Regularization methods for large-scale problems. (English) Zbl 0805.65058

Hanke, Martin; Hansen, Per Christian

Surv. Math. Ind. 3, No. 4, 253-315 (1993).

This 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.

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

Keywords:

large-scale problems; ill-conditioned system; regularization; linear inverse problems; Tikhonov regularization; iterative regularization; helioseismology; computerized tomography; compact operators; singular value decomposition; parametric choice; conjugate gradient methods; stopping criteria