arXiv subjects
Publications and source records attributed to Mourad Choulli.
We establish Hölder stability of an inverse hyperbolic obstacle problem. Mainly, we study the problem of reconstructing an unknown function defined on the boundary of the obstacle from two measurements taken on the boundary of a domain surrounding the obstacle.
This work is composed of two parts. We prove in the first part the uniqueness of the determination of the unbounded zero-order coefficient in a parabolic equation from boundary measurements. The novelty of our result is that it covers the largest class of unbounded zero-order coefficients. We establish in the second part a logarithmic stability inequality for the problem of determining the initial condition from a single interior measurement. As by-product, we obtain an observability inequality for a parabolic equation with unbounded zero-order coefficient. The proof of this observability inequality is based on a new global quantitative unique continuation for the Schrödinger equation with unbounded potential. For the sake of completeness, we provide in Appendix A a full proof of this result.
We establish stability inequalities of an inverse obstacle problem for the magnetic Schrödinger equation. We mainly study the problem of reconstructing an unknown function defined on the obstacle boundary from two measurements performed on the boundary of a domain surrounding the obstacle. We show for the inverse problem a Lipschitzian stability locally in time and a logarithmic stability globally in time.
We consider the problem of determining the unknown boundary values of a solution of an elliptic equation outside a bounded open set $B$ from the knowledge of the values of this solution on a boundary of an arbitrary Lipschitz bounded domain surrounding $B$. We obtain for this inverse problem Lipschitz stability for an admissible class of unknown boundary functions. Our analysis applies as well to an interior problem. We also give an extension to the parabolic case.
We present a simple and self-contained approach to establish the unique continuation property for some classical evolution equations of second order in a cylindrical domain. We namely discuss this property for wave, parabolic and Schödinger operators with time-independent principal part. Our method is builds on two-parameter Carleman inequalities combined with unique continuation across a pseudo-convex hypersurface with respect to the space variable. The most results we demonstrate in this work are more or less classical. Some of them are not stated exactly as in their original form.
We establish near-optimal quantitative uniqueness of continuation for solutions of evolution equations vanishing on the lateral boundary. These results were obtained simply by combining existing observability inequalities and energy estimates.
This article deals with the uniqueness and stability issues in the inverse problem of determining the unbounded potential of the Schrödinger operator in a bounded domain of dimension 3 or greater, endowed with Robin boundary condition, from knowledge of its boundary spectral data. These data are defined by the pairs formed by the eigenvalues and either full or partial Dirichlet measurement of the eigenfunctions on the boundary of the domain.
We establish a double logarithmic stability inequality for the problem of determining the initial data in an IBVP for the wave equation outside a non-trapping obstacle from two localized measurements.
We establish stability inequalities for the problem of determining a Dirichlet-Laplace-Beltrami operator from its boundary spectral data. We study the case of complete spectral data as well as the case of partial spectral data.
We prove that the metric tensor $g$ of a complete Riemannian manifold is uniquely determined, up to isometry, from the knowledge of a local source-to-solution operator. This later is associated to a fractional power of the Laplace-Belrami operator $Δ_g$. Our result holds under the condition that the metric tensor $g$ is known in an arbitrary small subdomain. We also consider the case of closed manifolds and provide an improvement of the main result in \cite{FGKU}
We establish in dimension $3$ a stability inequality for the problem of determining the potential in the Schrödinger equation from boundary measurements in the case where the potential belongs to $L^s$ with $s\in (2,3)$.
We are concerned with the problem of determining the nonlinear term in a semilinear elliptic equation by boundary measurements. Precisely, we improve [5, Theorem 1.3], where a logarithmic type stability estimate was proved. We show actually that we have Hölder stability estimate with less boundary measurents and less regular nonlinearities. We establish our stability inequality by following the same method as in [4]. This method consists in constructing special solutions vanishing at a subboundary of the domain.
We establish that the potential appearing in a fractional Schrödinger operator is uniquely determined by an internal spectral data.
We establish a Lipschitz stability inequality for the problem of determining the nonlinear term in a quasilinear elliptic equation by boundary measurements. We give a proof based on a linearization procedure together with special solutions constructed from the fundamental solution of the linearized problem.
We quantify the uniqueness of continuation from Cauchy or interior data. Our approach consists in extending the existing results in the linear case. As by product we obtain a new stability estimate in the linear case. We also show the so-called strong uniqueness of continuation and the uniqueness of continuation from a set of positive measure. These results are derived by using a linearization procedure.
This paper is concerned with the stability issue in determining absorption and diffusion coefficients in quantitative photoacoustic imaging. Assuming that the optical wave is generated by point sources in a region where the optical coefficients are known, we derive pointwise H{ö}lder stability estimate of the inversion. This result shows that the reconstruction of the optical coefficients is stable in the region close to the optical illumination sources and deteriorate exponentially far away. Our stability estimate is therefore in accordance with known experimental observations. Mathematics subject classification : 35R30.
We revisit the stability issue of determining the conductivity at the boundary from the corresponding Dirichlet-to-Neumann map. We discuss both the method based on singular solutions and the one built on the localized oscillating solutions. Our primary objective is not establishing new results on this subject even if the present work contains some new ones. We mainly clarify some points in the existing proofs and make some comments. We also derive some consequences of the stability inequality of the determination of the conductivity at the boundary from the Dirichlet-to-Neumann map.
We study some hybrid inverse problems associated to BVP's for Schrödinger and Helmholtz type equations. The inverse problems we consider consist in the determination of coefficients from the knowledge of internal energies. We establish local Lipschitz stability inequalities as well as conditional Hölder stability inequalities.