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Inverse modeling. An introduction to the theory and methods of inverse problems and data assimilation. (English) Zbl 1346.65029

Bristol: IOP Publishing (ISBN 978-0-7503-1219-6/hbk; 978-0-7503-1218-9/ebook). not consecutively paged. (2015).
The authors, Professors Gen Nakamura and Roland Potthast, are rooted in mathematical analysis and for many years they are very well known to the international inverse problems community as experts in inverse and ill-posed problems. Moreover, since 2010, the second author holds a post as a director at the German Weather Service. Also therefore, the book differs from other books on inverse problems theory, methods and applications, because the aspect of data assimilation, which is of great importance for atmospheric sciences and in particular for weather forecast systems, plays also a prominent role. It is a specialty of the publisher IOP Publishing that the book is not consecutively paged and only the individual chapters have their own numbering. Therefore it should be mentioned that this substantial and interesting introduction to the theory and methods of inverse problems and data assimilation has 492 pages spread over 17 chapters and an appendix collecting important formulas and definitions which are needed in single chapters.
The authors introduce the reader to basic concepts and ingredients of the inverse problems theory by presenting a collection of applied inverse problems as well as of deterministic and stochastic regularization approaches for their stable approximate solution. Mathematical analysis, functional analysis and probability theory provide the main tools for the investigations and algorithms presented in the book with respect to both inverse problems and data assimilation. In this context, the authors take claim for themselves to include deep mathematical studies of ill-posendess phenomena, convergence, consistency and stability. There are given lists of relevant monographs and articles at the end of each chapter.
Chapter 1 introduces to types of inverse problems and their mathematical set-up in abstract Hilbert and Banach spaces. Inverse source problems, inverse scattering problems, the dynamical systems inversion and spectral inverse problems serve as illustrative examples. Furthermore, the introductory chapter provides short characterizations of the content of each chapter. With focus on operator equations and to understand the impact of ill-posedness on inverse problems, Chapter 2 summarizes functional analysis tools required in the subsequent chapters. All serious approaches to inverse problems are in some sense regularization approaches, i.e., the original ill-posed problem is substituted by a well-posed auxiliary problem, where a regularization parameter controls this process which is based on an appropriate trade-off between stability and approximation. Chapter 3 introduces the reader to these concepts relevant for the successful treatment of inverse and data assimilation problems. Classical regularization methods like Tikhonov regularization and spectral-off are explained as usual for linear inverse problems. The annoying but unavoidable question of choosing the regularization parameter is briefly touched. Discussions on iterative regularization and stopping rules complete the chapter. A glimpse of stochastic aspects for inverse problems is given in Chapter 4. In particular, Bayesian methods have become rather popular in the past years.
The Chapter 5 gives a comprehensive introduction into the modern field of dynamical systems inversion and data assimilation. The theory has grown from applications in atmospheric sciences and oceanography and a series of recent publications form a substantial basis for the presented ideas and methods. The focus of data assimilation is on the estimation of initial conditions and on the determination of required parameter functions of dynamical systems from measurement data. In the literature, the concept of 3D-VAR for state estimation by Tikhonov regularization at different subsequent time steps of the dynamical system and its four-dimensional extension 4D-VAR are important for this scenery like also the ensemble-based Kalman filter (EnKF). Chapter 6 refers to MATLAB and OCTAVE programming and presents some illustrative material with codes for toy problems concerning inverse theory and data assimilation. Chapters 7 and 8 are devoted to neural field inversion and inverse problems for acoustic and electromagnetic waves, respectively. Although the theory is mostly introduced by means of linear inverse problems, the majority of inverse real-life problems is nonlinear. This means that the forward operator mapping the unknown functions to the observable quantity is a nonlinear one. Even if the partial differential equations under consideration are linear, the inverse partial differential equation problems aimed at finding parameter functions in the differential equation or in the boundary condition are in general nonlinear. To handle such problems in infinite-dimensional function spaces, the tools of nonlinear functional analysis including Fréchet derivatives and associated linearizations of the nonlinear operators must be exploited. Chapter 9 presents some material on nonlinearity.
As mentioned, the book is a mixture of theory and practice in the world of inverse problems, which means that chapters devoted to specific applications alternate with those on the mathematical theory of inverse problems and data assimilation. Chapter 10 is designed as a theory section devoted to the basics of inverse problems analysis: uniqueness, stability, convergence and convergence rates. With hints to the relevant literature it introduces the reader to the associated concepts, where the notation may be somewhat different in inverse problems theory and data assimilation. The concepts of uniqueness, local uniqueness and \(\epsilon\)-uniqueness are presented. Moreover, uniquenss and stability are illustrated by means of inverse obstacle scattering problems. Inverse problems formulated as operator equations in infinite-dimensional normed spaces are ill-posed and one approach of stabilization is by discretization. In this context, one section of Chapter 10 performs first steps to projection methods. It is a speciality of this book that there is also a section on stability of cycled data assimilation, where the authors give answers to the question what happens with respect to stability when schemes of data assimilation methods like 3D-VAR are cycled. A final section of this chapter gives an overview of usual convergence concepts both in deterministic and stochastic set-ups and with applications to inverse scattering.
In medical applications as well as in engineering applications, magnetic tomography is of importance. The goal of this method discussed in Chapter 11 is the reconstruction of currents from their magnetic field. On the other hand, several approaches for finding not directly observable fields in acoustics, electromagnetics and fluid dynamics are presented in Chapter 12. They are based on Fourier-Hankel and Fourier-plane-wave expansions and moreover on the work by Kirsch and Kress who essentially influenced the progress in this area. Sampling methods and probing methods are considered in Chapters 13 and 14. Studies of the range of operators as a tool for solving inverse problems can be found in Chapter 15 which is devoted to analytic continuation tests. Chapter 17, however, takes into account the task of optimizing the experimental design for handling the inverse problems. The authors call it meta-inverse problems. This means that one has the freedom to choose parts of the set-up, the location of sensors or to add additional observations in order to improve the situation. For example, it is possible to use an experimental design which contains a maximum of information about the parameters to be determined and hence to suppress the ill-posedness and ill-conditioning effect in a best possible way. For data assimilation, targeted measurements and different types of sensitivity analysis play an important role and are parts of this area of research.

MSC:

65J22 Numerical solution to inverse problems in abstract spaces
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
47A52 Linear operators and ill-posed problems, regularization
65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs
00A71 General theory of mathematical modeling
62-07 Data analysis (statistics) (MSC2010)
35R30 Inverse problems for PDEs
45Q05 Inverse problems for integral equations
65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization
65J10 Numerical solutions to equations with linear operators
65J15 Numerical solutions to equations with nonlinear operators
76Q05 Hydro- and aero-acoustics
78A40 Waves and radiation in optics and electromagnetic theory
86A05 Hydrology, hydrography, oceanography
92C55 Biomedical imaging and signal processing
62K99 Design of statistical experiments

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