Inverse problems for partial differential equations. 2nd ed. (English) Zbl 1092.35001

Isakov, Victor

Applied Mathematical Sciences 127. New York, NY: Springer (ISBN 0-387-25364-5/hbk). xiii, 344 p. (2006).

The first edition of this excellent book appeared in 1998 (see Zbl 0908.35134) see and became a standard reference for everyone interested in analysis and numerics of inverse problems in partial differential equations. The topic is of substantial and growing interest for mathematicians, physicists and engineers, and to graduate students in these areas as well. The book consists of the following Chapters: 1) Inverse problems, 2) Ill-posed problems and regularization, 3) Uniqueness and stability in the Cauchy problem, 4) Elliptic equations: single boundary measurements, 5) Elliptic equations: many boundary measurements, 6) Scattering problems, 7) Integral geometry and tomography, 8) Hyperbolic equations, 9) Inverse parabolic problems, 10) Some numerical methods.
The second edition is considerably expanded and reflects important recent developments in the field, including new uniqueness and stability results for basic inverse problems, emerging financial applications, and new efficient reconstruction algorithms. Some of the research problems from the first edition have been solved, while most of them still await solutions.

Reviewer: Johannes Elschner (Berlin)

MSC:

35-02	Research exposition (monographs, survey articles) pertaining to partial differential equations
35R30	Inverse problems for PDEs
65N21	Numerical methods for inverse problems for boundary value problems involving PDEs
35J15	Second-order elliptic equations
35K30	Initial value problems for higher-order parabolic equations
35P25	Scattering theory for PDEs

Keywords:

inverse problem of gravimetry; inverse conductivity problem; inverse scattering problems; tomography; inverse seismic problem; inverse spectral problems; regularization; inverse problems of elliptic equations; Dirichlet-to-Neumann map; inverse parabolic problems; numerical methods