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Methods for solving incorrectly posed problems. Transl. from the Russian by A. B. Aries, ed. by Z. Nashed. (English) Zbl 0549.65031

New York etc.: Springer-Verlag. XVIII, 257 p. DM 118 (1984).
Many physical problems are such that they can be formulated in terms of the solution for u of the operator equation \(Au=f\) where A is an operator and f is given. Roughly, the problem is well-posed if the solution exists uniquely and the inverse operator \(A^{-1}\) is continuous. Many physical problems - for example the Cauchy problem for the Laplace equation - do not fall into this category and are ill-posed. The author purports in this book to give the abstract Hilbert space theory necessary for the solution of ill-posed problems.
There are five chapters in the book. The first one is entitled ”The regularization method”. The problem as formulated concerns the necessity of the existence of a common solution of the two operator equations \(Au=f\), \(Lu=g\). The determination of the solution of the problem is associated with the problem of finding regularized solutions by minimizing the quantity \(\Phi_{\alpha}(u)=\| Au-f\|^ 2+\alpha\| Lu-g\|^ 2, \alpha\), the regularization parameter being positive. Estimates are given for the accuracy of approximation, and the stability of solutions with respect to f and g and there is a discussion of the choice of a suitable basis.
The second chapter deals with the criteria for selection of regularization parameters. Three separate criteria are developed and their association with variational principles indicated. The third chapter considers regular methods for ill-posed problems. A treatment is given of the convergence and the accuracy of regular methods and this is illustrated by formulae for the accuracy of representation of observed data by the corresponding Fourier series. The connection with the method of least squares for nonlinear operators is also discussed.
The fourth chapter is entitled ”The problem of computation and the theory of splines”. This commences with a discussion of the theoretical problems associated with smoothing including a definition of optimality. These ideas are applied to the problem of differentiation of, and approximation to, experimental data, and through the use of piecewise cubic functions, the method of splines is arrived at.
The fifth chapter is concerned with algorithms for choosing the regularization parameter in special cases. For this purpose, the author introduces the concept of a pseudoinverse operator, associated with a pseudosolution (or least squares solution). The eigenvalues and eigenelements are discussed and consequent optimal approximations arrived at. Methods of evaluating the parameter are indicated and the chapter closes with a discussion of the adequacy of mathematical models. The book closes with a list of 112 references, nearly all of which are from the Soviet Union, including 31 by the author himself.
The book has been reproduced from typescript, and the few errors noted are not such as to cause inconvenience, and the translation on the whole reads well. The book concerns an interesting and important subject about which comparatively little has been written. It does nevertheless suffer from one very grave defect. The treatment is entirely involved with operators and abstract spaces. Nowhere is there an indication of how the abstract analysis formulated could be applied to even the simplest physical ill-posed problem, and this will prevent the fuller utilization of the methods developed by the author.
Reviewer: Ll.G.Chambers

MSC:

65J10 Numerical solutions to equations with linear operators
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
47A50 Equations and inequalities involving linear operators, with vector unknowns
65J15 Numerical solutions to equations with nonlinear operators
47A10 Spectrum, resolvent
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
41A15 Spline approximation