arXiv subjects

Yaroslav Kurylev

Publications and source records attributed to Yaroslav Kurylev.

At least 19 recordsLinked to original sources

arXiv2024

Inverse Spectral Problems for Collapsing Manifolds I: Uniqueness and Stability

We consider the geometric inverse problem of determining a closed Riemannian manifold from measurements of the heat kernel in an open subset of the manifold. In this paper we analyze the stability of this problem in the class of $n$-dimensional Riemannian manifolds with bounded diameter and sectional curvature. It is well-known that a sequence in this class of manifolds can collapse to a lower dimensional stratified space when the injectivity radius of the sequence of manifolds goes to zero. We prove the uniqueness of the inverse problem on the limiting spaces of the collapsing manifolds. As a result, we obtain stability results for the inverse problem in the class of manifolds with bounded diameter and sectional curvature.

math.DG↗

arXiv2024

Homogenization on parallelizable Riemannian manifolds

We consider the problem of finding the homogenization limit of oscillating linear elliptic equations on an arbitrary parallelizable manifold $(M,g,Γ)$. We replicate the concept of two-scale convergence by pulling back tensors $T$ defined on the torus bundle $\mathbb{T}M$ to $M$. The process consist of two steps: localization in the slow variable through Voronoi domains, and inducing local periodicity in the fast variable from the local exponential map in combination with the geometry of the torus bundle. The procedure yields explicit cell formulae for the homogenization limit and as a byproduct a theory of two-scale convergence of tensors of arbitrary order.

math.AP↗

arXiv2022

Approximations of the connection Laplacian spectra

We consider a convolution-type operator on vector bundles over metric-measure spaces. This operator extends the analogous convolution Laplacian on functions in our earlier work to vector bundles, and is a natural extension of the graph connection Laplacian. We prove that for Euclidean or Hermitian connections on closed Riemannian manifolds, the spectrum of this operator and that of the graph connection Laplacian both approximate the spectrum of the connection Laplacian.

math.AP↗

arXiv2019

Reconstruction and stability in Gel'fand's inverse interior spectral problem

Assume 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 $Δ_g$ on $M$ as well as the corresponding eigenfunctions restricted on an open set in $M$. We then construct a stable approximation to the manifold $(M,g)$. Namely, we construct a metric space and a Riemannian manifold which differ, in a proper sense, just a little from $M$ when the above data are given with a small error. We give an explicit $\log\log$-type stability estimate on how the constructed manifold and the metric on it depend on the errors in the given data. Moreover a similar stability estimate is derived for the Gel'fand's inverse problem. The proof is based on methods from geometric convergence, a quantitative stability estimate for the unique continuation and a new version of the geometric Boundary Control method.

math.AP↗

arXiv2019

Reconstruction and interpolation of manifolds I: The geometric Whitney problem

We 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 $S\subset {\mathbb R}^m$, $m>n$ needs to be constructed to approximate a point cloud in ${\mathbb R}^m$. These questions are encountered in differential geometry, machine learning, and in many inverse problems encountered in applications. The determination of a Riemannian manifold includes the construction of its topology, differentiable structure, and metric. We give constructive solutions to the above problems. Moreover, we characterize the metric spaces that can be approximated, by Riemannian manifolds with bounded geometry: We give sufficient conditions to ensure that a metric space can be approximated, in the Gromov-Hausdorff or quasi-isometric sense, by a Riemannian manifold of a fixed dimension and with bounded diameter, sectional curvature, and injectivity radius. Also, we show that similar conditions, with modified values of parameters, are necessary. As an application of the main results we give a new characterisation of Alexandrov spaces with two-sided curvature bounds. Moreover, we characterise the subsets of Euclidean spaces that can be approximated in the Hausdorff metric by submanifolds of a fixed dimension and with bounded principal curvatures and normal injectivity radius. We develop algorithmic procedures that solve the geometric Whitney problem for a metric space and the manifold reconstruction problem in Euclidean space, and estimate the computational complexity of these procedures.

math.DG↗

arXiv2018

Spectral stability of metric-measure Laplacians

We consider a "convolution mm-Laplacian" operator on metric-measure spaces and study its spectral properties. The definition is based on averaging over small metric balls. For reasonably nice metric-measure spaces we prove stability of convolution Laplacian's spectrum with respect to metric-measure perturbations and obtain Weyl-type estimates on the number of eigenvalues.

math.SP↗

arXiv2018

Hyperbolic inverse problem with data on disjoint sets

We 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 traces are restricted on another subset. We show that the restricted Dirichlet-to-Neumann map determines the lower order terms in the wave equation, up the natural gauge invariances, along a convex foliation of the manifold. We allow the lower order terms to be non-self-adjoint, and in particular, the corresponding physical system may have dissipation of energy.

math.AP↗

arXiv2018

Inverse problem for Einstein-scalar field equations

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.

math.AP↗

arXiv2017

The linearized Calderon problem in transversally anisotropic geometries

In this article we study the linearized anisotropic Calderon problem. In a compact manifold with boundary, this problem amounts to showing that products of harmonic functions form a complete set. Assuming that the manifold is transversally anisotropic, we show that the boundary measurements determine an FBI type transform at certain points in the transversal manifold. This leads to recovery of transversal singularities in the linearized problem. The method requires a geometric condition on the transversal manifold related to pairs of intersecting geodesics, but it does not involve the geodesic X-ray transform which has limited earlier results on this problem.

math.AP↗

arXiv2017

Inverse problems for Lorentzian manifolds and non-linear hyperbolic equations

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 $μ$. 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 $μ$ 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 $μ$ and return back to $μ$.

math.DG↗

arXiv2017

Inverse problems for the connection Laplacian

We reconstruct a Riemannian manifold and a Hermitian vector bundle with compatible connection from the hyperbolic Dirichlet-to-Neumann operator associated with the wave equation of the connection Laplacian. The boundary data is local and the reconstruction is up to the natural gauge transformations of the problem. As a corollary we derive an elliptic analogue of the main result which solves a Calderon problem for connections on a cylinder.

math.AP↗

arXiv2015

Stability of the unique continuation for the wave operator via Tataru inequality and applications

In 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 calculated explicitly in terms of the $C^1$-norm of the coefficients and on the other geometric properties of the problem. We use the Carleman-type estimate proved by Tataru in 1995 and an iteration for locals stability. We apply the result to the case of a wave equation with data on a cylinder an we get a stable estimate for any positive time, also after the first conjugate point for the geodesics of the metric related to the variable coefficients.

math.AP↗

arXiv2014

A graph discretization of the Laplace-Beltrami operator

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.

math.AP↗

arXiv2014

Superdimensional Metamaterial Resonators

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 Schrödinger 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.

physics.optics↗

arXiv2014

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

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 $μ=μ([s_-,s_+])$, determine the conformal structure of the spacetime in the minimal causal diamond-type set $V_g=J_g^+(μ(s_-))\cap J_g^-(μ(s_+))\subset M$ containing $μ$? More precisely, we consider the Einstein equations coupled with the scalar field equations and study the system $Ein(g)=T$, $T=T(g,ϕ)+F_1$, and $\square_gϕ-\mathcal V^\prime(ϕ)=F_2$, 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 $(g,ϕ,F)$ are solutions of this system and satisfy the conservation law $\nabla_jT^{jk}=0$. Let $(\hat g,\hat ϕ)$ be the background fields corresponding to the vanishing source $F$. We prove that the observation of the solutions $(g,ϕ)$ 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.

math.AP↗

arXiv2014

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

We study the Einstein equations coupled with the scalar field equations, $\hbox{Ein}(g)=T$, $T=T(g,ϕ)+F^1$, and $\square_gϕ^\ell-m^2ϕ^\ell= F^2$, where the sources $F=(F^1, F^2)$ correspond to perturbations of the physical fields which we control. Here $ϕ=(ϕ^\ell)_{\ell=1}^L$ and $(M,g)$ is a 4-dimensional globally hyperbolic Lorentzian manifold. The sources $F$ need to be such that the fields $(g,ϕ,F)$ satisfy the conservation law $\hbox{div}_g(T)=0$. If $(g_ε,ϕ_ε)$ solves the above equations, $\dot g=\partial_εg_ε|_{ε=0}$, $\dotϕ=ϕ_ε|_{ε=0}$, and $f=(f^1,f^2)= \partial_εF_ε|_{ε=0}$ solve the linearized Einstein equations and the linearized conservation law $$ \frac 12 \hat g^{pk}\hat \nabla_p f^1_{kj}+ \sum_{\ell=1}^L f^2_\ell \, \partial_j\hatϕ_\ell=0, $$ where $\hat g= g_ε|_{ε=0}$ and $\hat ϕ= ϕ_ε|_{ε=0}$. Then $(\hat g,\hat ϕ)$ and $f$ have the linearization stability property. Here ask the converse: If $\dot g$, $\dot ϕ$, and $f$ solve the linearized Einstein equations and the linearized conservation law, are there $F_ε=(F^1_ε,F^2_ε)$ and $(g_ε,ϕ_ε)$ depending on $ε\in [0,ε_0)$, $ε_0>0$, such that $(g_ε,ϕ_ε)$ solves the Einstein-scalar field equations and the conservation law. When $\hat g$ and $\hat ϕ$ vary enough and $L\geq 5$, we prove a microlocal version of this: When $Y\subset M$ is a 2-surface and $(y,η)\in N^*Y$, there is $f$ that is a conormal distibutions wrt. the surface $Y$ with a given principal symbol at $(y,η)$ such that $(\hat g,\hat ϕ)$ and $f$ have the linearization stability property.

math-ph↗

arXiv2014

The Calderon problem in transversally anisotropic geometries

We 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 class is uniquely determined by boundary measurements under two conditions: (1) the metric is conformally transversally anisotropic (CTA), and (2) the transversal manifold is simple. In this paper we will consider geometries satisfying (1) but not (2). The first main result states that the boundary measurements uniquely determine a mixed Fourier transform / attenuated geodesic ray transform (or integral against a more general semiclassical limit measure) of an unknown coefficient. In particular, one obtains uniqueness results whenever the geodesic ray transform on the transversal manifold is injective. The second result shows that the boundary measurements in an infinite cylinder uniquely determine the transversal metric. The first result is proved by using complex geometrical optics solutions involving Gaussian beam quasimodes, and the second result follows from a connection between the Calderon problem and Gel'fand's inverse problem for the wave equation and the boundary control method.

math.AP↗

arXiv2013

Spectral theory and inverse problem on asymptotically hyperbolic orbifolds

We 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 $S$-matrix, and then show that it determines the manifolds with its Riemannian metric and the orbifold structure.

math.AP↗