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E-raamat: Inverse Problems for Partial Differential Equations

  • Formaat: EPUB+DRM
  • Sari: Applied Mathematical Sciences 127
  • Ilmumisaeg: 24-Feb-2017
  • Kirjastus: Springer International Publishing AG
  • Keel: eng
  • ISBN-13: 9783319516585
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  • Formaat: EPUB+DRM
  • Sari: Applied Mathematical Sciences 127
  • Ilmumisaeg: 24-Feb-2017
  • Kirjastus: Springer International Publishing AG
  • Keel: eng
  • ISBN-13: 9783319516585

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This third edition expands upon the earlier edition by adding nearly 40 pages of new material reflecting the analytical and numerical progress in inverse problems in last 10 years.  As in the second edition, the emphasis is on new ideas and methods rather than technical improvements. These new ideas include use of the stationary phase method in the two-dimensional elliptic problems and of multi frequencies\temporal data to improve stability and numerical resolution. There are also numerous corrections and improvements of the exposition throughout.

This book is intended for mathematicians working with partial differential equations and their applications, physicists, geophysicists, and financial, electrical, and mechanical engineers involved with nondestructive evaluation, seismic exploration, remote sensing, and various kinds of tomography.







Review of the second edition:

"The first edition of this excellent book appeared in 1998 and became a standard reference for everyone interested in analysis and numerics of inverse problems in partial differential equations. The second edition is considerably expanded and reflects important recent developments in the field . Some of the research problems from the first edition have been solved ." (Johannes Elschner, Zentralblatt MATH, Vol. 1092 (18), 2006)

Arvustused

The book in its present third edition thus continues to serve as a valuable introduction and reference work on the contemporary analytical and numerical methods for treating inverse problems for PDE and it will guide its readers straight to forefront of current mathematical research questions in this field. (Aleksandar Perovi, zbMATH, Vol. 1366.65087, 2017) 

Preface to the Third Edition vii
Preface to the Second Edition ix
Preface to the First Edition xi
Chapter 1 Inverse Problems
1(22)
1.1 The Inverse Problem of Gravimetry
1(5)
1.2 The Inverse Conductivity Problem
6(2)
1.3 Inverse Scattering
8(4)
1.4 Tomography and the Inverse Seismic Problem
12(4)
1.5 Inverse Spectral Problems
16(7)
Chapter 2 Ill-Posed Problems and Regularization
23(24)
2.1 Well- and Ill-Posed Problems
23(4)
2.2 Conditional Correctness: Regularization
27(3)
2.3 Construction of Regularizes
30(7)
2.4 Convergence of Regularization Algorithms
37(5)
2.5 Iterative Algorithms
42(5)
Chapter 3 Uniqueness and Stability in the Cauchy Problem
47(58)
3.1 The Backward Parabolic Equation
48(10)
3.2 General Carleman Estimates and the Cauchy Problem
58(7)
3.3 Elliptic and Parabolic Equations
65(8)
3.4 Hyperbolic and Schrodinger Equations
73(22)
3.5 Systems of Partial Differential Equations
95(6)
3.6 Open Problems
101(4)
Chapter 4 Elliptic Equations: Single Boundary Measurements
105(44)
4.0 Results on Elliptic Boundary Value Problems
105(3)
4.1 Inverse Gravimetry
108(6)
4.2 Reconstruction of Lower-Order Terms
114(5)
4.3 The Inverse Conductivity Problem
119(11)
4.4 Methods of the Theory of One Complex Variable
130(5)
4.5 Linearization of the Coefficient Problem
135(4)
4.6 Some Problems of Detection of Defects
139(7)
4.7 Open Problems
146(3)
Chapter 5 Elliptic Equations: Many Boundary Measurements
149(62)
5.0 The Dirichlet-to-Neumann Map and the Cauchy Set
149(5)
5.1 Boundary Reconstruction
154(5)
5.2 Completeness of Products of Solutions of PDE
159(6)
5.3 Uniqueness and Stability in Ω
165(7)
5.4 Recovery of Several Coefficients
172(6)
5.5 The Plane Case
178(8)
5.6 Nonlinear Equations
186(7)
5.7 Transmission Problems
193(9)
5.8 Maxwell's and Elasticity Systems
202(6)
5.9 Open Problems
208(3)
Chapter 6 Scattering Problems and Stationary Waves
211(30)
6.0 Direct Scattering
211(3)
6.1 From A to Near Field
214(8)
6.2 The Inverse Source Problem with Many Wave Numbers
222(3)
6.3 Scattering by a Medium
225(6)
6.4 Scattering by Obstacles
231(7)
6.5 Open Problems
238(3)
Chapter 7 Integral Geometry and Tomography
241(28)
7.1 The Radon Transform and its Inverse
241(10)
7.2 The Energy Integral Methods
251(4)
7.3 Boman's Counterexample
255(4)
7.4 The Transport Equation
259(7)
7.5 Open Problems
266(3)
Chapter 8 Hyperbolic Problems
269(40)
8.0 Introduction
269(4)
8.1 The One-Dimensional Case
273(8)
8.2 Single Boundary Measurements
281(9)
8.3 Many Measurements: The Use of Beam Solutions
290(6)
8.4 Many measurements: Methods of Boundary Control
296(7)
8.5 Recovery of Discontinuity of the Speed of Propagation
303(4)
8.6 Open Problems
307(2)
Chapter 9 Inverse Parabolic Problems
309(46)
9.0 Introduction
309(4)
9.1 Final Overdetermination
313(6)
9.2 Lateral Overdetermination: Single Measurements
319(6)
9.3 The Inverse Problem of Option Pricing
325(6)
9.4 Lateral Overdetermination: Many Measurements
331(4)
9.5 Discontinuous Principal Coefficient and Recovery of a Domain
335(10)
9.6 Nonlinear Equations
345(5)
9.7 Interior Sources
350(2)
9.8 Open Problems
352(3)
Chapter 10 Some Numerical Methods
355(26)
10.1 Numerics for the Cauchy Problem
356(6)
10.2 Linearization
362(5)
10.3 Relaxation Methods
367(3)
10.4 Layer Stripping
370(3)
10.5 Range Test Algorithms
373(4)
10.6 Discrete Methods
377(4)
Appendix. Function Spaces 381(4)
References 385(20)
Index 405