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Yavar Kian

Publications and source records attributed to Yavar Kian.

At least 91 records · Page 5Linked to original sources
arXiv2017

Inverse source problems in elastodynamics

We are concerned with time-dependent inverse source problems in elastodynamics. The source term is supposed to be the product of a spatial function and a temporal function with compact support. We present frequency-domain and time-domain approaches to show uniqueness in determining the spatial function from wave fields on a large sphere over a finite interval. Stability estimate of the temporal function from the data of one receiver and uniqueness result using partial boundary data are proved. Our arguments rely heavily on the use of the Fourier transform, which motivated inversion schemes that can be easily implemented. A Landweber iterative algorithm for recovering the spatial function and a non-iterative inversion scheme based on the uniqueness proof for recovering the temporal function are proposed. Numerical examples are demonstrated in both two and three dimensions.

math.AP↗
arXiv2017

Determination of singular time-dependent coefficients for wave equations from full and partial data

We study the problem of determining uniquely a time-dependent singular potential $q$, appearing in the wave equation $\partial_t^2u-Δ_x u+q(t,x)u=0$ in $Q=(0,T)\timesΩ$ with $T>0$ and $Ω$ a $ \mathcal C^2$ bounded domain of $\mathbb R^n$, $n\geq2$. We start by considering the unique determination of some singular time-dependent coefficients from observations on $\partial Q$. Then, by weakening the singularities of the set of admissible coefficients, we manage to reduce the set of data that still guaranties unique recovery of such a coefficient. To our best knowledge, this paper is the first claiming unique determination of unbounded time-dependent coefficients, which is motivated by the problem of determining general nonlinear terms appearing in nonlinear wave equations.

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arXiv2017

Hölder stably determining the time-dependent electromagnetic potential of the Schrödinger equation

We consider the inverse problem of determining the time and space dependent electromagnetic potential of the Schrödinger equation in a bounded domain of $\mathbb R^n$, $n\geq 2$, by boundary observation of the solution over the entire time span. Assuming that the divergence of the magnetic potential is fixed, we prove that the electric potential and the magnetic potential can be Hölder stably retrieved from these data, whereas stability estimates for inverse time-dependent coefficients problems of evolution partial differential equations are usually of logarithmic type.

math.AP↗
arXiv2017

On time-fractional diffusion equations with space-dependent variable order

We investigate diffusion equations with time-fractional derivatives of space-dependent variable order. We examine the well-posedness issue and prove that the space-dependent variable order coefficient is uniquely determined among other coefficients of these equations, by the knowledge of a suitable time-sequence of partial Dirichlet-to-Neumann maps.

math.AP↗
arXiv2017

Application of the boundary control method to partial data Borg-Levinson inverse spectral problem

We consider the multidimensional Borg-Levinson problem of determining a potential $q$, appearing in the Dirichlet realization of the Schrödinger operator $A_q=-Δ+q$ on a bounded domain $Ω\subset \mathbb{R}^n$, $n\geq2$, from the boundary spectral data of $A_q$ on an arbitrary portion of $\partialΩ$. More precisely, for $γ$ an open and non-empty subset of $\partialΩ$, we consider the boundary spectral data on $γ$ given by $\mathrm{BSD}(q,γ):=\{(λ_{k},{\partial_νϕ_{k}}_{|\overlineγ}):\ k \geq1\}$, where $\{ λ_k:\ k \geq1\}$ is the non-decreasing sequence of eigenvalues of $A_q$, $\{ ϕ_k:\ k \geq1 \}$ an associated Hilbertian basis of eigenfunctions, and $ν$ is the unit outward normal vector to $\partialΩ$. We prove that the data $\mathrm{BSD}(q,γ)$ uniquely determine a bounded potential $q\in L^\infty(Ω)$. Previous uniqueness results, with arbitrarily small $γ$, assume that $q$ is smooth. Our approach is based on the Boundary Control method, and we give a self-contained presentation of the method, focusing on the analytic rather than geometric aspects of the method.

math.AP↗
arXiv2016

A multidimensional Borg-Levinson theorem for magnetic Schrödinger operators with partial spectral data

We consider the multidimensional Borg-Levinson theorem of determining both the magnetic field $dA$ and the electric potential $V$, appearing in the Dirichlet realization of the magnetic Schrödinger operator $H=(-{\rm i}\nabla+A)^2+V$ on a bounded domain $Ω\subset\mathbb R^n$, $n\geq2$, from partial knowledge of the boundary spectral data of $H$. The full boundary spectral data are given by the set $\{(λ_{k},{\partial_νϕ_{k}}_{|\partialΩ}):\ k\geq1\}$, where $\{ λ_k:\ k\in \mathbb N^* \}$ is the non-decreasing sequence of eigenvalues of $H$, $\{ ϕ_k:\ k\in \mathbb N^* \}$ an associated Hilbertian basis of eigenfunctions and $ν$ is the unit outward normal vector to $\partialΩ$. We prove that some asymptotic knowledge of $(λ_{k},{\partial_νϕ_{k}}_{|\partialΩ})$ with respect to $k\geq1$ determines uniquely the magnetic field $dA$ and the electric potential $V$.

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arXiv2016

Recovery of time-dependent damping coefficients and potentials appearing in wave equations from partial data

We consider the inverse problem of determining a time-dependent damping coefficient $a$ and a time-dependent potential $q$, appearing in the wave equation $\partial_t^2u-Δ_x u+a(t,x)\partial_tu+q(t,x)u=0$ in $Q=(0,T)\timesΩ$, with $T>0$ and $Ω$ a $ \mathcal C^2$ bounded domain of $\mathbb R^n$, $n\geq2$, from partial observations of the solutions on $\partial Q$. More precisely, we look for observations on $\partial Q$ that allow to determine uniquely a large class of time-dependent damping coefficients $a$ and time-dependent potentials $q$ without involving an important set of data. We prove global unique determination of $a\in W^{1,p}(Q)$, with $p>n+1$, and $q\in L^\infty(Q)$ from partial observations on $\partial Q$.

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arXiv2016

Recovery of time-dependent coefficient on Riemanian manifold for hyperbolic equations

Given $(M,g)$, a compact connected Riemannian manifold of dimension $d \geq 2$, with boundary $\partial M$, we study the inverse boundary value problem of determining a time-dependent potential $q$, appearing in the wave equation $\partial_t^2u-Δ_g u+q(t,x)u=0$ in $\bar M=(0,T)\times M$ with $T>0$. Under suitable geometric assumptions we prove global unique determination of $q\in L^\infty(\bar M)$ given the Cauchy data set on the whole boundary $\partial \bar M$, or on certain subsets of $\partial \bar M$. Our problem can be seen as an analogue of the Calderón problem on the Lorentzian manifold $(\bar M, dt^2 - g)$.

math.AP↗
arXiv2016

Logarithmic stability in determining the time-dependent zero order coefficient in a parabolic equation from a partial Dirichlet-to-Neumann map. Application to the determination of a nonlinear term

We give a new stability estimate for the problem of determining the time-dependent zero order coefficient in a parabolic equation from a partial parabolic Dirichlet-to-Neumann map. The novelty of our result is that, contrary to the previous works, we do not need any measurement on the final time. We also show how this result can be used to establish a stability estimate for the problem of determining the nonlinear term in a semilinear parabolic equation from the corresponding "linearized" Dirichlet-to-Neumann map. The key ingredient in our analysis is a parabolic version of an elliptic Carleman inequality due to Bukhgeim and Uhlmann. This parabolic Carleman inequality enters in an essential way in the construction of complex geometric optics solutions that vanish at a part of the lateral boundary.

math.AP↗
arXiv2016

An inverse problem for the magnetic Schrödinger equation in infinite cylindrical domains

We study the inverse problem of determining the magnetic field and the electric potential entering the Schrödinger equation in an infinite 3D cylindrical domain, by Dirichlet-to-Neumann map. The cylindrical domain we consider is a closed waveguide in the sense that the cross section is a bounded domain of the plane. We prove that the knowledge of the Dirichlet-to-Neumann map determines uniquely, and even Hölder-stably, the magnetic field induced by the magnetic potential and the electric potential. Moreover, if the maximal strength of both the magnetic field and the electric potential, is attained in a fixed bounded subset of the domain, we extend the above results by taking finitely extended boundary observations of the solution, only.

math.AP↗
arXiv2016

Stability result for elliptic inverse periodic coefficient problem by partial Dirichlet-to-Neumann map

We study the inverse problem of identifying a periodic potential perturbation of the Dirichlet Laplacian acting in an infinite cylindrical domain, whose cross section is assumed to be bounded. We prove log-log stable determination of the potential with respect to the partial Dirichlet-to-Neumann map, where the Neumann data is taken on slightly more than half of the boundary of the domain.

math.AP↗
arXiv2016

Global uniqueness in an inverse problem for time fractional diffusion equations

Given $(M,g)$, a compact connected Riemannian manifold of dimension $d \geq 2$, with boundary $\partial M$, we consider an initial boundary value problem for a fractional diffusion equation on $(0,T) \times M$, $T>0$, with time-fractional Caputo derivative of order $α\in (0,1) \cup (1,2)$. We prove uniqueness in the inverse problem of determining the smooth manifold $(M,g)$ (up to an isometry), and various time-independent smooth coefficients appearing in this equation, from measurements of the solution on a subset of $\partial M$ at fixed time. In the "flat" case where $M$ is a compact subset of $\mathbb R^d$, two out the three coefficients $ρ$ (weight), $a$ (conductivity) and $q$ (potential) appearing in the equation $ρ\partial_t^αu-\textrm{div}(a \nabla u)+ q u=0$ on $(0,T)\times Ω$ are recovered simultaneously.

math.AP↗
arXiv2015

On existence and uniqueness of solutions for semilinear fractional wave equations

Let $Ω$ be a $\mathcal C^2$-bounded domain of $\mathbb R^d$, $d=2,3$, and fix $Q=(0,T)\timesΩ$ with $T\in(0,+\infty]$. In the present paper we consider a Dirichlet initial-boundary value problem associated to the semilinear fractional wave equation $\partial_t^αu+\mathcal A u=f_b(u)$ in $Q$ where $1<α<2$, $\partial_t^α$ corresponds to the Caputo fractional derivative of order $α$, $\mathcal A$ is an elliptic operator and the nonlinearity $f_b\in \mathcal C^1( \mathbb R)$ satisfies $f_b(0)=0$ and $|f_b'(u)|\leq C|u|^{b-1}$ for some $b>1$. We first provide a definition of local weak solutions of this problem by applying some properties of the associated linear equation $\partial_t^αu+\mathcal A u=f(t,x)$ in $Q$. Then, we prove existence of local solutions of the semilinear fractional wave equation for some suitable values of $b>1$. Moreover, we obtain an explicit dependence of the time of existence of solutions with respect to the initial data that allows longer time of existence for small initial data.

math.AP↗
arXiv2015

An inverse stability result for non compactly supported potentials by one arbitrary lateral Neumann observation

In this paper we investigate the inverse problem of determining the time independent scalar potential of the dynamic Schrödinger equation in an infinite cylindrical domain, from partial measurement of the solution on the boundary. Namely, if the potential is known in a neighborhood of the boundary of the spatial domain, we prove that it can be logarithmic stably determined in the whole waveguide from a single observation of the solution on any arbitrary strip-shaped subset of the boundary.

math.AP↗
arXiv2015

Unique determination of a time-dependent potential for wave equations from partial data

We consider the inverse problem of determining a time-dependent potential $q$, appearing in the wave equation $\partial_t^2u-Δu+q(t,x)u=0$ in $Q=(0,T)\timesΩ$ with $Ω$ a $C^2$ bounded domain of $\mathbb R^n$, $n\geq2$, from partial observations of the solutions on $\partial Q$. We prove global unique determination of a coefficient $q\in L^\infty(Q)$ from these observations.

math.AP↗
arXiv2015

Stability in the determination of a time-dependent coefficient for wave equations from partial data

We consider the stability in the inverse problem consisting of the determination of a time-dependent coefficient of order zero $q$, appearing in a Dirichlet initial-boundary value problem for a wave equation $\partial_t^2u-Δu+q(t,x)u=0$ in $Q=(0,T)\timesΩ$ with $Ω$ a $C^2$ bounded domain of $\mathbb R^n$, $n\geq2$, from partial observations on $\partial Q$. The observation is given by a boundary operator associated to the wave equation. Using suitable complex geometric optics solutions and Carleman estimates, we prove a stability estimate in the determination of $q$ from the boundary operator.

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