Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems - Beilina, Larisa; Klibanov, Michael Victor; SpringerLink (Online service) (Boston, MA : Springer US, 2012)
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.
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.
Keywords: Mathematics; Global analysis; Differential equations, partial; Numerical analysis; Engineering mathematics; Mathematics; Partial Differential Equations; Numerical and Computational Physics; Appl.Mathematics/Computational Methods of Engineering; Numerical Analysis; Global Analysis and Analysis on Manifolds