Isakov V. Inverse Problems for Partial Differential Equations

3rd Edition. — Springer International Publishing AG, 2017. — 414 p. — (Applied Mathematical Sciences 127) — ISBN: 3319516574This 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. Mathematically, these problems are relatively new and quite challenging due to the lack of conventional stability and to nonlinearity and non-convexity. Applications include recovery of inclusions from anomalies of their gravitational fields; reconstruction of the interior of the human body from exterior electrical, ultrasonic, and magnetic measurements; recovery of interior structural parameters of detail of machines and of the underground from similar data (nondestructive evaluation); and locating flying or navigated objects from their acoustic or electromagnetic fields. Currently, there are hundreds of publications containing new and interesting results. A purpose of the book is to collect and present many of them in a readable and informative form.
In 10 years after the publication of the second edition of this book, the changing field of inverse problems witnessed further new developments. Parts of the look were used at several universities, and many colleagues and students as well as myself observed several misprints and imprecisions. Some of the research problems from the previous editions have been solved. We comment on major changes.Inverse Problems
Ill-Posed Problems and Regularization
Uniqueness and Stability in the Cauchy Problem
Elliptic Equations: Single Boundary Measurements
Elliptic Equations: Many Boundary Measurements
Scattering Problems and Stationary Waves
Integral Geometry and Tomography
Hyperbolic Problems
Inverse Parabolic Problems
Some Numerical Methods
Appendix. Function Spaces