Iterative methods for ill-posed problems : an introduction / Anatoly B. Bakushinsky, Mikhail Yu. Kokurin, Alexandra Smirnova.

Includes bibliographical references and index.

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Machine generated contents note: 1. The regularity condition. Newton's method -- 1.1. Preliminary results -- 1.2. Linearization procedure -- 1.3. Error analysis -- Problems -- 2. The Gauss -- Newton method -- 2.1. Motivation -- 2.2. Convergence rates -- Problems -- 3. The gradient method -- 3.1. The gradient method for regular problems -- 3.2. Ill-posed case -- Problems -- 4. Tikhonov's scheme -- 4.1. The Tikhonov functional -- 4.2. Properties of a minimizing sequence -- 4.3. Other types of convergence -- 4.4. Equations with noisy data -- Problems -- 5. Tikhonov's scheme for linear equations -- 5.1. The main convergence result -- 5.2. Elements of spectral theory -- 5.3. Minimizing sequences for linear equations.

5.4. A priori agreement between the regularization parameter and the error for equations with perturbed right-hand sides -- 5.5. The discrepancy principle -- 5.6. Approximation of a quasi-solution -- Problems -- 6. The gradient scheme for linear equations -- 6.1. The technique of spectral analysis -- 6.2. A priori stopping rule -- 6.3. A posteriori stopping rule -- Problems -- 7. Convergence rates for the approximation methods in the case of linear irregular equations -- 7.1. The source-type condition (STC) -- 7.2. STC for the gradient method -- 7.3. The saturation phenomena -- 7.4. Approximations in case of a perturbed STC -- 7.5. Accuracy of the estimates -- Problems -- 8. Equations with a convex discrepancy functional by Tikhonov's method -- 8.1. Some difficulties associated with Tikhonov's method in case of a convex discrepancy functional.

8.2. An illustrative example -- Problems -- 9. Iterative regularization principle -- 9.1. The idea of iterative regularization -- 9.2. The iteratively regularized gradient method -- Problems -- 10. The iteratively regularized Gauss -- Newton method -- 10.1. Convergence analysis -- 10.2. Further properties of IRGN iterations -- 10.3. A unified approach to the construction of iterative methods for irregular equations -- 10.4. The reverse connection control -- Problems -- 11. The stable gradient method for irregular nonlinear equations -- 11.1. Solving an auxiliary finite dimensional problem by the gradient descent method -- 11.2. Investigation of a difference inequality -- 11.3. The case of noisy data -- Problems -- 12. Relative computational efficiency of iteratively regularized methods -- 12.1. Generalized Gauss -- Newton methods -- 12.2. A more restrictive source condition.

12.3. Comparison to iteratively regularized gradient scheme -- Problems -- 13. Numerical investigation of two-dimensional inverse gravimetry problem -- 13.1. Problem formulation -- 13.2. The algorithm -- 13.3. Simulations -- Problems -- 14. Iteratively regularized methods for inverse problem in optical tomography -- 14.1. Statement of the problem -- 14.2. Simple example -- 14.3. Forward simulation -- 14.4. The inverse problem -- 14.5. Numerical results -- Problems -- 15. Feigenbaum's universality equation -- 15.1. The universal constants -- 15.2. Ill-posedness -- 15.3. Numerical algorithm for 2 & le; z & le; 12 -- 15.4. Regularized method for z & ge; 13 -- Problems -- 16. Conclusion.

Ill-posed problems are encountered in countless areas of real world science and technology. A variety of processes in science and engineering is commonly modeled by algebraic, differential, integral and other equations. In a more difficult case, it can be systems of equations combined with the associated initial and boundary conditions. Frequently, the study of applied optimization problems is also reduced to solving the corresponding equations. These equations, encountered both in theoretical and applied areas, may naturally be classified as operator equations. The current textbook will focus on iterative methods for operator equations in Hilbert spaces.

English.

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