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M. I. Belishev

Publications and source records attributed to M. I. Belishev.

At least 37 records · Page 2Linked to original sources
arXiv2021

Electric Impedance Tomography problem for surfaces with internal holes

Let $(M,g)$ be a smooth compact Riemann surface with the multicomponent boundary $Γ=Γ_0\cupΓ_1\cup\dots\cupΓ_m=:Γ_0\cup\tildeΓ$. Let $u=u^f$ obey $Δu=0$ in $M$, $u|_{Γ_0}=f,\,\,u|_{\tildeΓ}=0$ (the grounded holes) and $v=v^h$ obey $Δv=0$ in $M$, $v|_{Γ_0}=h,\,\,\partial_νv|_{\tildeΓ}=0$ (the isolated holes). Let $Λ_{g}^{\rm gr}: f\mapsto\partial_νu^f|_{Γ_{0}}$ and $Λ_{g}^{\rm is}: h\mapsto\partial_νv^h|_{Γ_{0}}$ be the corresponding DN-maps. The EIT problem is to determine $M$ from $Λ_{g}^{\rm gr}$ or $Λ_{g}^{\rm is}$. To solve it, an algebraic version of the BC-method is applied. The main instrument is the algebra of holomorphic functions on the ma\-ni\-fold ${\mathbb M}$, which is obtained by gluing two examples of $M$ along $\tildeΓ$. We show that this algebra is determined by $Λ_{g}^{\rm gr}$ (or $Λ_{g}^{\rm is}$) up to isometric isomorphism. Its Gelfand spectrum (the set of characters) plays the role of the material for constructing a relevant copy $(M',g',Γ_{0}')$ of $(M,g,Γ_{0})$. This copy is conformally equivalent to the original, provides $Γ_{0}'=Γ_{0},\,\,Λ_{g'}^{\rm gr}=Λ_{g}^{\rm gr},\,\,Λ_{g'}^{\rm is}=Λ_{g}^{\rm is}$, and thus solves the problem.

math-ph↗
arXiv2021

On characterization of Dirichlet-to-Neumann map of Riemannian surface with boundary

Let $(M,g)$ be a smooth compact orientable two-dimensional Riemannian manifold ({\it surface}) with a smooth metric tensor $g$ and smooth connected boundary $Γ$. Its {\it DN-map} $Λ_g:{C^\infty}(Γ)\to{C^\infty}(Γ)$ is associated with the (forward) elliptic problem $ Δ_gu=0 \,\,\, {\rm in}\,\,M\setminusΓ,\,\,u=f \,\,\, {\rm on}\,\,\,Γ$, and acts by $ Λ_g f:=\partial_νu^f \,\,\, {\rm on}\,\,\,Γ, $ where $Δ_g$ is the Beltrami-Laplace operator, $u=u^f(x)$ is the solution, $ν$ is the outward normal to $Γ$. The corresponding {\it inverse problem} is to determine the surface $(M,g)$ from its DN-map $Λ_g$. We provide the necessary and sufficient conditions on an operator acting in ${C^\infty}(Γ)$ to be the DN-map of a surface. In contrast to the known conditions by G.Henkin and V.Michel in terms of multidimensional complex analysis, our ones are based on the connections of the inverse problem with commutative Banach algebras.

math.AP↗
arXiv2020

On the EIT problem for nonorientable surfaces

Let $(Ω,g)$ be a smooth compact two-dimensional Riemannian manifold with boundary, $Λ_g: f\mapsto \partial_νu|_{\partialΩ}$ its DN map, where $u$ obeys $Δ_g u=0$ in $Ω$ and $u|_{\partial Ω}=f$. The Electric Impedance Tomography problem is to determine $Ω$ from $Λ_g$. A criterion is proposed that enables one to detect (via $Λ_g$) whether $Ω$ is orientable or not. The algebraic version of the BC-method is applied to solve the EIT problem for the Moebius band. The main instrument is the algebra of holomorphic functions on the double covering ${\mathbb M}$ of $M$, which is determined by $Λ_g$ up to an isometric isomorphism. Its Gelfand spectrum (the set of characters) plays the role of the material for constructing a relevant copy $(M',g')$ of $(M,g)$. This copy is conformally equivalent to the original, provides $\partial M'=\partial M,\,\,Λ_{g'}=Λ_g$, and thus solves the problem.

math-ph↗
arXiv2020

On an evolutionary dynamical system of the first order with boundary control

The work is carried out as part of the program to construct a new functio\-nal (so-called {\it wave}) model of symmetric operators. It is shown that an abstract evolutionary dynamic system of the first order (with respect to time) with boundary control, which is determined by a symmetric operator $L_0:{\mathscr H}\to{\mathscr H}$, is controllable if and only if $L_0$ has no maximal symmetric parts in~${\mathscr H}$.

math.FA↗
arXiv2020

Eikonal algebra on a graph of simple structure

An eikonal algebra ${\mathfrak E}(Ω)$ is a C*-algebra related to a metric graph $Ω$. It is determined by trajectories and reachable sets of a dynamical system associated with the graph. The system describes the waves, which are initiated by boundary sources (controls) and propagate into the graph with finite velocity. Motivation and interest to eikonal algebras comes from the inverse problem of reconstruction of the graph via its dynamical and/or spectral boundary data. Algebra ${\mathfrak E}(Ω)$ is determined by these data. In the mean time, its structure and algebraic invariants (irreducible representations) are connected with topology of $Ω$. We demonstrate such connections and study ${\mathfrak E}(Ω)$ by the example of $Ω$ of a simple structure. Hopefully, in future, these connections will provide an approach to reconstruction.

math-ph↗
arXiv2019

The wave model of metric spaces

Let $Ω$ be a metric space, $A^t$ denote the metric neighborhood of the set $A\subsetΩ$ of the radius $t$; ${\mathfrak O}$ be the lattice of open sets in $Ω$ with the partial order $\subseteq$ and the order convergence. The lattice of $\mathfrak O$-valued functions of $t\in(0,\infty)$ with the point-wise partial order and convergence contains the family ${I\mathfrak O}=\{A(\cdot)\,|\,\,A(t)=A^t,\,\,A\in{\mathfrak O}\}$. Let $\widetildeΩ$ be the set of atoms of the order closure $\overline{I\mathfrak O}$. We describe a class of spaces for which the set $\widetildeΩ$, equipped with an appropriate metric, is isometric to the original space $Ω$. The space $\widetildeΩ$ is the key element of the construction of the wave spectrum of a symmetric operator semi-bounded from below, which was introduced in a work of one of the authors. In that work, a program of constructing a functional model of operators of the aforementioned class was devised. The present paper is a step in realization of this program.

math.FA↗
arXiv2017

Wave model of the Sturm-Liouville operator on the half-line

The notion of the wave spectrum of a semi-bounded symmetric operator was introduced by one of the authors in 2013. The wave spectrum is a topological space determined by the operator in a canonical way. The definition uses a dynamical system associated with the operator: the wave spectrum is constructed from its reachable sets. In the paper we give a description of the wave spectrum of the operator $L_0=-\frac{d^2}{dx^2}+q$ which acts in the space $L_2(0,\infty)$ and has defect indices $(1,1)$. We construct a functional (wave) model of the operator $L_0^*$ in which the elements of the original $L_2(0,\infty)$ are realized as functions on the wave spectrum. It turns out to be identical to the original $L_0^*$. The latter is fundamental in solving inverse problems: the wave model is determined by their data, which allows for reconstruction of the original.

math.SP↗
arXiv2017

On algebras of three-dimensional quaternionic harmonic fields

A quaternionic field is a pair $p=\{α,u\}$ of function $α$ and vector field $u$ given on a 3d Riemannian maifold $Ω$ with the boundary. The field is said to be harmonic if $\nabla α={\rm rot\,}u$\, in $Ω$. The linear space of harmonic fields is not an algebra w.r.t. quaternion multiplication. However, it may contain the commutative algebras, what is the subject of the paper. Possible application of these algebras to the impedance tomography problem is touched on.

math-ph↗
arXiv2016

The reconstruction of sound speed in the Marmousi model by the boundary control method

We present the results on numerical testing of the Boundary Control Method in the sound speed determination for the acoustic equation on semiplane. This method for solving multidimensional inverse problems requires no a priory information about the parameters under reconstruction. The application to the realistic Marmousi model demonstrates that the boundary control method is workable in the case of complicated and irregular field of acoustic rays. By the use of the chosen boundary controls, an `averaged' profile of the sound speed is recovered (the relative error is about $10-15\%$). Such a profile can be further utilized as a starting approximation for high resolution iterative reconstruction methods.

physics.geo-ph↗
arXiv2015

Numerical testing in determination of sound speed from a part of boundary by the BC-method

We present the results of numerical testing on determination of the sound speed $c$ in the acoustic equation $u_{tt}-c^2Δu=0$ by the ${\it boundary}$ ${control}$ ${method}$. The inverse data is a response operator (a hyperbolic Dirichlet-to-Neumann map) given on controls, which are supported on ${\it a part}$ of the boundary. The speed is determined in the subdomain covered by acoustic rays, which are emanated from the points of this part orthogonally to the boundary. The determination is ${\it time-optimal}$: the longer is the observation time, the larger is the subdomain, in which $c$ is recovered. The numerical results are preceded with brief exposition of the relevant variant of the BC-method.

math.OC↗
arXiv2013

Non-smooth unobservable states in control problem for the wave equation in ${\mathbb R}^3$ (corrected)

The paper deals with a dynamical system \begin{align*} &u_{tt}-Δu=0, \qquad (x,t) \in {\mathbb R}^3 \times (-\infty,0) \\ &u \mid_{|x|<-t} =0 , \qquad t<0\\ &\lim_{s \to \infty} su((s+τ)ω,-s)=f(τ,ω), \qquad (τ,ω) \in [0,\infty)\times S^2\,, \end{align*} where $u=u^f(x,t)$ is a solution ({\it wave}), $f \in {\cal F} :=L_2\left([0,\infty);L_2\left(S^2\right)\right)$ is a {\it control}. For the reachable sets ${\cal U}^ξ:=\{u^f(\cdot, -ξ)\,|\,\, f \in {\cal F}\}\,\,(ξ\geqslant 0)$, the embedding ${\cal U}^ξ\subset {\cal H}^ξ:=\{y \in L_2({\mathbb R}^3)\,|\,\,\,y|_{|x|<ξ}=0\}$ holds, whereas the subspaces ${\cal D}^ξ:={\cal H}^ξ\ominus {\cal U}^ξ$ of unreachable ({\it unobservable}) states are nonzero for $ξ> 0$. There was a conjecture motivated by some geometrical optics arguments that the elements of ${\cal D}^ξ$ are $C^\infty$-smooth with respect to $|x|$. We provide rather unexpected counterexamples of $h\in {\cal D}^ξ$ with ${\rm sing\,supp\,}h \subset \{x\in{\mathbb R}^3|\,\,|x|=ξ_0>ξ\}$.

math-ph↗
arXiv2013

A C*-algebra associated with dynamics on a graph of strings

A C*-algebra $\mathfrak E$ associated with a dynamical system on a metric graph is introduced. The system is governed by the wave equation and controlled from boundary vertices. Algebra $\mathfrak E$ is generated by the so-called {\it eikonals}, which are self-adjoint operators related with reachable sets of the system. Its structure is the main subject of the paper. We show that $\mathfrak E$ is a direct sum of "elementary blocks". Each block is an algebra of operators multiplying ${\mathbb R}^n$-valued functions by continuous matrix-valued functions of special kind. The eikonal algebra is determined by the boundary inverse data. This shows promise of its possible applications to inverse problems.

math-ph↗
arXiv2013

Elements of noncommutative geometry in inverse problems on manifolds

We deal with two dynamical systems associated with a Riemannian manifold with boundary. The first one is a system governed by the scalar wave equation, the second is governed by the Maxwell equations. Both of the systems are controlled from the boundary. The inverse problem is to recover the manifold via the relevant measurements at the boundary (inverse data). We show that the inverse data determine a C*-algebras, whose (topologized) spectra are identical to the manifold. By this, to recover the manifold is to determine a proper algebra from the inverse data, find its spectrum, and provide the spectrum with a Riemannian structure. The paper develops an algebraic version of the boundary control method, which is an approach to inverse problems based on their relations to control theory.

math-ph↗
arXiv2012

Dynamical System with Boundary Control Associated with Symmetric Semi-Bounded Operator

Let $L_0$ be a closed densely defined symmetric semi-bounded operator with nonzero defect indexes in a separable Hilbert space $\cal H$. It determines a {\it Green system} $\{{\cal H}, {\cal B}; L_0, Γ_1, Γ_2\}$, where ${\cal B}$ is a Hilbert space, and $Γ_i: {\cal H} \to \cal B$ are the operators related through the Green formula $$(L_0^*u, v)_{\cal H}-(u,L_0^*v)_{\cal H}=(Γ_1 u, Γ_2 v)_{\cal B} - (Γ_2 u, Γ_1 v)_{\cal B}.$$ The {\it boundary operators} $Γ_i$ are chosen canonically in the framework of the Vishik theory. With the Green system one associates a {\it dynamical system with boundary control} (DSBC) {align*} & u_{tt}+L_0^*u = 0 && {\rm in}\,\,\,{\cal H}, \,\,\,t>0 & u|_{t=0}=u_t|_{t=0}=0 && {\rm in}\,\,\,{\cal H} & Γ_1 u = f && {\rm in}\,\,\,{\cal B},\,\,\,t \geqslant 0. {align*} We show that this system is {\it controllable} if and only if the operator $L_0$ is completely non-self-adjoint. A version of the notion of a {\it wave spectrum} of $L_0$ is introduced. It is a topological space determined by $L_0$ and constructed from reachable sets of the DSBC.

math.FA↗
arXiv2012

A unitary invariant of semi-bounded operator in reconstruction of manifolds

With a densely defined symmetric semi-bounded operator of nonzero defect indexes $L_0$ in a separable Hilbert space ${\cal H}$ we associate a topological space $Ω_{L_0}$ ({\it wave spectrum}) constructed from the reachable sets of a dynamical system governed by the equation $u_{tt}+(L_0)^*u=0$. Wave spectra of unitary equivalent operators are homeomorphic. In inverse problems, one needs to recover a Riemannian manifold $Ω$ via dynamical or spectral boundary data. We show that for a generic class of manifolds, $Ω$ is isometric to the wave spectrum $Ω_{L_0}$ of the minimal Laplacian $L_0=-Δ|_{C^\infty_0(Ω\backslash \partial Ω)}$ acting in ${\cal H}=L_2(Ω)$, whereas $L_0$ is determined by the inverse data up to unitary equivalence. Hence, the manifold can be recovered (up to isometry) by the scheme `data $\Rightarrow L_0 \Rightarrow Ω_{L_0} \overset{\rm isom}= Ω$'. The wave spectrum is relevant to a wide class of dynamical systems, which describe the finite speed wave propagation processes. The paper elucidates the operator background of the boundary control method (Belishev`1986), which is an approach to inverse problems based on their relations to control theory.

math.FA↗
arXiv2012

C*-algebras and inverse problem of electrodynamics

We consider the dynamical inverse problem for the Maxwell system on a Riemannian 3-manifold with boundary in a time-optimal set-up. Using BC-method we show that the data of the inverse problem (electromagnetic measurements on the boundary) determine a $C$*-algebra, which has a spectrum homeomorphic to a part of the manifold. This part depends on the duration of measurements.

math-ph↗