Showing papers by "Yaroslav Kurylev published in 2014"

A graph discretization of the laplace-beltrami operator

Abstract: We show that eigenvalues and eigenfunctions of the Laplace–Beltrami operator on a Riemannian manifold are approximated by eigenvalues and eigenvectors of a (suitably weighted) graph Laplace operator of a proximity graph on an epsilon-net.

Inverse problems for Lorentzian manifolds and non-linear hyperbolic equations

Abstract: We 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, we reconstruct the conformal type of the unknown open, relatively compact set $W\subset M$, when we are given $V$, the conformal class of $g|_V$, and the light observations sets $P_V(q)$ corresponding to all source points $q$ in $W$. The light observation set $P_V(q)$ is the intersection of $V$ and the light-cone emanating from the point $q$, i.e., the points in the set $V$ where light from a point source at $q$ is observed. 2. Active measurements in spacetime: We develop a new method for inverse problems for non-linear hyperbolic equations that utilizes the non-linearity as a tool. This enables us to solve inverse problems for non-linear equations for which the corresponding problems for linear equations are still unsolved. To illustrate this method, we solve an inverse problem for semilinear wave equations with quadratic non-linearities. We assume that we are given the neighborhood $V$ of the time-like geodesic $\mu$ and the source-to-solution operator that maps the source supported on $V$ to the restriction of the solution of the wave equation in $V$. When $M$ is 4-dimensional, we show that these data determine the topological, differentiable, and conformal structures of the spacetime in the maximal set where waves can propagate from $\mu$ and return back to $\mu$.

Introduction to Spectral Theory and Inverse Problem on Asymptotically Hyperbolic Manifolds

Abstract: This manuscript is devoted to a rigorous and detailed exposition of the spectral theory and associated forward and inverse scattering problems for the Laplace-Beltrami operators on asymptotically hyperbolic manifolds. Based upon the classical stationary scattering theory in n, the key point of the approach is the generalized Fourier transform, which serves as the basic tool to introduce and analyse the time-dependent wave operators and the S-matrix. The crucial role is played by the characterization of the space of the scattering solutions for the Helmholtz equations utilizing a properly defined Besov-type space. After developing the scattering theory, we describe, for some cases, the inverse scattering on the asymptotically hyperbolic manifolds by adopting, for the considered case, the boundary control method for inverse problems.The manuscript is aimed at graduate students and young mathematicians interested in spectral and scattering theories, analysis on hyperbolic manifolds and theory of inverse problems. We try to make it self-consistent and, to a large extent, not dependent on the existing treatises on these topics. To our best knowledge, it is the first comprehensive description of these theories in the context of the asymptotically hyperbolic manifolds.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets

Inverse problem for Einstein-scalar field equations

Abstract: The 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 be determined by making active measurements near the world line of an observer. We show that such measurements determine the topological, differential and conformal structure of the spacetime in the optimal chronological diamond type set containing the world line. In the case when the unknown part of the spacetime is vacuum, we can also determine the metric itself. We exploit the non-linearity of the equation to obtain a rich set of propagating singularities, produced by a non-linear interaction of singularities that propagate initially as for linear wave equations. This non-linear effect is then used as a tool to solve the inverse problem for the non-linear system. The method works even in cases where the corresponding inverse problems for linear equations remain open, and it can potentially be applied to a large class of inverse problems for non-linear hyperbolic equations encountered in practical imaging problems.

Inverse problems in spacetime I: Inverse problems for Einstein equations - Extended preprint version

Abstract: We 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 $U\subset M$ of a freely falling observed $\mu=\mu([s_-,s_+])$, determine the conformal structure of the spacetime in the minimal causal diamond-type set $V_g=J_g^+(\mu(s_-))\cap J_g^-(\mu(s_+))\subset M$ containing $\mu$? More precisely, we consider the Einstein equations coupled with the scalar field equations and study the system $Ein(g)=T$, $T=T(g,\phi)+F_1$, and $\square_g\phi-\mathcal V^\prime(\phi)=F_2$, where the sources $F=(F_1,F_2)$ correspond to perturbations of the physical fields which we control. The sources $F$ need to be such that the fields $(g,\phi,F)$ are solutions of this system and satisfy the conservation law $ abla_jT^{jk}=0$. Let $(\hat g,\hat \phi)$ be the background fields corresponding to the vanishing source $F$. We prove that the observation of the solutions $(g,\phi)$ in the set $U$ corresponding to sufficiently small sources $F$ supported in $U$ determine $V_{\hat g}$ as a differentiable manifold and the conformal structure of the metric $\hat g$ in the domain $V_{\hat g}$. The methods developed here have potential to be applied to a large class of inverse problems for non-linear hyperbolic equations encountered e.g. in various practical imaging problems.

Inverse problems in spacetime II: Reconstruction of a Lorentzian manifold from light observation sets

Abstract: We 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, we reconstruct the conformal type of the unknown open, relatively compact set $W\subset M$, when we are given $V$, the conformal class of $g|_V$, and the light observations sets $P_V(q)$ corresponding to all source points $q$ in $W$. The light observation set $P_V(q)$ is the intersection of $V$ and the light-cone emanating from the point $q$, i.e., the points in the set $V$ where light from a point source at $q$ is observed. 2. Active measurements in spacetime: We develop a new method for inverse problems for non-linear hyperbolic equations that utilizes the non-linearity as a tool. This enables us to solve inverse problems for non-linear equations for which the corresponding problems for linear equations are still unsolved. To illustrate this method, we solve an inverse problem for semilinear wave equations with quadratic non-linearities. We assume that we are given the neighborhood $V$ of the time-like geodesic $\mu$ and the source-to-solution operator that maps the source supported on $V$ to the restriction of the solution of the wave equation in $V$. When $M$ is 4-dimensional, we show that these data determine the topological, differentiable, and conformal structures of the spacetime in the maximal set where waves can propagate from $\mu$ and return back to $\mu$.

Seeing through spacetime

Abstract: We study two inverse problems on a globally hyperbolic Lorentzian manifold (M,g). We consider measurements done in a neighborhood V of a time-like, future directed geodesic µ that connects p − to p + . The studied problems are: 1. Active measurements in spacetime: We consider inverse problems for non-linear hyperbolic equations and develop a method that utilizes the non-linearity as a benefit. The method is demonstrated for the non-linear wave equationgu + au 2 = F. We consider the source-to- solution map LV : F → u|V that maps the source F supported on V to the restriction of the solution u on V. When M is 4-dimensional, we show that the set V , the metric g|V on it, and the map LV determine the topological, differentiable, and conformal structures of the spacetime in the maximal set where waves can propagate from µ and return back to µ. We use the non-linearity of the wave equation to reduce the problem of active measurements to passive measurements. 2. Passive observations in spacetime: We assume that we are given V , g|V, and the light observations sets PV (q) corresponding to all (or a dense subset of) source points q in W ⊂ M. The light observation set PV (q) is the intersection of V and the light-cone emanating from the point q. When W ⊂ M is a relatively compact, connected, open set which all points are in the chronological past of the point p + but not in the past of the point p − , we show that the given data determine the conformal type of (W,g|W). Under assumption that the space-time is Ricci-flat we can also determine the whole metric tensor in W. The methods developed here have the potential to be applied to a large class of inverse problems for non-linear hyperbolic equations en- countered in various practical imaging problems and other problems in mathematical physics.

Linearization stability results and active measurements for the einstein-scalar field equations

Abstract: We consider linearization stability results for the cou- pled Einstein equations and the scalar field equations for the metric g and scalar fields φ = (φ l ) L=1 on a 4-dimensional globally hyperbolic Lorentzian manifold (M,g). More precisely, we study the Einstein equations coupled with the scalar field equations and study the sys- tem Ein(g) = T, T = T(g,φ) + F 1 , andgφ l − m 2 φ l = F 2 , where the sources F = (F 1 ,F 2 ) correspond to perturbations of the physi- cal fields which we control. The sources F need to be such that the fields (g,φ,F) are solutions of this system and satisfy the conservation law divg(T) = 0. If (ge,φe) solves the above equations, the derivatives u

Superdimensional Metamaterial Resonators

Abstract: We 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 asymptotics of solutions to Schrodinger wave equations with harmonic and degenerate potentials. These resonators behave superdimensionally, with a higher local density of eigenvalues and greater concentration of waves than expected from the physical dimension, e.g., planar resonators function as 3- or higher-dimensional media, and bulk material as effectively of dimension 4 or higher. Applications include antennas with a high density of resonant frequencies and giant focussing, and are potentially broadband.

Recent progress of inverse scattering theory on non-compact manifolds

Abstract: We give a brief survey for the recent development of inverse scattering theory on non-compact Riemannian manifolds. The main theme is the reconstruction of the manifold and the metric from the scattering matrix.