An inverse source problem for the heat equation and the enclosure method

An inverse source problem for the heat equation is considered. Extraction formulae for information about the time and location when and where the unknown source of the equation firstly appeared are given from a single lateral boundary measurement. New roles of the plane progressive wave solutions or their complex versions for the backward heat equation are given.Comment: 23page

Probe method and a Carleman function

A Carleman function is a special fundamental solution with a large parameter for the Laplace operator and gives a formula to calculate the value of the solution of the Cauchy problem in a domain for the Laplace equation. The probe method applied to an inverse boundary value problem for the Laplace equation in a bounded domain is based on the existence of a special sequence of harmonic functions which is called a {\it needle sequence}. The needle sequence blows up on a special curve which connects a given point inside the domain with a point on the boundary of the domain and is convergent locally outside the curve. The sequence yields a reconstruction formula of unknown discontinuity, such as cavity, inclusion in a given medium from the Dirichlet-to-Neumann map. In this paper, an explicit needle sequence in {\it three dimensions} is given in a closed form. It is an application of a Carleman function introduced by Yarmukhamedov. Furthermore, an explicit needle sequence in the probe method applied to the reduction of inverse obstacle scattering problems with an {\it arbitrary} fixed wave number to inverse boundary value problems for the Helmholtz equation is also given.Comment: 2 figures, final versio

Linear sampling method for identifying cavities in a heat conductor

We consider an inverse problem of identifying the unknown cavities in a heat conductor. Using the Neumann-to-Dirichlet map as an input data, we develop a linear sampling type method for the heat equation. A new feature is that there is a freedom to choose the time variable, which suggests that we have more data than the linear sampling methods for the inverse boundary value problem associated with EIT and inverse scattering problem with near field data

Radiating and non-radiating sources in elasticity

In this work, we study the inverse source problem of a fixed frequency for the Navier's equation. We investigate that nonradiating external forces. If the support of such a force has a convex or non-convex corner or edge on their boundary, the force must be vanishing there. The vanishing property at corners and edges holds also for sufficiently smooth transmission eigenfunctions in elasticity. The idea originates from the enclosure method: The energy identity and new type exponential solutions for the Navier's equation.Comment: 17 page

Computing Volume Bounds of Inclusions by EIT Measurements

The size estimates approach for Electrical Impedance Tomography (EIT) allows for estimating the size (area or volume) of an unknown inclusion in an electrical conductor by means of one pair of boundary measurements of voltage and current. In this paper we show by numerical simulations how to obtain such bounds for practical application of the method. The computations are carried out both in a 2D and a 3D setting.Comment: 20 pages with figure

Inverse problems with partial data for a magnetic Schr\"odinger operator in an infinite slab and on a bounded domain

In this paper we study inverse boundary value problems with partial data for the magnetic Schr\"odinger operator. In the case of an infinite slab in $R^n$, $n\ge 3$, we establish that the magnetic field and the electric potential can be determined uniquely, when the Dirichlet and Neumann data are given either on the different boundary hyperplanes of the slab or on the same hyperplane. This is a generalization of the results of [41], obtained for the Schr\"odinger operator without magnetic potentials. In the case of a bounded domain in $R^n$, $n\ge 3$, extending the results of [2], we show the unique determination of the magnetic field and electric potential from the Dirichlet and Neumann data, given on two arbitrary open subsets of the boundary, provided that the magnetic and electric potentials are known in a neighborhood of the boundary. Generalizing the results of [31], we also obtain uniqueness results for the magnetic Schr\"odinger operator, when the Dirichlet and Neumann data are known on the same part of the boundary, assuming that the inaccessible part of the boundary is a part of a hyperplane

Application of mixed formulations of quasi-reversibility to solve ill-posed problems for heat and wave equations: the 1d case

International audienceIn this paper we address some ill-posed problems involving the heat or the wave equation in one dimension, in particular the backward heat equation and the heat/wave equation with lateral Cauchy data. The main objective is to introduce some variational mixed formulations of quasi-reversibility which enable us to solve these ill-posed problems by using some classical La-grange finite elements. The inverse obstacle problems with initial condition and lateral Cauchy data for heat/wave equation are also considered, by using an elementary level set method combined with the quasi-reversibility method. Some numerical experiments are presented to illustrate the feasibility for our strategy in all those situations. 1. Introduction. The method of quasi-reversibility has now a quite long history since the pioneering book of Latt es and Lions in 1967 [1]. The original idea of these authors was, starting from an ill-posed problem which satisfies the uniqueness property, to introduce a perturbation of such problem involving a small positive parameter ε. This perturbation has essentially two effects. Firstly the perturbation transforms the initial ill-posed problem into a well-posed one for any ε, secondly the solution to such problem converges to the solution (if it exists) to the initial ill-posed problem when ε tends to 0. Generally, the ill-posedness in the initial problem is due to unsuitable boundary conditions. As typical examples of linear ill-posed problems one may think of the backward heat equation, that is the initial condition is replaced by a final condition, or the heat or wave equations with lateral Cauchy data, that is the usual Dirichlet or Neumann boundary condition on the boundary of the domain is replaced by a pair of Dirichlet and Neumann boundary conditions on the same subpart of the boundary, no data being prescribed on the complementary part of the boundary

Application of retinoic acid to obtain osteocytes cultures from primary mouse osteoblasts

The need for osteocyte cultures is well known to the community of bone researchers; isolation of primary osteocytes is difficult and produces low cell numbers. Therefore, the most widely used cellular system is the osteocyte-like MLO-Y4 cell line. The method here described refers to the use of retinoic acid to generate a homogeneous population of ramified cells with morphological and molecular osteocyte features. After isolation of osteoblasts from mouse calvaria, all-trans retinoic acid (ATRA) is added to cell medium, and cell monitoring is conducted daily under an inverted microscope. First morphological changes are detectable after 2 days of treatment and differentiation is generally complete in 5 days, with progressive development of dendrites, loss of the ability to produce extracellular matrix, down-regulation of osteoblast markers and up-regulation of osteocyte-specific molecules. Daily cell monitoring is needed because of the inherent variability of primary cells, and the protocol can be adapted with minimal variation to cells obtained from different mouse strains and applied to transgenic models. The method is easy to perform and does not require special instrumentation, it is highly reproducible, and rapidly generates a mature osteocyte population in complete absence of extracellular matrix, allowing the use of these cells for unlimited biological applications

Conformal mapping for cavity inverse problem: an explicit reconstruction formula

International audienceIn this paper, we address a classical case of the Calder\'on (or conductivity) inverse problem in dimension two. We aim to recover the location and the shape of a single cavity $\omega$ (with boundary $\gamma$) contained in a domain $\Omega$ (with boundary $\Gamma$) from the knowledge of the Dirichlet-to-Neumann (DtN) map $\Lambda_\gamma: f \longmapsto \partial_n u^f|_{\Gamma}$, where $u^f$ is harmonic in $\Omega\setminus\overline{\omega}$, $u^f|_{\Gamma}=f$ and $u^f|_{\gamma}=c^f$, $c^f$ being the constant such that $\int_{\gamma}\partial_n u^f\,{\rm d}s=0$. We obtain an explicit formula for the complex coefficients $a_m$ arising in the expression of the Riemann map $z\longmapsto a_1 z + a_0 + \sum_{m\leqslant -1} a_m z^{m}$ that conformally maps the exterior of the unit disk onto the exterior of $\omega$. This formula is derived by using two ingredients: a new factorization result of the DtN map and the so-called generalized P\'olia-Szeg\"o tensors (GPST) of the cavity. As a byproduct of our analysis, we also prove the analytic dependence of the coefficients $a_m$ with respect to the DtN. Numerical results are provided to illustrate the efficiency and simplicity of the method

A novel model of in vitro osteocytogenesis induced by retinoic acid treatment

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