Partial Boundary Measurements Research Articles

Numerical solution of an inverse initial boundary value problem for the wave equation in the presence of conductivity imperfections of small volume

We consider the numerical solution, in two- and three-dimensional bounded domains, of the inverse problem for identifying the location of small-volume, conductivity imperfections in a medium with homogeneous background. A dynamic approach, based on the wave equation, permits us to treat the important case of “limited-view” data. Our numerical algorithm is based on the coupling of a finite element solution of the wave equation, an exact controllability method and finally a Fourier inversion for localizing the centers of the imperfections. Numerical results, in 2- and 3-D, show the robustness and accuracy of the approach for retrieving randomly placed imperfections from both complete and partial boundary measurements.

Circular resistor networks for electrical impedance tomography with partial boundary measurements

We introduce an algorithm for the numerical solution of electrical impedance tomography (EIT) in two dimensions, with partial boundary measurements. The algorithm is an extension of the one in Borcea et al (2008 Inverse Problems 24 035013 (31pp)) and Vasquez (2006 PhD Thesis Rice University, Houston, TX, USA) for EIT with full boundary measurements. It is based on resistor networks that arise in finite volume discretizations of the elliptic partial differential equation for the potential on so-called optimal grids that are computed as part of the problem. The grids are adaptively refined near the boundary, where we measure and expect better resolution of the images. They can be used very efficiently in inversion, by defining a reconstruction mapping that is an approximate inverse of the forward map, and acts therefore as a preconditioner in any iterative scheme that solves the inverse problem via optimization. The main result in this paper is the construction of optimal grids for EIT with partial measurements by extremal quasiconformal (Teichmüller) transformations of the optimal grids for EIT with full boundary measurements. We present the algorithm for computing the reconstruction mapping on such grids, and we illustrate its performance with numerical simulations. The results show an interesting trade-off between the resolution of the reconstruction in the domain of the solution and distortions due to artificial anisotropy induced by the distribution of the measurement points on the accessible boundary.

Convergence analysis of the Newton algorithm and a pseudo-time marching scheme for diffuse correlation tomography

We propose a self-regularized pseudo-time marching scheme to solve the ill-posed, nonlinear inverse problem associated with diffuse propagation of coherent light in a tissuelike object. In particular, in the context of diffuse correlation tomography (DCT), we consider the recovery of mechanical property distributions from partial and noisy boundary measurements of light intensity autocorrelation. We prove the existence of a minimizer for the Newton algorithm after establishing the existence of weak solutions for the forward equation of light amplitude autocorrelation and its Fréchet derivative and adjoint. The asymptotic stability of the solution of the ordinary differential equation obtained through the introduction of the pseudo-time is also analyzed. We show that the asymptotic solution obtained through the pseudo-time marching converges to that optimal solution provided the Hessian of the forward equation is positive definite in the neighborhood of optimal solution. The superior noise tolerance and regularization-insensitive nature of pseudo-dynamic strategy are proved through numerical simulations in the context of both DCT and diffuse optical tomography.

On an Inverse Problem of Reconstructing an Unknown Coefficient in a Second Order Hyperbolic Equation from Partial Boundary Measurements

On an Inverse Problem of Reconstructing an Unknown Coefficient in a Second Order Hyperbolic Equation from Partial Boundary Measurements

On a Hyperbolic Coefficient Inverse Problem via Partial Dynamic Boundary Measurements

This paper is devoted to the identification of the unknown smooth coefficient c entering the hyperbolic equation in a bounded smooth domain in ℝd from partial (on part of the boundary) dynamic boundary measurements. In this paper, we prove that the knowledge of the partial Cauchy data for this class of hyperbolic PDE on any open subset Γ of the boundary determines explicitly the coefficient c provided that c is known outside a bounded domain. Then, through construction of appropriate test functions by a geometrical control method, we derive a formula for calculating the coefficient c from the knowledge of the difference between the local Dirichlet‐to‐Neumann maps.

A stability estimate for an inverse problem for the Schrödinger equation in a magnetic field from partial boundary measurements

In this paper, we study the inverse boundary value problem of determining the magnetic field and the electric potential of the magnetic Schrödinger equation from the Dirichlet to Neumann map when the boundary measurement is taken on any open subset Γ0 of the boundary. Assuming that in a neighborhood of the boundary, the potentials coefficients are known, we prove the logarithmic stability estimate. The proofs involve the use of the complex geometrical optics solutions of the magnetic Schrödinger equation.

Stability Estimates for Coefficients of Magnetic Schrödinger Equation from Full and Partial Boundary Measurements

In this paper we establish a log log-type estimate which shows that in dimension n ≥ 3 the magnetic field and the electric potential of the magnetic Schrodinger equation depends stably on the Diric...

An inverse problem for a linear Schrödinger equation in the presence of inhomogeneities of small volumes

AbstractIn this paper we prove that the knowledge of the partial (on part of the boundary) dynamic boundary measurements for the Schrödinger equation on any open subset Γ of the boundary determines uniquely the potential. Using a global Carleman estimate, we also derive a stability result with appropriate norms provided that Γ satisfies a suitable geometrical condition.

Determination of small amplitude perturbations for the electric permittivity from partial dynamic boundary measurements

The paper deals with reconstruction of small perturbation of the uniform isotropic complex electric permittivity from dynamic measurements of the tangential component of the magnetic field on the boundary (or a part of the boundary) of a domain. The method is based on derived asymptotic inverse Fourier transform of the perturbation in the complex permittivity. Through construction of appropriate test functions by a geometrical control method, we provide a rigorous derivation of the inverse Fourier transform of the perturbations in the electric permittivity as the leading order of an appropriate averaging of the partial dynamic boundary perturbations.

Determining nonsmooth first order terms from partial boundary measurements

We extend results of Dos Santos Ferreira-Kenig-Sjöstrand-Uhlmann (Comm. Math. Phys., 2007) to less smooth coefficients, and we show that measurements on part of the boundary for the magnetic Schrödinger operator determine uniquely the magnetic field related to a Hölder continuous potential. We give a similar result for determining a convection term. The proofs involve Carleman estimates, a smoothing procedure, and an extension of the Nakamura-Uhlmann pseudodifferential conjugation method to logarithmic Carleman weights.

Three spheres inequalities for a two-dimensional elliptic system and its application

In this paper we prove three spheres inequalities for a two-dimensional strongly elliptic system. We then give an application of these three spheres inequalities to the inverse problem of identifying cavities by partial boundary measurements.

A linear integral equation approach to the Robin inverse problem

We consider the numerical solution of an inverse problem for the Laplace equation where the Robin coefficient is to be recovered from a partial boundary measurement. We first formulate the problem by integral equations on the boundary. By introducing a new variable, the key integral equation involved becomes linear, and the ill-posedness of the inverse problem is clearly revealed. On the basis of this linearity, we then design linear least-squares-based methods for reconstruction of the Robin coefficient. A proper form of regularization is chosen, and both direct (non-iterative) and iterative methods are investigated. Numerical examples are presented to show the effectiveness of these methods in providing excellent estimates for the Robin coefficient from data with and without noise.

Identification of small amplitude perturbations in the electromagnetic parameters from partial dynamic boundary measurements

We consider the inverse problem of reconstructing small amplitude perturbations in the conductivity for the wave equation from partial (on part of the boundary) dynamic boundary measurements. Through construction of appropriate test functions by a geometrical control method we provide a rigorous derivation of the inverse Fourier transform of the perturbations in the conductivity as the leading order of an appropriate averaging of the partial dynamic boundary perturbations. This asymptotic formula is generalized to the full time-dependent Maxwell's equations. Our formulae may be expected to lead to very effective computational identification algorithms, aimed at determining electromagnetic parameters of an object based on partial dynamic boundary measurements.

Recovery of small electromagnetic inhomogeneities from partial boundary measurements

We consider for the inverse problem of identifying locations and certain properties of the shapes of small dielectric inhomogeneities in a homogeneous background medium from boundary measurements on part of the boundary or dynamic boundary measurements for a finite time interval. Using as weights particular background solutions we develop asymptotic methods based on appropriate averaging of the data. To cite this article: H. Ammari, A.G. Ramm, C. R. Mecanique 330 (2002) 199–205.

An Inverse Initial Boundary Value Problem for the Wave Equation in the Presence of Imperfections of Small Volume

We consider for the wave equation the inverse problem of identifying locations and certain properties of the shapes of small conductivity inhomogeneities in a homogeneous background medium from dynamic boundary measurements on part of the boundary and for a finite interval in time. Using as weights particular background solutions constructed by a geometrical control method, we develop an asymptotic method based on appropriate averaging of the partial dynamic boundary measurements. Our approach is expected to lead to very effective computational identification algorithms.

Coefficient identification in some partial differential equations from partial boundary measurements

We consider the problem of finding a finite number of controls such that the overposed data uniquely determine the coefficient q(x) in the parabolic equation subject to homogeneous Dirichlet boundary conditions. Where the sources form a complete set in and is an arbitrary small region of the boundary assumed to be controlled by a positive function .