×

The theory of Tikhonov regularization for Fredholm equations of the first kind. (English) Zbl 0545.65034

This is an extremely well readable introduction to the theory of Tikhonov regularization for Fredholm integral equations of the first kind. In an abstract setting, these problems may be written as \(Kf=g\), where K is a given bounded linear operator from one Hilbert space, \(H_ 1\), to another, \(H_ 2\), and g is a given element of \(H_ 2\). \(Kf=g\) is said to be ill-posed if K has no bounded inverse on \(H_ 2\). The theory and the numerical solution of ill-posed problems is a topic that is of great importance to a wide field of applications in physics and engineering. The text contains chapters on linear operators, generalized inverses and ill-posed problems; general regularization methods for the computation of minimal norm least squares solutions, convergence proofs and convergence rates; Tikhonov regularization, saturation and converse results, the discrepancy principle and the use of differential operators in the regularization functionals; finite dimensional approximation of the minimal norm least squares solution, algorithms for the appropriate choice of regularization parameters, the methods of moment discretization and cross validation. By using singular systems for K the author manages to present the different regularization methods and its convergence proofs in a very elegant and unifying way. This small volume is best suited for lectures or seminars given to applied mathematicians or engineering students knowing some prerequisites of functional analysis.
Reviewer: J.T.Marti

MSC:

65J10 Numerical solutions to equations with linear operators
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
65R20 Numerical methods for integral equations
47A50 Equations and inequalities involving linear operators, with vector unknowns
45B05 Fredholm integral equations