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 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.
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Chapter 2 Approximately Globally Convergent Numerical Method
Chapter 3 Numerical Implementation of the Approximately Globally Convergent Method
Chapter 4 The Adaptive Finite Element Technique and Its Synthesis with the Approximately Globally Convergent Numerical Method
Chapter 5 Blind Experimental Data
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algorithm of Sect approximate mathematical model approximately globally convergent assume Banach space Beilina and M.V. boundary condition boundary value problem Carleman estimate Cauchy problem cglob Chap CIPs coefficient inverse problem compact set Consider convergent numerical method convex cube number defined Denote dielectric constant Dirichlet boundary condition equation exact solution exists experimental data finite elements forward problem Fr´echet derivative function q globally convergent algorithm globally convergent numerical grid step Hence hyperbolic ill-posed problems imaged inclusions imply inequality IOP Publishing iterations with respect L2-norms Laplace transform Lemma located M.V. Klibanov maximal value minimizer Neumann boundary condition nonlinear norms obtain operator plane wave problem 2.1 proof pseudo frequency reconstruction refractive indices regularization parameter Reprinted with permission satisfying conditions solve space stage stopping rule tail function target Test Theorem Tikhonov functional two-stage numerical procedure