Regularization of inverse problems.

*(English)*Zbl 0859.65054
Mathematics and its Applications (Dordrecht). 375. Dordrecht: Kluwer Academic Publishers. viii, 321 p. (1996).

“Driven by the needs of applications both in sciences and in industry, the field of inverse problems has certainly been one of the fastest growing areas in applied mathematics recently. Inverse problems typically lead to mathematical models that are ill-posed in the sense of Hadamard. Especially, their solution is unstable under data perturbations, so that special numerical methods that can cope with these instabilities, so-called regularization methods, have to be developed.

This book is devoted to the mathematical theory of regularization methods and is intended to give an up-to-date account of the currently available results about regularization methods.” (From the cover text.)

Roughly speaking the book is divided into two parts: Secs. 2 to 9 where a rather complete theory for linear problems is treated, and Secs. 10 and 11 concerning nonlinear problems.

In the (introductory) Sec. l the authors discuss examples of various inverse problems from different application fields (as computerized tomography, physical chemistry, signal processing, heat conduction, parameter identification, inverse scattering). A lot of linear inverse problems can be formulated as first kind integral equations.

Sec. 2 treats linear operator equations from the view points of ill-posedness, generalized inverses, spectral theory and functional calculus. In Sec. 3 abstract regularization theory, order optimality and regularization by projection are considered. A priori parameter choice rules, saturation and converse results and a posteriori parameter choice rules are treated in Sec. 4. Moreover, in this section error free, i.e. heuristic, parameter choice rules as the L-curve method, and mollifier methods, where only a ‘mollified version’ of the solution is looked for, are investigated. Then, Sec. 5 is devoted to Tikhonov regularization from the view points of a posteriori rules, iteration, combination with projection. Besides, maximum entropy regularization and problems with convex constraints are studied. In Sec. 6, the authors consider iteration methods, such as Landweber iteration and \(\nu\)-methods with respect to their self-regularization property. In this section, certain properties of residual polynomials, special orthogonal polynomials and Christoffel functions are needed which are presented in the Appendices A, B and C, respectively. In Sec. 7 the conjugate gradient algorithm for the normal equation is studied with respect to stability, convergence, discrepancy principle, number of iterations. Sec. 8, entitled ‘Regularization with differential operators’, is devoted to regularization with different norms from Hilbert scales or seminorms generated by densely defined operators. Here, the ‘weighted generalized inverse’ is a useful notion.

The part on linear inverse problems culminates in Sec. 9, where effective numerical realizations of Tikhonov regularization and iterative regularization (combined with discretization and standard form transformation) are developed. This section seems to be extremely valuable for concrete applications of the theory.

As to the ‘nonlinear part’ of the book, in Sec. l0 the Tikhonov regularization of nonlinear problems with a continuous, weakly closed, FrĂ©chet differentiable operator is investigated under the aspects of a posteriori rules, regularization in Hilbert scales and maximum entropy regularization. Finally, in Sec. 11 the nonlinear Landweber iteration and Newton type methods are considered.

The bibliography of 290 items covers nearly all important major works up till 1995 concerning the mathematical regularization theory for inverse problems.

Again from the cover text: “This book, which can be read by students with a basic knowledge of functional analysis, should be useful both to mathematicians and to scientists and engineers who deal with inverse problems in their fields. It can be used as a text for a graduate course on inverse problems and will also be useful to specialists in the field as a reference work”.

This book is devoted to the mathematical theory of regularization methods and is intended to give an up-to-date account of the currently available results about regularization methods.” (From the cover text.)

Roughly speaking the book is divided into two parts: Secs. 2 to 9 where a rather complete theory for linear problems is treated, and Secs. 10 and 11 concerning nonlinear problems.

In the (introductory) Sec. l the authors discuss examples of various inverse problems from different application fields (as computerized tomography, physical chemistry, signal processing, heat conduction, parameter identification, inverse scattering). A lot of linear inverse problems can be formulated as first kind integral equations.

Sec. 2 treats linear operator equations from the view points of ill-posedness, generalized inverses, spectral theory and functional calculus. In Sec. 3 abstract regularization theory, order optimality and regularization by projection are considered. A priori parameter choice rules, saturation and converse results and a posteriori parameter choice rules are treated in Sec. 4. Moreover, in this section error free, i.e. heuristic, parameter choice rules as the L-curve method, and mollifier methods, where only a ‘mollified version’ of the solution is looked for, are investigated. Then, Sec. 5 is devoted to Tikhonov regularization from the view points of a posteriori rules, iteration, combination with projection. Besides, maximum entropy regularization and problems with convex constraints are studied. In Sec. 6, the authors consider iteration methods, such as Landweber iteration and \(\nu\)-methods with respect to their self-regularization property. In this section, certain properties of residual polynomials, special orthogonal polynomials and Christoffel functions are needed which are presented in the Appendices A, B and C, respectively. In Sec. 7 the conjugate gradient algorithm for the normal equation is studied with respect to stability, convergence, discrepancy principle, number of iterations. Sec. 8, entitled ‘Regularization with differential operators’, is devoted to regularization with different norms from Hilbert scales or seminorms generated by densely defined operators. Here, the ‘weighted generalized inverse’ is a useful notion.

The part on linear inverse problems culminates in Sec. 9, where effective numerical realizations of Tikhonov regularization and iterative regularization (combined with discretization and standard form transformation) are developed. This section seems to be extremely valuable for concrete applications of the theory.

As to the ‘nonlinear part’ of the book, in Sec. l0 the Tikhonov regularization of nonlinear problems with a continuous, weakly closed, FrĂ©chet differentiable operator is investigated under the aspects of a posteriori rules, regularization in Hilbert scales and maximum entropy regularization. Finally, in Sec. 11 the nonlinear Landweber iteration and Newton type methods are considered.

The bibliography of 290 items covers nearly all important major works up till 1995 concerning the mathematical regularization theory for inverse problems.

Again from the cover text: “This book, which can be read by students with a basic knowledge of functional analysis, should be useful both to mathematicians and to scientists and engineers who deal with inverse problems in their fields. It can be used as a text for a graduate course on inverse problems and will also be useful to specialists in the field as a reference work”.

Reviewer: G.Bruckner (Berlin)

##### MSC:

65J20 | Numerical solutions of ill-posed problems in abstract spaces; regularization |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

65J10 | Numerical solutions to equations with linear operators (do not use 65Fxx) |

47A50 | Equations and inequalities involving linear operators, with vector unknowns |

65R20 | Numerical methods for integral equations |

65J15 | Numerical solutions to equations with nonlinear operators (do not use 65Hxx) |

47J25 | Iterative procedures involving nonlinear operators |

45B05 | Fredholm integral equations |

44A12 | Radon transform |

92C55 | Biomedical imaging and signal processing |

81U40 | Inverse scattering problems in quantum theory |

94A12 | Signal theory (characterization, reconstruction, filtering, etc.) |

35R30 | Inverse problems for PDEs |