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Determination of a Riemannian manifold from the distance difference functionsOct 21 2015Aug 24 2017Let $(N,g)$ be a Riemannian manifold with the distance function $d(x,y)$ and an open subset $M\subset N$. For $x\in M$ we denote by $D_x$ the distance difference function $D_x:F\times F\to \mathbb R$, given by $D_x(z_1,z_2)=d(x,z_1)-d(x,z_2)$, $z_1,z_2\in ... More

Hierarchical models in statistical inverse problems and the Mumford--Shah functionalAug 24 2009Sep 14 2009The Bayesian methods for linear inverse problems is studied using hierarchical Gaussian models. The problems are considered with different discretizations, and we analyze the phenomena which appear when the discretization becomes finer. A hierarchical ... More

Discrete regularization and convergence of the inverse problem for 1+1 dimensional wave equationMar 28 2018An inverse boundary value problem for the 1+1 dimensional wave equation $(\partial_t^2 - c(x)^2 \partial_x^2)u(x,t)=0,\quad x\in\mathbb{R}_+$ is considered. We give a discrete regularization strategy to recover wave speed $c(x)$ when we are given the ... More

Maxwell's Equations with Scalar Impedance: Direct and Inverse ProblemsDec 20 2002The article deals with electrodynamics in the presence of anisotropic materials having scalar wave impedance. Maxwell's equations written for differential forms over a 3-manifold are analysed. The system is extended to a Dirac type first order elliptic ... More

Analysis of regularized inversion of data corrupted by white Gaussian noiseNov 25 2013Jun 02 2016Tikhonov regularization is studied in the case of linear pseudodifferential operator as the forward map and additive white Gaussian noise as the measurement error. The measurement model for an unknown function $u(x)$ is \begin{eqnarray*} m(x) = Au(x) ... More

Inverse acoustic scattering problem in half-space with anisotropic random impedanceJul 09 2014Aug 15 2014We study an inverse acoustic scattering problem in half-space with a probabilistic impedance boundary value condition. The Robin coefficient (surface impedance) is assumed to be a Gaussian random function $\lambda = \lambda(x)$ with a pseudodifferential ... More

Regularization strategy for inverse problem for 1+1 dimensional wave equationSep 15 2015An inverse boundary value problem for a 1+1 dimensional wave equation with wave speed $c(x)$ is considered. We give a regularisation strategy for inverting the map $\mathcal A:c\mapsto \Lambda,$ where $\Lambda$ is the hyperbolic Neumann-to-Dirichlet map ... More

Inverse problems for heat equation and space-time fractional diffusion equation with one measurementMar 11 2019Given a connected compact Riemannian manifold $(M,g)$ without boundary, $\dim M\ge 2$, we consider a space--time fractional diffusion equation with an interior source that is supported on an open subset $V$ of the manifold. The time-fractional part of ... More

Inverse problem for compact Finsler manifolds with the boundary distance mapJan 12 2019We prove that the boundary distance map of a smooth compact Finsler manifold with smooth boundary determines its topological and differential structures. We construct the optimal fiberwise open subset of its tangent bundle and show that the boundary distance ... More

Inverse problems for elliptic equations with power type nonlinearitiesMar 29 2019We introduce a method for solving Calder\'on type inverse problems for semilinear equations with power type nonlinearities. The method is based on higher order linearizations, and it allows one to solve inverse problems for certain nonlinear equations ... More

Equivalence of time-domain inverse problems and boundary spectral problemsFeb 22 2002We consider inverse problems for wave, heat and Schr\"odinger-type operators and corresponding spectral problems on domains of ${\bf R}^n$ and compact manifolds. Also, we study inverse problems where coefficients of partial differential operator have ... More

An inverse problem for a hyperbolic system on a vector bundle and energy measurementsSep 22 2009Nov 09 2011A uniqueness result in the inverse problem for an inhomogeneous hyperbolic system on a real vector bundle over a smooth compact manifold, based on energy measurements for improperly known sources, is established.

Inverse Problems and Index Formulae for Dirac OperatorsJan 04 2005Oct 23 2006We consider a Dirac-type operator $D_P$ on a vector bundle $V$ over a compact Riemannian manifold $(M,g)$ with a nonempty boundary. The operator $D_P$ is specified by a boundary condition $P(u|_{\p M})=0$ where $P$ is a projector which may be a non-local, ... More

The blow-up of electromagnetic fields in 3-dimensional invisibility cloaking for Maxwell's equationsSep 13 2015Transformation optics constructions have allowed the design of cloaking devices that steer electromagnetic, acoustic and quantum waves around a region without penetrating it, so that this region is hidden from external observations. The proposed material ... More

Inverse problem for wave equation with sources and observations on disjoint setsJan 27 2010We consider an inverse problem for a hyperbolic partial differential equation on a compact Riemannian manifold. Assuming that $\Gamma_1$ and $\Gamma_2$ are two disjoint open subsets of the boundary of the manifold we define the restricted Dirichlet-to-Neumann ... More

Determination of a Riemannian manifold from the distance difference functionsOct 21 2015Let $(N,g)$ be a Riemannian manifold with the distance function $d(x,y)$ and an open subset $M\subset N$. For $x\in M$ we denote by $D_x$ the distance difference function $D_x:F\times F\to \mathbb R$, given by $D_x(z_1,z_2)=d(x,z_1)-d(x,z_2)$, $z_1,z_2\in ... More

Inverse problem for the Riemannian wave equation with Dirichlet data and Neumann data on disjoint setsAug 10 2012We consider the inverse problem to determine a smooth compact Riemannian manifold with boundary $(M, g)$ from a restriction $\Lambda_{\Src, \Rec}$ of the Dirichlet-to-Neumann operator for the wave equation on the manifold. Here $\Src$ and $\Rec$ are open ... More

Two dimensional invisibility cloaking for Helmholtz equation and non-local boundary conditionsNov 09 2010Jan 03 2011Transformation optics constructions have allowed the design of cloaking devices that steer electromagnetic, acoustic and quantum waves around a region without penetrating it, so that this region is hidden from external observations. The material pa- rameters ... More

Stability and Reconstruction in Gel'fand Inverse Boundary Spectral ProblemJan 30 2002We consider stability and approximate reconstruction of Riemannian manifold when the finite number of eigenvalues of the Laplace-Beltrami operator and the boundary values of the corresponding eigenfunctions are given. The reconstruction can be done in ... More

Inverse problems for semilinear wave equations on Lorentzian manifoldsJun 20 2016We consider inverse problems in space-time $(M, g)$, a $4$-dimensional Lorentzian manifold. For semilinear wave equations $\square_g u + H(x, u) = f$, where $\square_g$ denotes the usual Laplace-Beltrami operator, we prove that the source-to-solution ... More

Inverse boundary value problems for the perturbed polyharmonic operatorFeb 27 2011We show that a first order perturbation $A(x)\cdot D+q(x)$ of the polyharmonic operator $(-\Delta)^m$, $m\ge 2$, can be determined uniquely from the set of the Cauchy data for the perturbed polyharmonic operator on a bounded domain in $R^n$, $n\ge 3$. ... More

Reconstruction of Betti numbers of manifolds for anisotropic Maxwell and Dirac systemsMay 14 2010Jun 29 2010We consider an invariant formulation of the system of Maxwell's equations for an anisotropic medium on a compact orientable Riemannian 3-manifold $(M,g)$ with nonempty boundary. The system can be completed to a Dirac type first order system on the manifold. ... More

Determination of the Spacetime from Local Time MeasurementsDec 03 2014Jul 13 2015We consider an inverse problem for a Lorentzian spacetime $(M,g)$, and show that time measurements, that is, the knowledge of the Lorentzian time separation function on a submanifold $\Sigma\subset M$ determine the $C^\infty$-jet of the metric in the ... More

Uniqueness and Stability in Inverse Spectral Problems for Collapsing ManifoldsSep 26 2012Sep 27 2012We consider a geometric inverse problems associated with interior measurements: Assume that on a closed Riemannian manifold $(M, h)$ we can make measurements of the point values of the heat kernel on some open subset $U \subset M$. Can these measurements ... More

On Absence and Existence of the Anomalous Localized Resonance without the Quasi-static ApproximationJun 24 2014Apr 14 2015The paper considers the transmission problems for Helmholtz equation with bodies that have negative material parameters. Such material parameters are used to model metals on optical frequencies and so-called metamaterials. As the absorption of the materials ... More

Linearization stability results and active measurements for the Einstein-scalar field equationsMay 14 2014We study the Einstein equations coupled with the scalar field equations, $\hbox{Ein}(g)=T$, $T=T(g,\phi)+F^1$, and $\square_g\phi^\ell-m^2\phi^\ell= F^2$, where the sources $F=(F^1, F^2)$ correspond to perturbations of the physical fields which we control. ... More

Focusing waves in unknown media by modified time reversal iterationAug 16 2007We study the wave equation in a bounded domain or on a compact Riemannian manifold with boundary. Assume that we are given the hyperbolic Neumann-to-Dirichlet map on the boundary corresponding to physical boundary measurements. We consider how to focus ... More

Reconstruction and stability in Gel'fand's inverse interior spectral problemFeb 25 2017Oct 01 2018Assume that $M$ is a compact Riemannian manifold of bounded geometry given by restrictions on its diameter, Ricci curvature and injectivity radius. Assume we are given, with some error, the first eigenvalues of the Laplacian $\Delta_g$ on $M$ as well ... More

Discretization-invariant Bayesian inversion and Besov space priorsJan 27 2009Bayesian solution of an inverse problem for indirect measurement $M = AU + {\mathcal{E}}$ is considered, where $U$ is a function on a domain of $R^d$. Here $A$ is a smoothing linear operator and $ {\mathcal{E}}$ is Gaussian white noise. The data is a ... More

Stability of the unique continuation for the wave operator via Tataru inequality and applicationsJun 13 2015Aug 14 2015In this paper we study the stability of the unique continuation in the case of the wave equation with variable coefficients independent of time. We prove a logarithmic estimate in a arbitrary domain of ${\mathbb R}^{n+1}$, where all the parameters are ... More

Reconstruction of a compact Riemannian manifold from the scattering data of internal sourcesAug 24 2017Mar 19 2018Given 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 ... More

Optimal Acoustic MeasurementsSep 04 2000Nov 28 2000We consider the problem of obtaining information about an inaccessible half-space from acoustic measurements made in the accessible half-space. If the measurements are of limited precision, some scatterers will be undetectable because their scattered ... More

On nonuniqueness for Calderon's inverse problemFeb 20 2003Jul 01 2003We construct anisotropic conductivities with the same Dirichlet-to-Neumann map as a homogeneous isotropic conductivity. These conductivities are singular close to a surface inside the body.

Multidimensional Borg-Levinson TheoremJun 28 2004We consider the inverse problem of the reconstruction of a Schr\"odinger operator on a unknown Riemannian manifold or a domain of Euclidean space. The data used is a part of the boundary $\Gamma$ and the eigenvalues corresponding to a set of impedances ... More

Inverse problems for Lorentzian manifolds and non-linear hyperbolic equationsMay 14 2014Aug 23 2016We study two inverse problems on a globally hyperbolic Lorentzian manifold $(M,g)$. The problems are: 1. Passive observations in spacetime: Consider observations in a neighborhood $V\subset M$ of a time-like geodesic $\mu$. Under natural causality conditions, ... More

Forward and inverse scattering on manifolds with asymptotically cylindrical endsMay 11 2009We study an inverse problem for a non-compact Riemannian manifold whose ends have the following properties : On each end, the Riemannian metric is assumed to be a short-range perturbation of the metric of the form $(dy)^2 + h(x,dx)$, $h(x,dx)$ being the ... More

Inverse problem for the wave equation with a white noise sourceAug 22 2013Aug 23 2013We consider a smooth Riemannian metric tensor $g$ on $\R^n$ and study the stochastic wave equation for the Laplace-Beltrami operator $\p_t^2 u - \Delta_g u = F$. Here, $F=F(t,x,\omega)$ is a random source that has white noise distribution supported on ... More

The borderlines of the invisibility and visibility for Calderon's inverse problemSep 13 2011We consider the determination of a conductivity function in a two-dimensional domain from the Cauchy data of the solutions of the conductivity equation on the boundary. We prove uniqueness results for this inverse problem, posed by Calderon, for conductivities ... More

Maxwell's Equations with Scalar Impedance: Inverse Problems with data given on a part of the boundaryApr 15 2005May 25 2005We study Maxwell's equations in time domain in an anisotropic medium. The goal of the paper is to solve an inverse boundary value problem for anisotropies characterized by scalar impedance $\alpha$. This means that the material is conformal, i.e., the ... More

Calderon's inverse problem for anisotropic conductivity in the planeJan 29 2004We study inverse conductivity problem for an anisotropic conductivity in $L^\infty$ in bounded and unbounded domains. Also, we give applications of the results in the case when Dirichlet-to-Neumann and Neumann-to-Dirichlet maps are given only on a part ... More

Conic singularities, generalized scattering matrix, and inverse scattering on asymptotically hyperbolic surfacesAug 07 2011We 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 $\mathcal M = \Gamma\backslash{\bf H}^2$ associated with a Fuchsian ... More

The inverse conductivity problem with an imperfectly known boundary in three dimensionsJun 26 2006We consider the inverse conductivity problem in a strictly convex domain whose boundary is not known. Usually the numerical reconstruction from the measured current and voltage data is done assuming the domain has a known fixed geometry. However, in practical ... More

Rigidity of broken geodesic flow and inverse problemsMar 17 2007Consider a broken geodesics $\alpha([0,l])$ 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 ... More

Inverse conductivity problem with an imperfectly known boundaryAug 18 2004We show how to eliminate the error caused by an incorrectly modeled boundary in electrical impedance tomography (EIT). In practical measurements, one usually lacks the exact knowledge of the boundary. Because of this the numerical reconstruction from ... More

Spectral theory and inverse problem on asymptotically hyperbolic orbifoldsDec 02 2013We consider an inverse problem associated with $n$-dimensional asymptotically hyperbolic orbifolds $(n \geq 2)$ having a finite number of cusps and regular ends. By observing solutions of the Helmholtz equation at the cusp, we introduce a generalized ... More

Inverse problems for Lorentzian manifolds and non-linear hyperbolic equationsMay 14 2014Sep 20 2017We study two inverse problems on a globally hyperbolic Lorentzian manifold $(M,g)$. The problems are: 1. Passive observations in spacetime: Consider observations in a neighborhood $V\subset M$ of a time-like geodesic $\mu$. Under natural causality conditions, ... More

Inverse problems in spacetime I: Inverse problems for Einstein equations - Extended preprint versionMay 18 2014We consider inverse problems for the coupled Einstein equations and the matter field equations on a 4-dimensional globally hyperbolic Lorentzian manifold $(M,g)$. We give a positive answer to the question: Do the active measurements, done in a neighborhood ... More

Inverse problems with partial data for a magnetic Schrödinger operator in an infinite slab and on a bounded domainApr 05 2011In this paper we study inverse boundary value problems with partial data for the magnetic Schr\"odinger operator. In the case of an infinite slab in $R^n$, $n\ge 3$, we establish that the magnetic field and the electric potential can be determined uniquely, ... More

Determining a first order perturbation of the biharmonic operator by partial boundary measurementsMar 01 2011We consider an operator $\Delta^2 + A(x)\cdot D+q(x)$ with the Navier boundary conditions on a bounded domain in $R^n$, $n\ge 3$. We show that a first order perturbation $A(x)\cdot D+q$ can be determined uniquely by measuring the Dirichlet--to--Neumann ... More

The Poisson embedding approach to the Calderón problemJun 13 2018We introduce a new approach to the anisotropic Calder\'on problem, based on a map called Poisson embedding that identifies the points of a Riemannian manifold with distributions on its boundary. We give a new uniqueness result for a large class of Calder\'on ... More

Determining electrical and heat transfer parameters using coupled boundary measurementsDec 14 2010Let $\Omega\subset\R^n$, $n\ge 3$, be a smooth bounded domain and consider a coupled system in $\Omega$ consisting of a conductivity equation $\nabla \cdot \gamma(x) \nabla u(t,x)=0$ and an anisotropic heat equation $\kappa^{-1}(x)\partial_t\psi(t,x)=\nabla\cdot ... More

Posterior consistency and convergence rates for Bayesian inversion with hypoelliptic operatorsJul 07 2015May 24 2016Bayesian approach to inverse problems is studied in the case where the forward map is a linear hypoelliptic pseudodifferential operator and measurement error is additive white Gaussian noise. The measurement model for an unknown Gaussian random variable ... More

Inverse problems in spacetime I: Inverse problems for Einstein equationsMay 20 2014We consider inverse problems for the coupled Einstein equations and the matter field equations on a 4-dimensional globally hyperbolic Lorentzian manifold $(M,g)$. We give a positive answer to the question: Do the active measurements, done in a neighborhood ... More

Hyperbolic inverse problem with data on disjoint setsFeb 11 2016We consider a restricted Dirichlet-to-Neumann map associated to a wave type operator on a Riemannian manifold with boundary. The restriction corresponds to the case where the Dirichlet traces are supported on one subset of the boundary and the Neumann ... More

Inverse scattering for a random potentialMay 27 2016Jul 12 2016In this paper we consider an inverse problem for the $n$-dimensional random Schr\"{o}dinger equation $(\Delta-q+k^2)u = 0$. We study the scattering of plane waves in the presence of a potential $q$ which is assumed to be a Gaussian random function such ... More

The Calderón problem for the conformal LaplacianDec 23 2016We consider a conformally invariant version of the Calder\'on problem, where the objective is to determine the conformal class of a Riemannian manifold with boundary from the Dirichlet-to-Neumann map for the conformal Laplacian. The main result states ... More

Inverse spectral problems on a closed manifoldSep 13 2007In this paper we consider two inverse problems on a closed connected Riemannian manifold $(M,g)$. The first one is a direct analog of the Gel'fand inverse boundary spectral problem. To formulate it, assume that $M$ is divided by a hypersurface $\Sigma$ ... More

A Direct Reconstruction Method for Anisotropic Electrical Impedance TomographyFeb 05 2014Jun 24 2015A novel computational, non-iterative and noise-robust reconstruction method is introduced for the planar anisotropic inverse conductivity problem. The method is based on bypassing the unstable step of the reconstruction of the values of the isothermal ... More

Inverse problems for differential forms on Riemannian manifolds with boundaryJul 06 2010Consider a real-analytic orientable connected complete Riemannian manifold $M$ with boundary of dimension $n\ge 2$ and let $k$ be an integer $1\le k\le n$. In the case when $M$ is compact of dimension $n\ge 3$, we show that the manifold and the metric ... More

An inverse problem for the wave equation with one measurement and the pseudorandom noiseNov 10 2010We consider the wave equation $(\p_t^2-\Delta_g)u(t,x)=f(t,x)$, in $\R^n$, $u|_{\R_-\times \R^n}=0$, where the metric $g=(g_{jk}(x))_{j,k=1}^n$ is known outside an open and bounded set $M\subset \R^n$ with smooth boundary $\p M$. We define a deterministic ... More

The Calderon problem for conormal potentials, I: Global uniqueness and reconstructionDec 12 2001Oct 04 2002In dimensions greater than or equal to three, we establish global uniqueness and obtain reconstruction in the Calderon problem for the Schrodinger equation with certain singular potentials. The potentials considered are conormal of order less than 1-k ... More

Inverse problems and invisibility cloaking for FEM models and resistor networksJul 05 2013Jul 09 2013In this paper we consider inverse problems for resistor networks and for models obtained via the Finite Element Method (FEM) for the conductivity equation. These correspond to discrete versions of the inverse conductivity problem of Calder\'on. We characterize ... More

Determination of structures in the space-time from local measurements: a detailed expositionMay 08 2013May 29 2013We consider inverse problems for the Einstein equation with a time-depending metric on a 4-dimensional globally hyperbolic Lorentzian manifold $(M,g)$. We formulate the concept of active measurements for relativistic models. We do this by coupling the ... More

Comment on "Scattering Theory Derivation of a 3D Acoustic Cloaking Shell"Jan 21 2008In a recent Letter, Cummer et al. give a description of material parameters for acoustic wave propagation giving rise to a 3D spherical cloak, and verify the cloaking phenomenon on the level of scattering coefficients. A similar configuration has been ... More

Reconstruction and interpolation of manifolds I: The geometric Whitney problemAug 04 2015We study the geometric Whitney problem on how a Riemannian manifold $(M,g)$ can be constructed to approximate a metric space $(X,d_X)$. This problem is closely related to manifold interpolation (or manifold learning) where a smooth $n$-dimensional surface ... More

Correlation based passive imaging with a white noise sourceSep 26 2016Nov 04 2016Passive imaging refers to problems where waves generated by unknown sources are recorded and used to image the medium through which they travel. The sources are typically modelled as a random variable and it is assumed that some statistical information ... More

The Calderon problem in transversally anisotropic geometriesMay 06 2013May 12 2014We consider the anisotropic Calderon problem of recovering a conductivity matrix or a Riemannian metric from electrical boundary measurements in three and higher dimensions. In the earlier work \cite{DKSaU}, it was shown that a metric in a fixed conformal ... More

Correlation based passive imaging with a white noise sourceSep 26 2016Passive imaging refers to problems where waves generated by unknown sources are recorded and used to image the medium through which they travel. The sources are typically modelled as a random variable and it is assumed that some statistical information ... More

Detection of Hermitian connections in wave equations with cubic non-linearityFeb 15 2019We consider the geometric non-linear inverse problem of recovering a Hermitian connection $A$ from the source-to-solution map of the cubic wave equation $\Box_{A}\phi+\kappa |\phi|^{2}\phi=f$, where $\kappa\neq 0$ and $\Box_{A}$ is the connection wave ... More

Partial data inverse problems and simultaneous recovery of boundary and coefficients for semilinear elliptic equationsMay 07 2019We study various partial data inverse boundary value problems for the semilinear elliptic equation $\Delta u+ a(x,u)=0$ in a domain in $\mathbb R^n$ by using the higher order linearization technique introduced in [LLS 19, FO19]. We show that the Dirichlet-to-Neumann ... More

Approximate quantum and acoustic cloakingDec 09 2008At any energy E > 0, we construct a sequence of bounded potentials $V^E_{n}, n\in\N$, supported in an annular region $B_{out}\setminus B_{inn}$ in three-space, which act as approximate cloaks for solutions of Schr\"odinger's equation: For any potential ... More

Electromagnetic wormholes via handlebody constructionsApr 06 2007Cloaking devices are prescriptions of electrostatic, optical or electromagnetic parameter fields (conductivity $\sigma(x)$, index of refraction $n(x)$, or electric permittivity $\epsilon(x)$ and magnetic permeability $\mu(x)$) which are piecewise smooth ... More

Superdimensional Metamaterial ResonatorsSep 11 2014We propose a fundamentally new method for the design of metamaterial arrays, valid for any waves modeled by the Helmholtz equation, including scalar optics and acoustics. The design and analysis of these devices is based on eigenvalue and eigenfunction ... More

On the inverse problem of finding cosmic strings and other topological defectsMay 12 2015We consider how microlocal methods developed for tomographic problems can be used to detect singularities of the Lorentzian metric of the Universe using measurements of the Cosmic Microwave Background radiation. The physical model we study is mathematically ... More

The Light Ray transform on Lorentzian manifoldsJul 04 2019We study the weighted light ray transform $L$ of integrating functions on a Lorentzian manifold over lightlike geodesics. We analyze $L$ as a Fourier Integral Operator and show that if there are no conjugate points, one can recover the spacelike singularities ... More

Full-wave invisibility of active devices at all frequenciesNov 07 2006Mar 16 2007There has recently been considerable interest in the possibility, both theoretical and practical, of invisibility (or "cloaking") from observation by electromagnetic (EM) waves. Here, we prove invisibility, with respect to solutions of the Helmholtz and ... More

Cloaking a sensor via transformation opticsDec 09 2009Nov 14 2010It is generally believed that transformation optics based cloaking, besides rendering the cloaked region invisible to detection by scattering of incident waves, also shields the region from those same waves. We demonstrate a coupling between the cloaked ... More

Effectiveness and improvement of cylindrical cloaking with the SHS liningJul 09 2007We analyze, both analytically and numerically, the effectiveness of cloaking an infinite cylinder from observations by electromagnetic waves in three dimensions. We show that, as truncated approximations of the ideal permittivity and permeability tensors ... More

Inverse problem for Einstein-scalar field equationsMay 20 2014Jan 05 2018The paper introduces a method to solve inverse problems for hyperbolic systems where the leading order terms are non-linear. We apply the method to the coupled Einstein-scalar field equations and study the question whether the structure of spacetime can ... More

Propagation and recovery of singularities in the inverse conductivity problemOct 06 2016Sep 12 2017The ill-posedness of Calder\'on's inverse conductivity problem, responsible for the poor spatial resolution of Electrical Impedance Tomography (EIT), has been an impetus for the development of hybrid imaging techniques, which compensate for this lack ... More

Schrodinger's Hat: Electromagnetic, acoustic and quantum amplifiers via transformation opticsJul 23 2011The advent of transformation optics and metamaterials has made possible devices producing extreme effects on wave propagation. Here we give theoretical designs for devices, Schr\"odinger hats, acting as invisible concentrators of waves. These exist for ... More

Invisibility and Inverse ProblemsOct 01 2008This survey of recent developments in cloaking and transformation optics is an expanded version of the lecture by Gunther Uhlmann at the 2008 Annual Meeting of the American Mathematical Society.

Electromagnetic wormholes and virtual magnetic monopolesMar 20 2007We describe new configurations of electromagnetic (EM) material parameters, the electric permittivity $\epsilon$ and magnetic permeability $\mu$, that allow one to construct from metamaterials objects that function as invisible tunnels. These allow EM ... More

Propagation and recovery of singularities in the inverse conductivity problemOct 06 2016The ill-posedness of Calder\'on's inverse conductivity problem, responsible for the poor spatial resolution of Electrical Impedance Tomography (EIT), has been an impetus for the development of hybrid imaging techniques, which compensate for this lack ... More

Reconstruction and interpolation of manifolds I: The geometric Whitney problemAug 04 2015Nov 17 2018We study the geometric Whitney problem on how a Riemannian manifold $(M,g)$ can be constructed to approximate a metric space $(X,d_X)$. This problem is closely related to manifold reconstruction where a smooth $n$-dimensional submanifold $S\subset {\mathbb ... More

Reconstruction of a Riemannian manifold from noisy intrinsic distancesMay 17 2019We consider reconstruction of a manifold, or, invariant manifold learning, where a smooth Riemannian manifold $M$ is determined from intrinsic distances (that is, geodesic distances) of points in a discrete subset of $M$. In the studied problem the Riemannian ... More

Time reversal methods in unknown medium and inverse problemsJan 04 2007A novel method to solve inverse problems for the wave equation is introduced. The method is a combination of the boundary control method and an iterative time reversal scheme, leading to adaptive imaging of coefficient functions of the wave equation using ... More

Correlation imaging in inverse scattering is tomography on probability distributionsMay 15 2018Scattering from a non-smooth random field on the time domain is studied for plane waves that propagate simultaneously through the potential in variable angles. We first derive sufficient conditions for stochastic moments of the field to be recovered from ... More

Hyperbolic inverse problem with data on disjoint setsFeb 11 2016Jun 14 2018We consider a restricted Dirichlet-to-Neumann map associated to a wave type operator on a Riemannian manifold with boundary. The restriction corresponds to the case where the Dirichlet traces are supported on one subset of the boundary and the Neumann ... More

The Novikov-Veselov Equation and the Inverse Scattering Method, Part I: AnalysisMay 19 2011The Novikov-Veselov (NV) equation is a (2+1)-dimensional nonlinear evolution equation that generalizes the (1+1)-dimensional Korteweg-deVries (KdV) equation. Solution of the NV equation using the inverse scattering method has been discussed in the literature, ... More

Integral equations and boundary-element solution for static potential in a general piece-wise homogeneous volume conductorApr 29 2016Sep 05 2016Boundary element methods (BEM) are used for forward computation of bioelectromagnetic fields in multi-compartment volume conductor models. Most BEM approaches assume that each compartment is in contact with at most one external compartment. In this work, ... More

Quantum kinetic theory with nonlocal coherenceJun 17 2009In this thesis we develop a novel approximation scheme (eQPA), where the effects of nonlocal coherence are included in the kinetic approach to nonequilibrium quantum dynamics. The key element in our formalism is the finding of new singular shell solutions, ... More

Theta angle in holographic QCDNov 22 2016V-QCD is a class of effective holographic models for QCD which fully includes the backreaction of quarks to gluon dynamics. The physics of the theta-angle and the axial anomaly can be consistently included in these models. We analyze their phase diagrams ... More

Unbiased estimators and multilevel Monte CarloDec 03 2015Dec 22 2015Multilevel Monte Carlo (MLMC) and unbiased estimators recently proposed by McLeish (Monte Carlo Methods Appl., 2011) and Rhee and Glynn (Oper. Res., 2015) are closely related. This connection is elaborated by presenting a new general class of unbiased ... More

Unbiased estimators and multilevel Monte CarloDec 03 2015Nov 29 2016Multilevel Monte Carlo (MLMC) and unbiased estimators recently proposed by McLeish (Monte Carlo Methods Appl., 2011) and Rhee and Glynn (Oper. Res., 2015) are closely related. This connection is elaborated by presenting a new general class of unbiased ... More

Grapham: Graphical Models with Adaptive Random Walk Metropolis AlgorithmsNov 25 2008Sep 02 2009Recently developed adaptive Markov chain Monte Carlo (MCMC) methods have been applied successfully to many problems in Bayesian statistics. Grapham is a new open source implementation covering several such methods, with emphasis on graphical models for ... More

Generalized Compactification in Heterotic String TheoryApr 15 2012In this thesis, we consider heterotic string vacua based on a warped product of a four-dimensional domain wall and a six-dimensional internal manifold preserving only two supercharges. Thus, they correspond to half-BPS states of heterotic supergravity. ... More

Geometric properties of quasiconformal mapsMar 23 2007Our goal is to provide a survey of some topics in quasiconformal analysis of current interest. We try to emphasize ideas and leave proofs and technicalities aside. Several easily stated open problems are given. Most of the results are joint work with ... More

Unbiased estimators and multilevel Monte CarloDec 03 2015May 11 2017Multilevel Monte Carlo (MLMC) and unbiased estimators recently proposed by McLeish (Monte Carlo Methods Appl., 2011) and Rhee and Glynn (Oper. Res., 2015) are closely related. This connection is elaborated by presenting a new general class of unbiased ... More

On the stability and ergodicity of adaptive scaling Metropolis algorithmsMar 24 2009Apr 05 2011The stability and ergodicity properties of two adaptive random walk Metropolis algorithms are considered. The both algorithms adjust the scaling of the proposal distribution continuously based on the observed acceptance probability. Unlike the previously ... More