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**Inverse problems for partial differential equations.**
*(English)*
Zbl 0908.35134

Applied Mathematical Sciences. 127. New York, NY: Springer. xi, 284 p. (1998).

Most mathematical problems in science, technology and medicine are inverse problems. In nature the sense organs of an animal or man have to interprete electromagnetic waves, acoustic waves, and chemical products. The interpretation is realized by the special system using physical and chemical processes. If one wishes to replace the sense organs by measuring instruments, one needs a mathematical model and must study inverse problems for the optimal use of measuring data. The necessary high-level models require profound mathematics. Such investigations of equally deep and broad character can be done only by highly qualified and experienced scientists. Mathematics is the basis for all interpretative sciences, especially for medical diagnostics, using machines and measuring instruments. 30-40 years ago the following argumentation was quite customary with physicists and medical researchers. Nature exists, hence it can be completely characterized by a mathematical model with only few empirical parameters. This leads to the mistakes in geophysics, medicine, astrophysics at large distances, and so on. Therefore in interpretative sciences the principle praxis cum theoria holds.

Studying inverse problems is the only way of completely analyzing experimental results. This book describes the contemporary state of the theory and some numerical aspects of inverse problems in particular differential equations. The topic is of substantial and growing interest for many scientists, engineers and medical doctors. Mathematically, these problems are relatively new and quite challenging due to the lack of conventional stability and to nonlinearity and nonconvexity. V. Isakov knows completely the papers on inverse problems published in Russia, the USA and so on. 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.

Chapter 1 contains some special examples of inverse problem: the inverse problem of gravimetry, the inverse conductivity problem, inverse scattering problems, tomography and the inverse seismic problem, inverse spectral problems, and so on. The solutions of boundary or initial-boundary value problems (so-called direct problems) have very good mathematical properties (compact operators). Following F. Riesz (1918) the inverse operator of a one-to-one mapping on Banach spaces is discontinuous. Therefore the inverse problem is a so-called ill-posed problem.

Chapter 2 deals with ill-posed problems and the regularization of such problems. Such methods are necessary in order to get useful results for inverse problems. In Chapter 3 one can find basic results of direct problems necessary for studying inverse problems: Uniqueness and stability in the Cauchy problem. Chapter 4 is devoted to inverse problems of elliptic equations using single boundary measurements, for instance in nondestructive evaluation. If one wishes to determine uniquely the conductivity coefficient \(a\) (Chapter 5) in the equation \(\text{div}(- a\nabla u)= 0\) one needs many boundary measurements. This is possible by using the so-called Dirichlet-to-Neumann map defined on the boundary \(\partial\Omega\) for sufficiently smooth Dirichlet data \(u\) by \(\Lambda(u)= a{\partial u/\partial\nu}\). If \(\Lambda(u)\) is known for all \(u\) then the conductivity \(a\) is uniquely determined, also in more general cases.

There are some important results concerning scattering problems (Chapter 6) and integral geometry and tomography. The computerized tomography and inverse seismic problems consist in the following: the unknown function \(f\) is determined by using integrals \(\int_\gamma fd\gamma\) over a family of manifolds \(\{\gamma\}\) (Chapter 7). In applications many problems are governed by hyperbolic equations. Several coefficients or the right-hand side of the equation are to be determined by using additional information (Chapter 8). Chapter 9 deals with inverse parabolic problems and Chapter 10 with some numerical results on inverse problems.

This book is an important contribution to the theory of inverse problems. However, in the future we also need textbooks containing the principles of interpretative sciences not only using (deep results of) mathematics. It is possible to write such books but very difficult to realize.

The following books on inverse problems were published independently from this book: M. M. Lavrent’ev and L. Ya. Savel’ev, Linear operators and ill-posed problems (Consultant Bureau, Plenum Publishing, New York 1995; Zbl 0849.35143), and V. G. Romanov and S. L. Kubanikhin, Inverse problems for Maxwell’s equations (VSP, Utrecht 1994; Zbl 0853.35001).

Studying inverse problems is the only way of completely analyzing experimental results. This book describes the contemporary state of the theory and some numerical aspects of inverse problems in particular differential equations. The topic is of substantial and growing interest for many scientists, engineers and medical doctors. Mathematically, these problems are relatively new and quite challenging due to the lack of conventional stability and to nonlinearity and nonconvexity. V. Isakov knows completely the papers on inverse problems published in Russia, the USA and so on. 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.

Chapter 1 contains some special examples of inverse problem: the inverse problem of gravimetry, the inverse conductivity problem, inverse scattering problems, tomography and the inverse seismic problem, inverse spectral problems, and so on. The solutions of boundary or initial-boundary value problems (so-called direct problems) have very good mathematical properties (compact operators). Following F. Riesz (1918) the inverse operator of a one-to-one mapping on Banach spaces is discontinuous. Therefore the inverse problem is a so-called ill-posed problem.

Chapter 2 deals with ill-posed problems and the regularization of such problems. Such methods are necessary in order to get useful results for inverse problems. In Chapter 3 one can find basic results of direct problems necessary for studying inverse problems: Uniqueness and stability in the Cauchy problem. Chapter 4 is devoted to inverse problems of elliptic equations using single boundary measurements, for instance in nondestructive evaluation. If one wishes to determine uniquely the conductivity coefficient \(a\) (Chapter 5) in the equation \(\text{div}(- a\nabla u)= 0\) one needs many boundary measurements. This is possible by using the so-called Dirichlet-to-Neumann map defined on the boundary \(\partial\Omega\) for sufficiently smooth Dirichlet data \(u\) by \(\Lambda(u)= a{\partial u/\partial\nu}\). If \(\Lambda(u)\) is known for all \(u\) then the conductivity \(a\) is uniquely determined, also in more general cases.

There are some important results concerning scattering problems (Chapter 6) and integral geometry and tomography. The computerized tomography and inverse seismic problems consist in the following: the unknown function \(f\) is determined by using integrals \(\int_\gamma fd\gamma\) over a family of manifolds \(\{\gamma\}\) (Chapter 7). In applications many problems are governed by hyperbolic equations. Several coefficients or the right-hand side of the equation are to be determined by using additional information (Chapter 8). Chapter 9 deals with inverse parabolic problems and Chapter 10 with some numerical results on inverse problems.

This book is an important contribution to the theory of inverse problems. However, in the future we also need textbooks containing the principles of interpretative sciences not only using (deep results of) mathematics. It is possible to write such books but very difficult to realize.

The following books on inverse problems were published independently from this book: M. M. Lavrent’ev and L. Ya. Savel’ev, Linear operators and ill-posed problems (Consultant Bureau, Plenum Publishing, New York 1995; Zbl 0849.35143), and V. G. Romanov and S. L. Kubanikhin, Inverse problems for Maxwell’s equations (VSP, Utrecht 1994; Zbl 0853.35001).

Reviewer: Gottfried Anger (Berlin)

### MSC:

35R30 | Inverse problems for PDEs |

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

35J15 | Second-order elliptic equations |

35K30 | Initial value problems for higher-order parabolic equations |

35P25 | Scattering theory for PDEs |