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Yernat M. Assylbekov

Publications and source records attributed to Yernat M. Assylbekov.

16 recordsLinked to original sources
arXiv2026

Inverting the geodesic ray transform with finite measurements: stability and reconstruction

We study the inversion of the geodesic ray transform from finitely many local averages of its data. Under suitable geometric assumptions, we prove Lipschitz stability on any fixed finite-dimensional reconstruction space when the measurement partition is sufficiently fine. We develop convergent reconstruction algorithms based on Steepest Gradient Descent and Conjugate Gradients and implement them using piecewise constant finite element spaces. Numerical experiments in 2D and 3D illustrate the influence of geometry and measurement discretization on reconstruction quality.

math.AP↗
arXiv2021

The attenuated ray transforms on Gaussian thermostats with negative curvature

In the present paper we consider a Gaussian thermostat on a compact Riemannian surface with negative thermostat curvature. In the case of surfaces with boundary, we show that the thermostat ray transform with attenuation given by a general connection and Higgs field is injective, modulo the natural obstruction, for tensors. We also prove that the connection and Higgs field can be determined, up to a gauge transformation, from the knowledge of the parallel transport between boundary points along all possible thermostat geodesics. In the case of closed surfaces, we obtain similar results for the ray transform with some additional conditions on the connection. Under the same condition, we study connections and Higgs fields whose parallel transport along periodic thermostat geodesics coincides with the ones for the flat connection.

math.DG↗
arXiv2020

Unique recovery of electrical conductivity and magnetic permeability from Magneto-Telluric data

We present a comprehensive mathematical study of the Magneto-Telluric (MT) method, on bounded domain in $\mathbb{R}^3$. We show that electrical conductivity and magnetic permeability, assumed to be $C^2$, can be uniquely recovered from MT data measured on the boundary of the domain. The proof is based on the construction of complex geometric optics solutions. Furthermore, we obtain a unique determination result in the case when the MT data are measured only on an open subset of the boundary. Here, we assume that the part of the boundary inaccessible for measurements is a subset of a sphere.

math.AP↗
arXiv2019

Determining rough first order perturbations of the polyharmonic operator

We show that the knowledge of Dirichlet to Neumann map for rough $A$ and $q$ in $(-Δ)^m +A\cdot D +q$ for $m \geq 2$ for a bounded domain in $\mathbb{R}^n$, $n \geq 3$ determines $A$ and $q$ uniquely. This unique identifiability is proved via construction of complex geometrical optics solutions with sufficient decay of remainder terms, by using property of products of functions in Sobolev spaces.

math.AP↗
arXiv2019

A note on time-harmonic Maxwell equations on Riemannian manifolds

In this note we consider boundary value problems in electromagnetism. We prove well-posedness results for the time-harmonic Maxwell equations in the setting of Riemannian manifolds. We also consider the eigenvalue problem the homogeneous time-harmonic Maxwell equations with zero boundary conditions.

math.AP↗
arXiv2018

The attenuated geodesic ray transform on tensors: generic injectivity and stability

We consider the attenuated geodesic ray transform defined on pairs of symmetric $2$-tensors and $1$-forms on a simple Riemannian manifold. We prove injectivity and stability results for a class of generic simple metrics and attenuations containing real analytic ones. In fact, methods used in this paper can be modified to generalize our results for a class of non-simple manifolds similar to Stefanov-Uhlmann [American Journal of Mathematics, {\bf 130} (1):239--268 (2008)].

math.AP↗
arXiv2017

Attenuated tensor tomography on simple surfaces

In this paper we consider the geodesic X-ray transform with attenuation coefficient as a combination of smooth complex function and 1-form. We show that attenuated X-ray transform applied to the pair of tensors is injective modulo the natural obstruction.

math.DG↗
arXiv2017

Inverse problems for the perturbed polyharmonic operator with coefficients in Sobolev spaces with non-positive order

We show that the knowledge of the Dirichlet-to-Neumann map on the boundary of a bounded open set in $\mathbb R^n$, $n\ge 3$, for the perturbed polyharmonic operator $(-Δ)^m+A\cdot D+q$, $m\ge 2$, with $n>m$, $A\in W^{-\frac{m-2}{2},\frac{2n}{m}}$ and $q\in W^{-\frac{m}{2}+δ,\frac{2n}{m}}$, with $0<δ<1/2$, determines the potentials $A$ and $q$ in the set uniquely. The proof is based on a Carleman estimate with linear weights and with a gain of two derivatives and on the property of products of functions in Sobolev spaces.

math.AP↗
arXiv2015

Determining the first order perturbation of a polyharmonic operator on admissible manifolds

We consider the inverse boundary value problem for the first order perturbation of the polyharmonic operator $\mathcal L_{g,X,q}$, with $X$ being a $W^{1,\infty}$ vector field and $q$ being an $L^\infty$ function on compact Riemannian manifolds with boundary which are conformally embedded in a product of the Euclidean line and a simple manifold. We show that the knowledge of the Dirichlet-to-Neumann determines $X$ and $q$ uniquely. The method is based on the construction of complex geometrical optics solutions using the Carleman estimate for the Laplace-Beltrami operator due to Dos Santos Ferreira, Kenig, Salo and Uhlmann. Notice that the corresponding uniqueness result does not hold for the first order perturbation of the Laplace-Beltrami operator.

math.AP↗
arXiv2015

The X-ray transform on a general family of curves on Finsler surfaces

We consider a general family of curves $Γ$ on a compact oriented Finsler surface $(M,F)$ with boundary $\partial M$. Let $φ\in C^{\infty}(M)$ and $ω$ a smooth 1-form on $M$. We show that $$\int_{γ(t)}\{φ(γ(t))+ω_{γ(t)}(\dotγ(t))\}\,dt=0$$ holds for every $γ\inΓ$ whose endpoints belong to $\partial M$, $γ(a)\in\partial M$, $γ(b)\in\partial M$ if and only if $φ=0$ and $ω$ is exact. Similar results were proved when $M$ is closed and some additional conditions on Gaussian curvature are imposed. We also study the cohomological equations of Anosov generelized thermostats on a closed Finsler surface. Finally, we gave conditions when thermostat is of Anosov type.

math.DG↗
arXiv2013

An inverse radiative transfer in refractive media equipped with a magnetic field

We study the reconstruction of the attenuation and absorption coefficients in a stationary linear transport equation from knowledge of albedo operator in dimension $n\geq 3$ on a Riemannian manifold in the presence of a magnetic field. We show the direct boundary value problem is well posed under two types of subcritical conditions. We obtain uniqueness and non-uniqueness results of the reconstruction under some restrictions. Finally, stability estimates are also established.

math.AP↗
arXiv2013

Boundary and scattering rigidity problems in the presence of a magnetic field and a potential

In this paper, we consider a compact Riemannian manifold with boundary, endowed with a magnetic potential $α$ and a potential $U$. For brevity, this type of systems are called $\MP$-systems. On simple $\MP$-systems, we consider both the boundary rigidity problem and scattering rigidity problem, see the introduction for details. We show that these two problems are equivalent on simple $\MP$-systems. Unlike the cases of geodesic or magnetic systems, knowing boundary action functions or scattering relations for only one energy level is insufficient to uniquely determine a simple $\MP$-system, even under the assumption that we know the restriction of the system on the boundary $\p M$, and we provide some counterexamples. These problems can only be solved up to an isometry and a gauge transformations of $α$ and $U$. We prove rigidity results for metrics in a given conformal class, for simple real analytic $\MP$-systems and for simple two-dimensional $\MP$-systems.

math.DG↗