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E-raamat: Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems

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  • Formaat: PDF+DRM
  • Ilmumisaeg: 09-Mar-2012
  • Kirjastus: Springer-Verlag New York Inc.
  • Keel: eng
  • ISBN-13: 9781441978059
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Approximate Global Convergence and Adaptivity for Coefficient Inverse ProblemsSuurem pilt
  • Formaat: PDF+DRM
  • Ilmumisaeg: 09-Mar-2012
  • Kirjastus: Springer-Verlag New York Inc.
  • Keel: eng
  • ISBN-13: 9781441978059
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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.

The first of its kind, this volume presents convergent numerical methods for coefficient inverse problems for partial differential equations. Readers will find globally convergent methods that are synthesized with the Adaptive Finite Element technique (adaptivity for brevity).
1 Two Central Questions of This Book and an Introduction to the Theories of Ill-posed and Coefficient Inverse Problems
1(94)
1.1 Two Central Questions of This Book
2(12)
1.1.1 Why the Above Two Questions Are the Central Ones for Computations of CIPs
4(2)
1.1.2 Approximate Global Convergence
6(5)
1.1.3 Some Notations and Definitions
11(3)
1.2 Some Examples of Ill-posed Problems
14(7)
1.3 The Foundational Theorem of A.N. Tikhonov
21(2)
1.4 Classical Correctness and Conditional Correctness
23(2)
1.5 Quasi-solution
25(2)
1.6 Regularization
27(4)
1.7 The Tikhonov Regularization Functional
31(4)
1.7.1 The Tikhonov Functional
32(2)
1.7.2 Regularized Solution
34(1)
1.8 The Accuracy of the Regularized Solution for a Single Value of α
35(4)
1.9 Global Convergence in Terms of Definition 1.1.2.4
39(7)
1.9.1 The Local Strong Convexity
40(5)
1.9.2 The Global Convergence
45(1)
1.10 Uniqueness Theorems for Some Coefficient Inverse Problems
46(33)
1.10.1 Introduction
46(2)
1.10.2 Carleman Estimate for a Hyperbolic Operator
48(8)
1.10.3 Estimating an Integral
56(1)
1.10.4 Cauchy Problem with the Lateral Data for a Hyperbolic Inequality with Volterra-Like Integrals
57(5)
1.10.5 Coefficient Inverse Problem for a Hyperbolic Equation
62(6)
1.10.6 The First Coefficient Inverse Problem for a Parabolic Equation
68(2)
1.10.7 The Second Coefficient Inverse Problem for a Parabolic Equation
70(6)
1.10.8 The Third Coefficient Inverse Problem for a Parabolic Equation
76(2)
1.10.9 A Coefficient Inverse Problem for an Elliptic Equation
78(1)
1.11 Uniqueness for the Case of an Incident Plane Wave in Partial Finite Differences
79(16)
1.11.1 Results
81(2)
1.11.2 Proof of Theorem 1.11.1.1
83(2)
1.11.3 The Carleman Estimate
85(5)
1.11.4 Proof of Theorem 1.11.1.2
90(5)
2 Approximately Globally Convergent Numerical Method
95(74)
2.1 Statements of Forward and Inverse Problems
97(1)
2.2 Parabolic Equation with Application in Medical Optics
98(2)
2.3 The Transformation Procedure for the Hyperbolic Case
100(3)
2.4 The Transformation Procedure for the Parabolic Case
103(3)
2.5 The Layer Stripping with Respect to the Pseudo Frequency s
106(3)
2.6 The Approximately Globally Convergent Algorithm
109(6)
2.6.1 The First Version of the Algorithm
111(1)
2.6.2 A Simplified Version of the Algorithm
112(3)
2.7 Some Properties of the Laplace Transform of the Solution of the Cauchy Problem (2.1) and (2.2)
115(7)
2.7.1 The Study of the Limit (2.12)
115(3)
2.7.2 Some Additional Properties of the Solution of the Problem (2.11) and (2.12)
118(4)
2.8 The First Approximate Global Convergence Theorem
122(18)
2.8.1 Exact Solution
123(2)
2.8.2 The First Approximate Global Convergence Theorem
125(12)
2.8.3 Informal Discussion of Theorem 2.8.2
137(1)
2.8.4 The First Approximate Mathematical Model
138(2)
2.9 The Second Approximate Global Convergence Theorem
140(26)
2.9.1 Estimates of the Tail Function
142(9)
2.9.2 The Second Approximate Mathematical Model
151(4)
2.9.3 Preliminaries
155(2)
2.9.4 The Second Approximate Global Convergence Theorem
157(9)
2.10 Summary
166(3)
3 Numerical Implementation of the Approximately Globally Convergent Method
169(24)
3.1 Numerical Study in 2D
170(16)
3.1.1 The Forward Problem
171(2)
3.1.2 Main Discrepancies Between the Theory and the Numerical Implementation
173(1)
3.1.3 Results of the Reconstruction
174(12)
3.2 Numerical Study in 3D
186(5)
3.2.1 Computations of the Forward Problem
186(2)
3.2.2 Result of the Reconstruction
188(3)
3.3 Summary of Numerical Studies
191(2)
4 The Adaptive Finite Element Technique and Its Synthesis with the Approximately Globally Convergent Numerical Method
193(102)
4.1 Introduction
193(3)
4.1.1 The Idea of the Two-Stage Numerical Procedure
193(1)
4.1.2 The Concept of the Adaptivity for CIPs
194(2)
4.2 Some Assumptions
196(2)
4.3 State and Adjoint Problems
198(1)
4.4 The Lagrangian
199(3)
4.5 A Posteriori Error Estimate for the Lagrangian
202(8)
4.6 Some Estimates of the Solution an Initial Boundary Value Problem for Hyperbolic Equation (4.9)
210(6)
4.7 Frechet Derivatives of Solutions of State and Adjoint Problems
216(6)
4.8 The Frechet Derivative of the Tikhonov Functional
222(3)
4.9 Relaxation with Mesh Refinements
225(10)
4.9.1 The Space of Finite Elements
226(3)
4.9.2 Minimizers on Subspaces
229(4)
4.9.3 Relaxation
233(2)
4.10 From the Abstract Scheme to the Coefficient Inverse Problem 2.1
235(2)
4.11 A Posteriori Error Estimates for the Regularized Coefficient and the Relaxation Property of Mesh Refinements
237(4)
4.12 Mesh Refinement Recommendations
241(3)
4.13 The Adaptive Algorithm
244(1)
4.13.1 The Algorithm In Brief
244(1)
4.13.2 The Algorithm
244(1)
4.14 Numerical Studies of the Adaptivity Technique
245(13)
4.14.1 Reconstruction of a Single Cube
246(2)
4.14.2 Scanning Acoustic Microscope
248(10)
4.15 Performance of the Two-Stage Numerical Procedure in 2D
258(8)
4.15.1 Computations of the Forward Problem
258(3)
4.15.2 The First Stage
261(3)
4.15.3 The Second Stage
264(2)
4.16 Performance of the Two-Stage Numerical Procedure in 3D
266(13)
4.16.1 The First Stage
275(2)
4.16.2 The Second Stage
277(2)
4.17 Numerical Study of the Adaptive Approximately Globally Convergent Algorithm
279(12)
4.17.1 Computations of the Forward Problem
285(3)
4.17.2 Reconstruction by the Approximately Globally Convergent Algorithm
288(1)
4.17.3 The Adaptive Part
289(2)
4.18 Summary of Numerical Studies of
Chapter 4
291(4)
5 Blind Experimental Data
295(40)
5.1 Introduction
295(2)
5.2 The Mathematical Model
297(1)
5.3 The Experimental Setup
298(3)
5.4 Data Simulations
301(1)
5.5 State and Adjoint Problems for Experimental Data
302(2)
5.6 Data Pre-Processing
304(5)
5.6.1 The First Stage of Data Immersing
304(3)
5.6.2 The Second Stage of Data Immersing
307(2)
5.7 Some Details of the Numerical Implementation of the Approximately
309(2)
5.7.1 Stopping Rule for
311(1)
5.8 Reconstruction by the Approximately Globally Convergent Numerical Method
311(8)
5.8.1 Dielectric Inclusions and Their Positions
311(1)
5.8.2 Tables and Images
312(2)
5.8.3 Accuracy of the Blind Imaging
314(2)
5.8.4 Performance of a Modified Gradient Method
316(3)
5.9 Performance of the Two-Stage Numerical Procedure
319(13)
5.9.1 The First Stage
319(1)
5.9.2 The Third Stage of Data Immersing
320(3)
5.9.3 Some Details of the Numerical Implementation of the Adaptivity
323(1)
5.9.4 Reconstruction Results for Cube Number 1
323(2)
5.9.5 Reconstruction Results for the Cube Number 2
325(2)
5.9.6 Sensitivity to the Parameters α and β
327(1)
5.9.7 Additional Effort for Cube Number 1
327(5)
5.10 Summary
332(3)
6 Backscattering Data
335(58)
6.1 Introduction
335(2)
6.2 Forward and Inverse Problems
337(2)
6.3 Laplace Transform
339(1)
6.4 The Algorithm
340(6)
6.4.1 Preliminaries
340(2)
6.4.2 The Sequence of Elliptic Equations
342(2)
6.4.3 The Iterative Process
344(1)
6.4.4 The Quasi-Reversibility Method
345(1)
6.5 Estimates for the QRM
346(8)
6.6 The Third Approximate Mathematical Model
354(4)
6.6.1 Exact Solution
354(2)
6.6.2 The Third Approximate Mathematical Model
356(2)
6.7 The Third Approximate Global Convergence Theorem
358(9)
6.8 Numerical Studies
367(9)
6.8.1 Main Discrepancies Between Convergence Analysis and Numerical Implementation
367(1)
6.8.2 A Simplified Mathematical Model of Imaging of Plastic Land Mines
368(1)
6.8.3 Some Details of the Numerical Implementation
369(3)
6.8.4 Numerical Results
372(2)
6.8.5 Backscattering Without the QRM
374(2)
6.9 Blind Experimental Data Collected in the Field
376(17)
6.9.1 Introduction
378(1)
6.9.2 Data Collection and Imaging Goal
379(2)
6.9.3 The Mathematical Model and the Approximately Globally Convergent Algorithm
381(4)
6.9.4 Uncertainties
385(3)
6.9.5 Data Pre-processing
388(3)
6.9.6 Results of Blind Imaging
391(1)
6.9.7 Summary of Blind Imaging
392(1)
References 393(8)
Index 401
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