In 8 years after publication of the first version of this book，the rapidly progressing field of inverse problems witnessed changes and new developments Parts of艾赛科威专著的《偏微分方程中的逆问题(第2版)》were used at several universities．and many colleagues and students as
well asmyselfobserved several misprintsandimprecisions Some ofthe research
problems from the first edition have been solved This edition serves the purposes
of reflecting these changes and making appropiate corrections 1 hope that these
additions and corrections resulted in not too many new errors and misprints
Chapters I and 2 contain only 2-3 Pages of new materiaIJike in sections 1．5．
2 5 Chapter 3 order equations and included bound……
Prefaee to the Second Edition
PrefacetotheFirstEdifion
Chapter l Inverse Problems
1．1 The inverse problem of gravimetry
1．2 The lnverse conductivity problem
1．3 Inverse scattering
1．4 Tomography and the inverse seismic problem
1．5 Inverse spectral problems
Chapter 2 Ill-Posed Problems and Regularization
2．1 Well．and ill—posed problems
2．2 Conditional correctness：Regularization
2．3 Construetion of regularizers
2．4 Convergence of regularization algorithms
2．5 herative algorithms
Chapter 3 Uniqueness and Stability in the Cauchy Problem
3．1 The backward parabolic equation
3．2 General Carleman estimates and the Cauchy problem
3．3 Elliptic and parabolic equations
3．4 Hyperbolic and Schr6dinger equations
3．5 Systems of partial differential equations
3．6 Open problems
Chapter 4 Elliptic Equations：Single Boundary Measurements
4．0 Results on elliptic boundary value problems
4．1 Inverse gravimetry
4．2 Reconstruction of lowcr-order terms
4．3 The jBVeFSC conductivity problem
4．4 Methods of the theory of one complex variable
4．5 Linearization of the coe佑cients problem
4．6 Some problems ofdetection ofdefects
4．7 Open problems
Chapter 5 Elliptic Equations：Many Boundary Measurements
5．O The Dirichlet．to—Neumann map
5．1 Boundary reconstruction
5．2 Reconstruction in Q
5．3 Completeness of products of solutions of PDE
5．4 Recovery of several coeffcients
5．5 The plane case
5．6 Nonlinear equations
5．7 Discontinuous conductivities
5．8 Maxwell’s and elasticity systems
5．9 Open problems
Chapter 6 Scattering Problems
6．0 Direct Scattering
6 l From A to nearfield
6 2 Scattering by a medium
6．3 Scattering by obstacles
6．4 Open problems
Chapter 7 Integral Geometry and Tomography
7．1 The Radon transfofin and its inverse
7．2 The energy integral methods
7 3 Boman’s counterexample
7．4 The transport equation
7．5 Open problems
Chapter 8 Hyperbolic Problems
8．0 Introduction
8．1 The one．dimensional case
8．2 Single boundary measurements
8．3 Many measurements：use of beam solutions
8．4 Many measurements：methods of boundary control
8．5 Recovery ofdiscontinuity ofthe speed ofpropagation
8．6 Open problems
Chapter 9 Inverse parabolic problems
9．0 Introduction
9．1 Final orerdetermination
9．2 Lateral orerdetermination：single measurements
9．3 ne inverse problem of option pricing
9．4 Lateral overdetermination：many measurements
9．5 Discontinuous principal coeMcient and recovery of a domain
9．6 Nonlinear equations
9．7 Interior sources
9．8 Open problems
Chapter 10 Some Numerical Methods
10．1 Linearization
10．2 Variational regularization of the Cauchy problem
1O．3 Relaxation methods
10．4 Layer-stripping
10．5 Range test algorithms
10．6 Discrete methods
Appendix．Funcfion~Spaces
References
Index
