偏微分方程中的逆问题(第2版)(影印版)
2014-3
世界图书出版公司
Victor Isakov
《偏微分方程中的逆问题（第2版）》全面讲述了目前偏微分方程中逆问题的理论和数值方面。逆问题这个话题十分宽泛，并且得到了众多科学家和工程人员的青睐。这是第二版，包括了逆问题领域的最新进展，给出了理论和计算方法，强调最新观点和技巧。书中也体现了和第一版的不同，做了许多修订，内容更加充实，增加了如伪凸的概念，简化了证明。新材料的增加反应了逆问题理论的最新进展。本书对象是偏微分方程及其应用领域的数学工作者、物理学家、几何物理学家和工程人员。
目次：逆问题；不适定问题和正则化；柯西问题中的唯一性和稳定性；椭圆方程：单边界测量；椭圆方程：多边界测量；散射问题；积分几何和x线断层摄影术；双曲问题；逆抛物问题；一些数值方法。
读者对象：偏微分方程及其应用领域的数学工作者、物理学家、几何物理学家和工程人员。
《偏微分方程中的逆问题(第2版)》
preface to the second edition
preface to the first edition
chapter i inverse problems
1.1 the inverse problem of gravimctry
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 regularlzation
2. i 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 canchy problem
3. l the backward parabolic equation
3.2 general carleman estimates and the cauchy problem
3.3 elliptic and parabolic equations
3.4 hyperbolic and schr'6dinger 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. l inverse gravimetry
4.2 reconstruction of lower-order terms
4.3 the inverse conductivity problem
4.4 methods of the theory of one complex variable
4.5 linearization of the coefficients problem
4.6 some problems of detection of defects
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 f2
5.3 completeness of products of solutions of pde
5.4 recovery of several coefficients
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
6.1 from .a to near field
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 transform 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 of discontinuity of the speed of propagation
8.6 open problems
chapter 9 inverse parabolic problems
9.0 introduction
9. l final overdetermination
9.2 lateral overdetermination: single measurements
9.3 the inverse problem of option pricing
9.4 lateral overdetermination: many measurements
9.5 discontinuous principal coefficient 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
10.3 relaxation methods
10.4 layer-stripping
10.5 range test algorithms
10.6 discrete methods
appendix. functional spaces
references
index
