arXiv subjects

Ali Feizmohammadi

Publications and source records attributed to Ali Feizmohammadi.

At least 19 recordsLinked to original sources

arXiv2026

The planar Calderón problem for $L^p$ potentials

We prove that the Dirichlet-to-Neumann map associated to the Schrödinger operator $-Δ+q$ on a planar bounded Lipschitz domain uniquely determines an arbitrary complex-valued potential $q\in L^p$, $p>1$, whenever zero is not a Dirichlet eigenvalue. Our argument uses Nachman's d-bar equation only at sufficiently large complex frequencies, where exceptional points are absent. A new annular frequency estimate converts the first asymptotic coefficient of the complex geometric optics solutions into a pointwise bound for a primitive of the potential difference, reducing the inverse problem to a unique-continuation argument.

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arXiv2026

Recovery of Sturm-Liouville operators from partial boundary spectral data and applications

We study the inverse Sturm-Liouville problem on a finite interval from partial knowledge of spectral data. Specifically, we show that the potential can be uniquely reconstructed from the knowledge of a fraction of Dirichlet eigenvalues together with the normal derivatives of the corresponding eigenfunctions at both endpoints. We present two novel applications of our spectral result in inverse coefficient determination problems for evolutionary PDEs that include passive wave-based imaging of a medium and active imaging for the time-dependent Schrödinger equation with unknown internal sources. Our results yield finite time measurement bounds for such inverse coefficient determination problems. A central innovation is the use of Kahane's interpolation theorem to analyze endpoint time traces of solutions, enabling the recovery without requiring analyticity assumptions or infinite-time data, as in previous approaches. Finally, in the appendix, we present a spectral interpolation theorem for one-dimensional Schrödinger operators, which may be of independent interest.

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arXiv2025

Reconstruction of 1-D evolution equations and their initial data from one passive measurement

We study formally determined inverse problems with passive measurements for one dimensional evolution equations where the goal is to simultaneously determine both the initial data as well as the variable coefficients in such an equation from the measurement of its solution at a fixed spatial point for a certain amount of time. This can be considered as a one-dimensional model of widely open inverse problems in photo-acoustic and thermo-acoustic tomography. We provide global uniqueness results for wave and heat equations stated on bounded or unbounded spatial intervals. Contrary to all previous related results on the subject, we do not impose any genericity assumptions on the coefficients or initial data. Our proofs are based on creating suitable links to the well understood spectral theory for 1D Schrödinger operators. In particular, in the more challenging case of a bounded spatial domain, our proof for the inverse problem partly relies on the following two ingredients, namely (i) a Paley-Wiener type theorem for Schrödinger operators due to Remling \cite{Remling2002SchrdingerOA} and a theorem of Levinson \cite{Levinson1940} on distribution of zeros of entire functions of regular growth that together provide a quantifiable link between support of a compactly supported function and the upper density of its vanishing Schrödinger spectral modes and (ii) a result of Gesztesy and Simon \cite{Gesztesy1999InverseSA} on partial data inverse spectral problems for reconstructing an unknown potential in a 1D Schrödinger operator from the knowledge of only a fraction of its spectrum.

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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 Schrödinger 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

Simultaneous Identification of Coefficients and Source in a Subdiffusion Equation from One Passive Measurement

This article is devoted to the detection of parameters in anomalous diffusion from a single passive measurement. More precisely, we consider the simultaneous identification of coefficients as well as a time-dependent source term appearing in a time-fractional diffusion equation from a single boundary or internal passive measurement. We obtain several uniqueness results in dimension one as well as a multidimensional extension under some symmetry assumptions. Our analysis relies on spectral representation of solutions, complex and harmonic analysis combined with some known inverse spectral results for Sturm-Liouville operators. The theoretical results are complemented by a corresponding reconstruction algorithm and numerical simulations.

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arXiv2025

Fractional anisotropic Calderón problem with external data

In this paper, we solve the fractional anisotropic Calderón 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

Entanglement principle for the fractional Laplacian with applications to inverse problems

We prove an entanglement principle for fractional Laplace operators on $\mathbb R^n$ for $n\geq 2$ as follows; if different fractional powers of the Laplace operator acting on several distinct functions on $\mathbb R^n$, which vanish on some nonempty open set $O$, are known to be linearly dependent on $O$, then all the functions must be globally zero. This remarkable principle was recently discovered to be true for smooth functions on compact Riemannian manifolds without boundary \cite{FKU24}. Our main result extends the principle to the noncompact Euclidean space stated for tempered distributions under suitable decay conditions at infinity. We also present applications of this principle to solve new inverse problems for recovering anisotropic principal terms as well as zeroth order coefficients in fractional polyharmonic equations. Our proof of the entanglement principle uses the heat semigroup formulation of fractional Laplacian to establish connections between the principle and the study of several topics including interpolation properties for holomorphic functions under certain growth conditions at infinity, meromorphic extensions of holomorphic functions from a subdomain, as well as support theorems for spherical mean transforms on $\mathbb R^n$ that are defined as averages of functions over spheres.

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arXiv2024

Calderón problem for fractional Schrödinger operators on closed Riemannian manifolds

We study an analog of the anisotropic Calderón problem for fractional Schrödinger operators $(-Δ_g)^α+ V$ with $α\in (0,1)$ on closed Riemannian manifolds of dimensions two and higher. We prove that the knowledge of a Cauchy data set of solutions of the fractional Schrödinger 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

Partial data inverse problems for reaction-diffusion and heat equations

We study partial data inverse problems for linear and nonlinear parabolic equations with unknown time-dependent coefficients. In particular, we prove uniqueness results for partial data inverse problems for semilinear reaction-diffusion equations where Dirichlet boundary data and Neumann measurements of solutions are restricted to any open subset of the boundary. We also prove injectivity of the Fréchet derivative of the partial Dirichlet-to-Neumann map associated to heat equations. Our proof consists of two crucial ingredients; (i) we introduce an asymptotic family of spherical quasimodes that approximately solve heat equations modulo an exponentially decaying remainder term and (ii) the asymptotic study of a weighted Laplace transform of the unknown coefficient along a straight line segment in the domain where the weight may be viewed as a semiclassical symbol that itself depends on the complex-valued frequency. The latter analysis will rely on Phragmén-Lindelöf principle and Grönwall inequality.

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arXiv2024

Two uniqueness results in the inverse boundary value problem for the weighted p-Laplace equation

In this paper we prove a general uniqueness result in the inverse boundary value problem for the weighted p-Laplace equation in the plane, with smooth weights. We also prove a uniqueness result in dimension 3 and higher, for real analytic weights that are subject to a smallness condition on one of their directional derivatives. Both results are obtained by linearizing the equation at a solution without critical points. This unknown solution is then recovered, together with the unknown weight.

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arXiv2023

Rigidity of inverse problems for nonlinear elliptic equations on manifolds

We consider the inverse problem of determining coefficients appearing in semilinear elliptic equations stated on Riemannian manifolds with boundary given the knowledge of the associated Dirichlet-to-Neumann map. We begin with a negative answer to this problem. Owing to this obstruction, we consider a new formulation of our inverse problem in terms of a rigidity problem. Precisely, we consider cases where the Dirichlet-to-Neumann map of a semilinear equation coincides with the one of a linear equation and ask whether this implies that the equation must indeed be linear. We give a positive answer to this rigidity problem under some assumptions imposed to the Riemannian manifold and to the semilinear term under consideration.

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arXiv2022

Remarks on the anisotropic Calderón problem

We show uniqueness results for the anisotropic Calderón problem stated on transversally anisotropic manifolds. Moreover, we give a convexity result for the range of Dirichlet-to-Neumann maps on general Riemannian manifolds near the zero potential. Finally, we present results for Calderón type inverse problems associated to semilinear elliptic equations on general Riemannian manifolds.

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arXiv2022

An inverse boundary value problem for isotropic nonautonomous heat flows

We study an inverse boundary value problem on the determination of principal order coefficients in isotropic nonautonomous heat flows stated as follows; given a medium, and in the absence of heat sources and sinks, can the time-dependent thermal conductivity and volumetric heat capacity of the medium be uniquely determined from the Cauchy data of temperature and heat flux measurements on its boundary? We prove uniqueness in all dimensions under an assumption on the thermal diffusivity of the medium, which is defined as the ratio of the thermal conductivity and volumetric heat capacity. As a corollary of our result for isotropic media, we also obtain a uniqueness result, up to a natural gauge, in two-dimensional anisotropic media. Our assumption on the thermal diffusivity is related to construction of certain families of exponential solutions to the heat equation.

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arXiv2021

An inverse problem for a semilinear elliptic equation on conformally transversally anisotropic manifolds

Given a conformally transversally anisotropic manifold $(M,g)$, we consider the semilinear elliptic equation $$(-Δ_{g}+V)u+qu^2=0\quad \text{on $M$}.$$ We show that an a priori unknown smooth function $q$ can be uniquely determined from the knowledge of the Dirichlet-to-Neumann map associated to the semilinear elliptic equation. This extends the previously known results of the works [FO20, LLLS21a]. Our proof is based on analyzing higher order linearizations of the semilinear equation with non-vanishing boundary traces and also the study of interactions of two or more products of the so-called Gaussian quasimode solutions to the linearized equation.

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arXiv2021

Fractional anisotropic Calderón problem on closed Riemannian manifolds

In this paper we solve the fractional anisotropic Calderón 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 Calderón problem in the setting of closed Riemannian manifolds, which is wide open in dimensions three and higher.

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arXiv2021

Lorentzian Calderón problem near the Minkowski geometry

We study a Lorentzian version of the well-known Calderón problem that is concerned with determination of lower order coefficients in a wave equation on a smooth Lorentzian manifold, given the associated Dirichlet-to-Neumann map. In the earlier work of the authors it was shown that zeroth order coefficients can be uniquely determined under a two-sided spacetime curvature bound and the additional assumption that there are no conjugate points along null or spacelike geodesics. In this paper we show that uniqueness for the zeroth order coefficient holds for manifolds satisfying a weaker curvature bound as well as spacetime perturbations of such manifolds. This relies on a new optimal unique continuation principle for the wave equation in the exterior regions of double null cones. In particular, we solve the Lorentzian Calderón problem near the Minkowski geometry.

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arXiv2021

An inverse problem for a quasilinear convection--diffusion equation

We study the inverse problem of recovering a semilinear diffusion term $a(t,λ)$ as well as a quasilinear convection term $\mathcal B(t,x,λ,ξ)$ in a nonlinear parabolic equation $$\partial_tu-\textrm{div}(a(t,u) \nabla u)+\mathcal B(t,x,u,\nabla u)\cdot\nabla u=0, \quad \mbox{in}\ (0,T)\timesΩ,$$ given the knowledge of the flux of the moving quantity associated with different sources applied at the boundary of the domain. This inverse problem that is modeled by the solution dependent parameters $a$ and $\mathcal B$ has many physical applications related to various classes of cooperative interactions or complex mixing in diffusion processes. Our main result states that, under suitable assumptions, it is possible to fully recover the nonlinear diffusion term $a$ as well as the nonlinear convection term $\mathcal B$. The recovery of the diffusion term is based on the idea of solutions to the linearized equation with singularities near the boundary $\partial Ω$. Our proof of the recovery of the convection term is based on the idea of higher order linearization to reduce the inverse problem to a density property for certain anisotropic products of solutions to the linearized equation. We show this density property by constructing sufficiently smooth geometric optic solutions concentrating on rays in $Ω$.

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arXiv2021

Fractional Calderón problem on a closed Riemannian manifold

Given a fixed $α\in (0,1)$, we study the inverse problem of recovering the isometry class of a smooth closed and connected Riemannian manifold $(M,g)$, given the knowledge of a source-to-solution map for the fractional Laplace equation $(-Δ_g)^αu=f$ on the manifold subject to an arbitrarily small observation region $\mathcal O$ where sources can be placed and solutions can be measured. This can be viewed as a non-local analogue of the well known anisotropic Calderón problem that is concerned with the limiting case $α=1$. While the latter problem is widely open in dimensions three and higher, we solve the non-local problem in broad geometric generality, assuming only a local property on the a priori known observation region $\mathcal O$ while making no geometric assumptions on the inaccessible region of the manifold, namely $M\setminus \mathcal O$. Our proof is based on discovering a hidden connection to a variant of Carlson's theorem in complex analysis that allows us to reduce the non-local inverse problem to the Gel'fand inverse spectral problem.

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