This is the first full-length book where globally convergent numerical methods for coefficient inverse problems for partial differential equations are presented. Conventional algorithms for these problems are known to converge locally. Globally convergent methods described in this book are synthesized with the Adaptive Finite Element technique (adaptivity for brevity). A broad range of applications is discussed, and a detailed convergence analysis is presented via a variety of numerical experiments. Computational studies include the work with both computationally simulated and experimental data.
A special focus of the book is on specifics of numerical implementations of algorithms and instructions for these implementations. The authors also describe the many details of numerical implementation of the globally convergent method as well as of the adaptivity technique in the two-stage numerical procedure.
Introduction to the theories of Ill-Posed and Coefficient Inverse Problems.- The Globally Convergent Numerical Method.- Numerical Implementation of the Globally Convergent Method.- Blind imaging of experimental data via the Globally Convergent Method.- The Adaptive Finite Element Technique and Its Synthesis with the Globally Convergent Method: The Two-Stage Numerical Procedure.- Analysis: Why the Adaptivity Refines the Image of the Globally Convergent Stage.- The Quasi-Reversibility Method and Global Convergence.-
Two Central Questions of This Book and an Introduction to the Theories of Ill-Posed and Coefficient Inverse Problems.- Approximately Globally Convergent Numerical Method.- Numerical Implementation of the Approximately Globally Convergent Method.- The Adaptive Finite Element Technique and its Synthesis with the Approximately Globally Convergent Numerical Method.- Blind Experimental Data.- Backscattering Data.
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 approximations 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.