Regularization methods for large-scale problems.

*(English)*Zbl 0805.65058This highly competent survey of the state of the art of computational methods for large-scale discretizations of linear inverse problems will be found to be very useful by engineers and scientists as well as mathematicians. It forms an excellent complement, stressing computational issues, to the more theoretical survey paper on regularization by H. W. Engl [Surv. Math. Ind. 3, No. 2, 71-143 (1993; Zbl 0776.65043)].

Regularization may be viewed as a philosophy rather than a technique and this survey contains much sage advice for the practitioner as well as an intuitive slant on the theory for the more mathematically inclined. The authors take the sensible approach of illustrating the main computational issues by concentrating on two types of regularization: Tikhonov regularization and iterative regularization. The methods and techniques are motivated by two problems in inverse helioseismology and computerized tomography.

Topics include: compact operators, the theory of regularization, discretization issues, standard forms for singular value decomposition, Tikhonov regularization, parameter choice strategies, iterative regularization methods, \(\nu\)-methods, conjugate gradient methods, and stopping criteria. The principal methods are illustrated numerically on several model problems.

Regularization may be viewed as a philosophy rather than a technique and this survey contains much sage advice for the practitioner as well as an intuitive slant on the theory for the more mathematically inclined. The authors take the sensible approach of illustrating the main computational issues by concentrating on two types of regularization: Tikhonov regularization and iterative regularization. The methods and techniques are motivated by two problems in inverse helioseismology and computerized tomography.

Topics include: compact operators, the theory of regularization, discretization issues, standard forms for singular value decomposition, Tikhonov regularization, parameter choice strategies, iterative regularization methods, \(\nu\)-methods, conjugate gradient methods, and stopping criteria. The principal methods are illustrated numerically on several model problems.

Reviewer: C.W.Groetsch (Cincinnati)

##### MSC:

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

65F20 | Numerical solutions to overdetermined systems, pseudoinverses |

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

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

65R20 | Numerical methods for integral equations |

65R30 | Numerical methods for ill-posed problems for integral equations |

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