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Katya Krupchyk

Publications and source records attributed to Katya Krupchyk.

At least 19 recordsLinked to original sources
arXiv2025

Inverse spectral problems with sparse data and applications to passive imaging on manifolds

Motivated by inverse problems with a single passive measurement, we introduce and analyze a new class of inverse spectral problems on closed Riemannian manifolds. Specifically, we establish two general uniqueness results for the recovery of a potential in the stationary Schr\"odinger operator from partial spectral data, which consists of a possibly sparse subset of its eigenvalues and the restrictions of the corresponding eigenfunctions to a nonempty open subset of the manifold. Crucially, the eigenfunctions are not assumed to be orthogonal, and no information about global norming constants is required. The partial data formulation of our inverse spectral problems is naturally suited to the analysis of inverse problems with passive measurements, where only limited observational access to the solution is available. Leveraging this structure, we establish generic uniqueness results for a broad class of evolutionary PDEs, in which both the coefficients and the initial or source data are to be recovered from knowledge of the solution restricted to a subset of spacetime. These results introduce a spectral framework for passive imaging and extend inverse spectral theory into a regime characterized by highly incomplete, yet physically realistic, data.

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arXiv2025

Fractional anisotropic Calder\'on problem with external data

In this paper, we solve the fractional anisotropic Calder\'on problem with external data in the Euclidean space, in dimensions two and higher, for smooth Riemannian metrics that agree with the Euclidean metric outside a compact set. Specifically, we prove that the knowledge of the partial exterior Dirichlet--to--Neumann map for the fractional Laplace-Beltrami operator, given on arbitrary open nonempty sets in the exterior of the domain in the Euclidean space, determines the Riemannian metric up to diffeomorphism, fixing the exterior. We provide two proofs of this result: one relies on the heat semigroup representation of the fractional Laplacian and a pseudodifferential approach, while the other is based on a variable-coefficient elliptic extension interpretation of the fractional Laplacian.

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arXiv2024

Calder\'{o}n problem for fractional Schr\"{o}dinger operators on closed Riemannian manifolds

We study an analog of the anisotropic Calder\'on problem for fractional Schr\"odinger operators $(-\Delta_g)^\alpha + V$ with $\alpha \in (0,1)$ on closed Riemannian manifolds of dimensions two and higher. We prove that the knowledge of a Cauchy data set of solutions of the fractional Schr\"odinger equation, given on an open nonempty a priori known subset of the manifold determines both the Riemannian manifold up to an isometry and the potential up to the corresponding gauge transformation, under certain geometric assumptions on the manifold as well as the observation set. Our method of proof is based on: (i) studying a new variant of the Gel'fand inverse spectral problem without the normalization assumption on the energy of eigenfunctions, and (ii) the discovery of an entanglement principle for nonlocal equations involving two or more compactly supported functions. Our solution to (i) makes connections to antipodal sets as well as local control for eigenfunctions and quantum chaos, while (ii) requires sharp interpolation results for holomorphic functions. We believe that both of these results can find applications in other areas of inverse problems.

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arXiv2024

Inverse problems for semilinear Schrödinger equations at large frequency via polynomial resolvent estimates on manifolds

We study inverse boundary problems for semilinear Schrödinger equations on smooth compact Riemannian manifolds of dimensions $\ge 2$ with smooth boundary, at a large fixed frequency. We show that certain classes of cubic nonlinearities are determined uniquely from the knowledge of the nonlinear Dirichlet--to--Neumann map at a large fixed frequency on quite general Riemannian manifolds. In particular, in contrast to the previous results available, here the manifolds need not satisfy any product structure, may have trapped geodesics, and the geodesic ray transform need not be injective. Only a mild assumption about the geometry of intersecting geodesics is required. We also establish a polynomial resolvent estimate for the Laplacian on an arbitrary smooth compact Riemannian manifold without boundary, valid for most frequencies. This estimate, along with the invariant construction of Gaussian beam quasimodes with uniform bounds for underlying constants and a stationary phase lemma with explicit control over all involved constants, constitutes the key elements in proving the uniqueness results for the considered inverse problems.

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arXiv2023

Inverse Problems For Third-Order Nonlinear Perturbations Of Biharmonic Operators

We study inverse boundary problems for third-order nonlinear tensorial perturbations of biharmonic operators on a bounded domain in $\mathbb{R}^n$, where $n\geq 3$. By imposing appropriate assumptions on the nonlinearity, we demonstrate that the Dirichlet-to-Neumann map, known on the boundary of the domain, uniquely determines the genuinely nonlinear tensorial third-order perturbations of the biharmonic operator. The proof relies on the inversion of certain generalized momentum ray transforms on symmetric tensor fields. Notably, the corresponding inverse boundary problem for linear tensorial third-order perturbations of the biharmonic operator remains an open question.

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arXiv2021

Fractional anisotropic Calder\'on problem on closed Riemannian manifolds

In this paper we solve the fractional anisotropic Calder\'on problem on closed Riemannian manifolds of dimensions two and higher. Specifically, we prove that the knowledge of the local source-to-solution map for the fractional Laplacian, given on an arbitrary small open nonempty a priori known subset of a smooth closed connected Riemannian manifold, determines the Riemannian manifold up to an isometry. This can be viewed as a nonlocal analog of the anisotropic Calder\'on problem in the setting of closed Riemannian manifolds, which is wide open in dimensions three and higher.

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arXiv2021

The Calderón inverse problem for isotropic quasilinear conductivities

We prove a global uniqueness result for the Calderón inverse problem for a general quasilinear isotropic conductivity equation on a bounded open set with smooth boundary in dimension $n\ge 3$. Performing higher order linearizations of the nonlinear Dirichlet--to--Neumann map, we reduce the problem of the recovery of the differentials of the quasilinear conductivity, which are symmetric tensors, to a completeness property for certain anisotropic products of solutions to the linearized equation. The completeness property is established using complex geometric optics solutions to the linearized conductivity equation, whose amplitudes concentrate near suitable two dimensional planes.

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arXiv2020

Partial data inverse problems for quasilinear conductivity equations

We show that the knowledge of the Dirichlet-to-Neumann maps given on an arbitrary open non-empty portion of the boundary of a smooth domain in $\mathbb{R}^n$, $n\ge 2$, for classes of semilinear and quasilinear conductivity equations, determines the nonlinear conductivities uniquely. The main ingredient in the proof is a certain $L^1$-density result involving sums of products of gradients of harmonic functions which vanish on a closed proper subset of the boundary.

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arXiv2020

Reconstruction in the Calderón problem on conformally transversally anisotropic manifolds

We show that a continuous potential $q$ can be constructively determined from the knowledge of the Dirichlet-to-Neumann map for the Schrödinger operator $-Δ_g+q$ on a conformally transversally anisotropic manifold of dimension $\geq 3$, provided that the geodesic ray transform on the transversal manifold is constructively invertible. This is a constructive counterpart of the uniqueness result of Dos Santos Ferreira-Kurylev-Lassas-Salo. A crucial role in our reconstruction procedure is played by a constructive determination of the boundary traces of suitable complex geometric optics solutions based on Gaussian beams quasimodes concentrated along non-tangential geodesics on the transversal manifold, which enjoy uniqueness properties. This is achieved by applying the simplified version of the approach of Nachman-Street to our setting. We also identify the main space introduced by Nachman-Street with a standard Sobolev space on the boundary of the manifold. Another ingredient in the proof of our result is a reconstruction formula for the boundary trace of a continuous potential from the knowledge of the Dirichlet-to-Neumann map.

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arXiv2020

Inverse problems for nonlinear magnetic Schrödinger equations on conformally transversally anisotropic manifolds

We study the inverse boundary problem for a nonlinear magnetic Schrödinger operator on a conformally transversally anisotropic Riemannian manifold of dimension $n\ge 3$. Under suitable assumptions on the nonlinearity, we show that the knowledge of the Dirichlet-to-Neumann map on the boundary of the manifold determines the nonlinear magnetic and electric potentials uniquely. No assumptions on the transversal manifold are made in this result, whereas the corresponding inverse boundary problem for the linear magnetic Schrödinger operator is still open in this generality.

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arXiv2020

Linearized Calderón problem and exponentially accurate quasimodes for analytic manifolds

In this article we study the linearized anisotropic Calderón problem on a compact Riemannian manifold with boundary. This problem amounts to showing that products of pairs of harmonic functions of the manifold form a complete set. We assume that the manifold is transversally anisotropic and that the transversal manifold is real analytic and satisfies a geometric condition related to the geometry of pairs of intersecting geodesics. In this case, we solve the linearized anisotropic Calderón problem. The geometric condition does not involve the injectivity of the geodesic X-ray transform. Crucial ingredients in the proof of our result are the construction of Gaussian beam quasimodes on the transversal manifold, with exponentially small errors, as well as the FBI transform characterization of the analytic wave front set.

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arXiv2019

Eigenvalue bounds for non-self-adjoint Schrödinger operators with non-trapping metrics

We study eigenvalues of non-self-adjoint Schrödinger operators on non-trapping asymptotically conic manifolds of dimension $n\ge 3$. Specifically, we are concerned with the following two types of estimates. The first one deals with Keller type bounds on individual eigenvalues of the Schrödinger operator with a complex potential in terms of the $L^p$-norm of the potential, while the second one is a Lieb-Thirring type bound controlling sums of powers of eigenvalues in terms of the $L^p$-norm of the potential. We extend the results of Frank (2011), Frank-Sabin (2017), and Frank-Simon (2017) on the Keller and Lieb-Thirring type bounds from the case of Euclidean spaces to that of non-trapping asymptotically conic manifolds. In particular, our results are valid for the operator $Δ_g+V$ on $\mathbb{R}^n$ with $g$ being a non-trapping compactly supported (or suitably short range) perturbation of the Euclidean metric and $V\in L^p$ complex valued.

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arXiv2019

Partial data inverse problems for semilinear elliptic equations with gradient nonlinearities

We show that the linear span of the set of scalar products of gradients of harmonic functions on a bounded smooth domain $Ω\subset \mathbb{R}^n$ which vanish on a closed proper subset of the boundary is dense in $L^1(Ω)$. We apply this density result to solve some partial data inverse boundary problems for a class of semilinear elliptic PDE with quadratic gradient terms.

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arXiv2019

Stability estimates for partial data inverse problems for Schrödinger operators in the high frequency limit

We consider the partial data inverse boundary problem for the Schrödinger operator at a frequency $k>0$ on a bounded domain in $\mathbb{R}^n$, $n\ge 3$, with impedance boundary conditions. Assuming that the potential is known in a neighborhood of the boundary, we first show that the knowledge of the partial Robin-to-Dirichlet map at the fixed frequency $k>0$ along an arbitrarily small portion of the boundary, determines the potential in a logarithmically stable way. We prove, as the principal result of this work, that the logarithmic stability can be improved to the one of Hölder type in the high frequency regime.

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arXiv2017

Inverse problems for advection diffusion equations in admissible geometries

We study inverse boundary problems for the advection diffusion equation on an admissible manifold, i.e. a compact Riemannian manifold with boundary of dimension $\ge 3$, which is conformally embedded in a product of the Euclidean real line and a simple manifold. We prove the unique identifiability of the advection term of class $H^1\cap L^\infty$ and of class $H^{2/3}\cap C^{0,1/3}$ from the knowledge of the associated Dirichlet-to-Neumann map on the boundary of the manifold. This seems to be the first global identifiability result for possibly discontinuous advection terms.

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