偏微分方程逆问题
2004-4
世界图书出版公司
Victor Isakov
284
This book describes the contemporary state of the theory and some numerical aspects of inverse problems in partial differential equations. The topic is of substantial and growing interest for many scientists and engineers, and accordingly to graduate students in these areas.
Preface Chapter 1 Inverse Problems 1.1 The inverse problem of gravimetry 1.2 The inverse 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 Construction of regularizers 2.4 Convergence of regularization algorithms 2.5 Iterative algorithms Chapter 3 Uniqueness and Stability in the Cauchy Problem 3.1 The backward parabolic equation 3.2 General Carleman type estimates and the Cauchy problem 3.3 Elliptic and parabolic equations 3.4 Hyperbolic and Schr6dinger equations 3.5 Open problems Chapter 4 Elliptic Equations:cSingle Boundary Measurements 4.0 Results on elliptic boundary value problems 4.1 Inverse gravimetry 4.2 Reconstruction of lower-ordercterms 4.3 The inverse conductivity problem 4.4 Methodscof the theory of one complex variable 4.5 Linearization of the coefficients problem 4.6 Some problems of nondestructive evaluation 4.7 Open problems Chapter 5 Elliptic Equations: Many Boundary Measurements 5.0 The Dirichlet-to-Neumann map 5.1 Boundary Reconstruction 5.2 Reconstruction in 5.3 Completeness of products of solutions of PDE 5.4 Thecplane case 5.5 Recovery of several coefficients 5.6 Nonlinear equations 5.7 Discontinuous conductivities 5.9 Open problems Chapter 6 Scattering Problems 6.0 Dire tcscattering 6.1 From A to near field 6.2 Scattering by a medium 6.3 Scattering by obstacles 6.4 Open problems Chapter 7 Integral Geometrycand Tomography 7.1 The Radon transform and its inverse 7.2 The energy integrals methods 7.3 Boman'sccounterexample 7.4 The transport equation 7.5 Open problems Chapter 8 Hyperboli Equations 8.0 Introduction 8.1 Thecone-dimensional case 8.2 Single boundary measurements 8.3 Many measurements:cuse of beam solutions 8.4 Many measurements:cmethods of boundary contro 8.5 Recovery of discontinuity of thecspeed of propagation 8.6 Opencproblems Chapter 9 Parabolic Equations 9.0 Introduction 9.1 Final overdetermination 9.2 Lateral overdetermination:csingle measurements 9.3 Lateral overdetermination:cmany measurements 9.4 Discontinuous principal coefficient 9.5 Nonlinear equations 9.6 Interior sources 9.7 Open problems Chapter 10 Some Numerical Methods 10.1 Linearization 10.2 Variational regularizationcof the Cauchycproblem 10.3 Relaxation methods 10.4 Layer-stripping 10.5 Discrete methods Appendix.cFun tional Spaces References Index
