10 research outputs found
Global convergence for Inverse Problems
A globally convergent numerical method for a multidimensional Coefficient Inverse Problem for a hyperbolic equation is presented. It is shown that this technique provides a good starting point for the finite element adaptive method (adaptivity). This leads to a natural two-stage numerical procedure, which synthesizes both these methods
Application of the Finite Element Method in a Quantitative Imaging technique
We present the Finite Element Method (FEM) for the numerical solution of the
multidimensional coefficient inverse problem (MCIP) in two dimensions. This
method is used for explicit reconstruction of the coefficient in the hyperbolic
equation using data resulted from a single measurement. To solve our MCIP we
use approximate globally convergent method and then apply FEM for the resulted
equation. Our numerical examples show quantitative reconstruction of the sound
speed in small tumor-like inclusions
Adaptive Reconstruction for Electrical Impedance Tomography with a Piecewise Constant Conductivity
In this work we propose and analyze a numerical method for electrical
impedance tomography of recovering a piecewise constant conductivity from
boundary voltage measurements. It is based on standard Tikhonov regularization
with a Modica-Mortola penalty functional and adaptive mesh refinement using
suitable a posteriori error estimators of residual type that involve the state,
adjoint and variational inequality in the necessary optimality condition and a
separate marking strategy. We prove the convergence of the adaptive algorithm
in the following sense: the sequence of discrete solutions contains a
subsequence convergent to a solution of the continuous necessary optimality
system. Several numerical examples are presented to illustrate the convergence
behavior of the algorithm.Comment: 26 pages, 12 figure
A posteriori error estimates for the adaptivity technique for the Tikhonov functional and global convergence for a coefficient inverse problem
A synthesis of a globally convergent numerical method for a coefficient inverse problem and the adaptivity technique is presented. \ud
First, the globally convergent method provides a good approximation for the unknown coefficient. Next, this approximation is refined via the adaptivity technique. \ud
The analytical effort is focused on a posteriori error estimates for the adaptivity. \ud
A numerical test is presented
A posteriori error estimation in biomedical imaging
We present an adaptive hybrid FEM/FDM method for an inverse scattering problem in scanning acoustic microscopy with a special focus on new application in medical imaging. The problem takes the form of reconstructing an unknown sound velocity c(x) from boundary displacement data measured in acoustic microscopy in order to obtain the pathological defects in bone. The inverse problem is formulated as an optimal control problem, where we solve the equations of optimality expressing stationarity of an associated Lagrangian by a quasi-Newton method. We present a posteriori error estimate for the error in the Lagrangian which couples residuals of the computed solution to weights of the reconstruction obtained by solving and associated linearized problem for the Hessian of the Lagrangian. The performance of the adaptive hybrid method and usefulness of the a posteriori error estimator are illustrated in numerical examples. \ua9 2007 IEEE