Larisa Beilina

Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems

Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems is the first book in which two new concepts of numerical solutions of multidimensional Coefficient Inverse Problems (CIPs) for a hyperbolic Partial Differential Equation (PDE) are presented: Approximate Global Convergence and the Adaptive Finite Element Method (adaptivity for brevity). Two central questions for CIPs are addressed: How to obtain a good approximation for the exact solution without any knowledge of a small neighborhood of this solution, and how to refine it given the approximation. The book also combines analytical convergence results with recipes for various numerical implementations of developed algorithms. The developed technique is applied to two types of blind experimental data, which are collected both in a laboratory and in the field. The result for the blind backscattering experimental data collected in the field addresses a real-world problem of imaging of shallow explosives.

A Globally Convergent Numerical Method for a Coefficient Inverse Problem

A new globally convergent numerical method is developed for a multidimensional coefficient inverse problem for a hyperbolic PDE with applications in acoustics and electromagnetics. On each iterative step the Dirichlet boundary value problem for a second-order elliptic equation is solved. The global convergence is rigorously established, and numerical experiments are presented.

A posteriori error estimation in computational inverse scattering

We prove an a posteriori error estimate for an inverse acoustic scattering problem, where the objective is to reconstruct an unknown wave speed coefficient inside a body from measured wave reflection data in time on parts of the surface of the body. The inverse problem is formulated as a problem of finding a zero of a Jacobian of a Lagrangian. The a posterori error estimate couples residuals of the computed solution to weights the reconstruction reflecting the sensitivity of the reconstruction obtained by solving an associated linaerized problem for the Hessian of the Lagrangian. We show concrete examples of reconstrution including a posteriori error estimation.

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. First, the globally convergent method provides a good approximation for the unknown coefficient. Next, this approximation is refined via the adaptivity technique. The analytical effort is focused on a posteriori error estimates for the adaptivity. A numerical test is presented.

Reconstruction of dielectrics from experimental data via a hybrid globally convergent/adaptive inverse algorithm

The validity of the synthesis of a globally convergent numerical method with the adaptive FEM technique for a coefficient inverse problem is verified on time-resolved experimental data. The refractive indices, locations and shapes of dielectric abnormalities are accurately imaged.

An Adaptive Hybrid FEM/FDM Method for an Inverse Scattering Problem in Scanning Acoustic Microscopy

Scanning acoustic microscopy based on focused ultrasound waves is a promising new tool in medical imaging. In this work we apply an adaptive hybrid FEM/FDM (finite element methods/finite difference methods) method to an inverse scattering problem for the time-dependent acoustic wave equation, where one seeks to reconstruct an unknown sound velocity c(x) from a single measurement of wave-reflection data on a small part of the boundary, e.g., to detect pathological defects in bone. Typically, this corresponds to identifying an unknown object (scatterer) in a surrounding homogeneous medium. The inverse problem is formulated as an optimal control problem, where we use an adjoint method to solve the equations of optimality expressing stationarity of an associated augmented Lagrangian by a quasi-Newton method. To treat the problem of multiple minima of the objective function, the optimization procedure is first performed on a coarse grid to smooth the high frequency error, generating a starting point for optimization steps on successively refined meshes. Local refinement based on the results of previous steps will improve computational efficiency of the method. As the main result then, an a posteriori error estimate is proved for the error in the Lagrangian, and a corresponding adaptive method is formulated, where the finite element mesh is refined from residual feedback. The performance of the adaptive hybrid method and the usefulness of the a posteriori error estimator for problems with limited boundary data are illustrated in three dimensional numerical examples.

Reconstruction of the Refractive Index from Experimental Backscattering Data Using a Globally Convergent Inverse Method

The problem to be studied in this work is within the context of coefficient identification problems for the wave equation. More precisely, we consider the problem of reconstruction of the refractive index (or equivalently, the dielectric constant) of an inhomogeneous medium using one backscattering boundary measurement. The goal of this paper is to analyze the performance of the globally convergent algorithm of Beilina and Klibanov on experimental data collected using a microwave scattering facility at the University of North Carolina at Charlotte. The main challenge in working with experimental data is the huge misfit between these data and computationally simulated data. We present data preprocessing steps to make the former somehow look similar to the latter. Results of both nonblind and blind targets are shown that indicate good reconstructions even for high contrasts between the targets and the background medium.

Blind backscattering experimental data collected in the field and an approximately globally convergent inverse algorithm

An approximately globally convergent numerical method for a 1D coefficient inverse problem for a hyperbolic PDE is applied to image dielectric constants of targets from blind experimental data. The data were collected in the field by the Forward Looking Radar of the US Army Research Laboratory. A posteriori analysis has revealed that computed and tabulated values of dielectric constants are in good agreement. Convergence analysis is presented.

Reconstruction of shapes and refractive indices from backscattering experimental data using the adaptivity

We consider the inverse problem of the reconstruction of the spatially distributed dielectric constant epsilon(r)(x), x is an element of R-3, which is an unknown coefficient in the Maxwells equations, from time-dependent backscattering experimental radar data associated with a single source of electric pulses. The refractive index is n(x) = root epsilon(r)(x). The coefficient epsilon(r)(x) is reconstructed using a two-stage reconstruction procedure. In the first stage an approximately globally convergent method proposed is applied to get a good first approximation of the exact solution. In the second stage a locally convergent adaptive finite element method is applied, taking the solution of the first stage as the starting point of the minimization of the Tikhonov functional. This functional is minimized on a sequence of locally refined meshes. It is shown here that all three components of interest of targets can be simultaneously accurately imaged: refractive indices, shapes and locations.

Hybrid FEM/FDM method for an inverse scattering problem

We apply an adaptive hybrid FEM/FDM method to an inverse scattering problem for the time-dependent acoustic wave equation in 2D and 3D, where we seek to reconstruct an unknown sound velocity c(x) from measured wave-reflection data. Typically,this corresponds to identifying an unknown object (scatterer) in a surrounding homogeneous medium.