arXiv research

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

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.

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

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

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…

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

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…

We discuss inverse resonance scattering for the Laplacian on a rotationally symmetric manifold $M = (0,\infty) \times Y$ whose rotation radius is constant outside some compact interval. The Laplacian on $M$ is unitarily equivalent to a direct sum of one-dimensional Schrödinger operators with compactly supported potenti…

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.

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…

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

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…

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…

By introducing a shape manifold as a solution set to solve inverse obstacle scattering problems we allow the reconstruction of general, not necessarily star-shaped curves. The bending energy is used as a stabilizing term in Tikhonov regularization to gain independence of the parametrization. Moreover, we discuss how se…

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 $\partial M$…

In vivo diffusion tensor imaging (DTI) is a promising technique to investigate noninvasively the fiber structures of the in vivo human heart. However, signal loss due to motions remains a persistent problem in in vivo cardiac DTI. We propose a novel motion-compensation method for investigating in vivo myocardium struct…

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…

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…

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…

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 …

Machine learning methods for computational imaging require uncertainty estimation to be reliable in real settings. While Bayesian models offer a computationally tractable way of recovering uncertainty, they need large data volumes to be trained, which in imaging applications implicates prohibitively expensive collectio…

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…

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 $…

We solve the problem of description for nonsingular pairs of compatible flat metrics in the general N-component case. The integrable nonlinear partial differential equations describing all nonsingular pairs of compatible flat metrics (or, in other words, nonsingular flat pencils of metrics) are found and integrated. Th…

On a fixed smooth compact Riemann surface with boundary $(M_0,g)$, we show that for the Schrödinger operator $Δ+V$ with potential $V\in C^{1,α}(M_0)$ for some $α>0$, the Dirichlet-to-Neumann map $N|_Γ$ measured on an open set $Γ\subset \partial M_0$ determines uniquely the potential $V$. We also discuss briefly the cor…

We study the perturbations of two classes of static black ellipsoid solutions of four dimensional vacuum Einstein equations. Such solutions are described by generic off--diagonal metrics which are generated by anholonomic transforms of diagonal metrics. The analysis is performed in the approximation of small eccentrici…

Motivated by the prediction of cell loads in cellular networks, we formulate the following new, fundamental problem of statistical learning of geometric marks of point processes: An unknown marking function, depending on the geometry of point patterns, produces characteristics (marks) of the points. One aims at learnin…

Twisted $U$- and twisted $U/K$-hierarchies are soliton hierarchies introduced by Terng to find higher flows of the generalized sine-Gordon equation. Twisted $\frac {O(J,J)}{O(J)\times O(J)}$-hierarchies are among the most important classes of twisted hierarchies. In this paper, interesting first and higher flows of twi…

Linear problems appear in a variety of disciplines and their application for the transmission matrix recovery is one of the most stimulating challenges in biomedical imaging. Its knowledge turns any random media into an optical tool that can focus or transmit an image through disorder. Here, converting an input-output …

We deal with the problem of description of nonsingular pairs of compatible flat metrics for the general $N$-component case. We describe the scheme of the integrating the nonlinear equations describing nonsingular pairs of compatible flat metrics (or, in other words, nonsingular flat pencils of metrics). It is based on …

The 1-d Schrodinger flow on 2-sphere, the Gauss-Codazzi equation for flat Lagrangian submanifolds in C^n, and the space-time monopole equation are all examples of geometric soliton equations. The linear systems with a spectral parameter (Lax pair) associated to these equations satisfy the reality condition associated t…

New proofs confirm travel time data determine simple metrics on a disc.

New method finds all thin film structures from reflectometry data.

A large dimensional characterization of robust M-estimators of covariance (or scatter) is provided under the assumption that the dataset comprises independent (essentially Gaussian) legitimate samples as well as arbitrary deterministic samples, referred to as outliers. Building upon recent random matrix advances in the…

We provide a comprehensive study of the convergence of the forward-backward algorithm under suitable geometric conditions, such as conditioning or Łojasiewicz properties. These geometrical notions are usually local by nature, and may fail to describe the fine geometry of objective functions relevant in inverse problems…