AN INVERSE SOURCE PROBLEM FOR THE DIFFUSION EQUATION WITH FINAL OBSERVATION

We investigate the inverse problem involving recovery of source temperature from the information of final temperature profile. We prove that we can uniquely recover the source of a n-dimensional heat equation from the measurement of the temperature at fixed time provided that the source is known in an arbitrary subdomain. The algorithm is based on the Carleman estimate. By using a Bukhgeim-Klibanov method, as a first step, we determine the source term by two measurements. A compacity and analyticity arguments procedure help to reduce the number of measurements

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

International audienceWe 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

Observability inequalities for transport equations through Carleman estimates

We consider the transport equation \ppp_t u(x,t) + H(t)\cdot \nabla u(x,t) = 0 in \OOO\times(0,T), where $T>0$ and \OOO\subset \R^d is a bounded domain with smooth boundary \ppp\OOO. First, we prove a Carleman estimate for solutions of finite energy with piecewise continuous weight functions. Then, under a further condition which guarantees that the orbits of $H$ intersect \ppp\OOO, we prove an energy estimate which in turn yields an observability inequality. Our results are motivated by applications to inverse problems.Comment: 18 pages, 3 figure

Thermoacoustic tomography arising in brain imaging

We study the mathematical model of thermoacoustic and photoacoustic tomography when the sound speed has a jump across a smooth surface. This models the change of the sound speed in the skull when trying to image the human brain. We derive an explicit inversion formula in the form of a convergent Neumann series under the assumptions that all singularities from the support of the source reach the boundary

A global Carleman estimate in a transmission wave equation and application to a one-measurement inverse problem

We consider a transmission wave equation in two embedded domains in $R^2$, where the speed is $a1 > 0$ in the inner domain and $a2 > 0$ in the outer domain. We prove a global Carleman inequality for this problem under the hypothesis that the inner domain is strictly convex and $a1 > a2$ . As a consequence of this inequality, uniqueness and Lip- schitz stability are obtained for the inverse problem of retrieving a stationary potential for the wave equation with Dirichlet data and discontinuous principal coefficient from a single time-dependent Neumann boundary measurement

Uniqueness, stability and numerical reconstruction of a time and space-dependent conductivity for an inverse hyperbolic problem

This paper is devoted to the reconstruction of the time and space-dependent coefficient in an inverse hyperbolic problem in a bounded domain. Using a local Carleman estimate we prove the uniqueness and a H\uf6lder stability in the determination of the conductivity by a single measurement on the lateral boundary. Our numerical examples show possibility of the determination of the location and the large contrast of the space-dependent function in three dimensions

Carleman estimates and applications to inverse problems for hyperbolic systems

[No abstract available

Carleman Estimates for Some Thermoelasticity Systems

In this chapter, we establish Carleman estimates for a thermoelastic plate system and a thermoelastic system with residual stress as applications of the Carleman estimate in Chap. 4. © 2017, Springer Science and Business Media Deutschland GmbH. All rights reserved

Inverse Heat Source Problem for the Thermoelasticity System

Using arguments similar to those presented in Chap. 5, we can apply the Carleman estimates obtained in Chap. 7 to the corrresponding inverse problems of determining coefficients and source terms. © 2017, Springer Science and Business Media Deutschland GmbH. All rights reserved