cauchy data space (or Dirichlet-to-Neumann map $\mc{N}$) of the Schrödinger operator $Δ+V$ with $V\in C^2(M_0)$ determines uniquely the potential $V$. We also discuss briefly the corresponding consequences for potential scattering at 0 …

PSC classifier improves HDLSS classification on class-imbalanced data.

The nonlinear equations describing all the nonsingular pencils of metrics of constant Riemannian curvature are derived and the integrability of these nonlinear equations by the method of inverse scattering problem is proved. It is proved that all the nonsingular pairs of compatible metrics of constant Riemannian curvat…

Novel framework uses few data for Bayesian inference in imaging.

On a fixed Riemann surface $(M_0,g_0)$ with $N$ Euclidean ends and genus $g$, we show that, under a topological condition, the scattering matrix $S_V(\la)$ at frequency $\la > 0$ for the operator $Δ+V$ determines the potential $V$ if $V\in C^{1,α}(M_0)\cap e^{-γd(\cdot,z_0)^j}L^\infty(M_0)$ for all $γ>0$ and for some $…