Methods for Solving Incorrectly Posed Problems

1. The Regularization Method. - Section 1. The Basic Problem for Linear Operators. - Section 2. The Approximation of the Solution of the Basic Problem. - Section 3. The Euler Variation Inequality. Estimation of Accuracy. - Section 4. Stability of Regularized Solutions. - Section 5. Approximation of the Admissible Set. Choice of the Basis. - 2. Criteria for Selection of Regularization Parameter. - Section 6. Some Properties of Regularized Solutions. - Section 7. Methods for Choosing the Parameter: Case of Exact Information. - Section 8. The Residual Method and the Method of Quasi-solutions: Case of Exact Information. - Section 9. Properties of the Auxiliary Functions. - Section 10. Criteria for the Choice of a Parameter: Case of Inexact Data. - 3. Regular Methods for Solving Linear and Nonlinear Ill-Posed Problems. - Section 11. Regularity of Approximation Methods. - Section 12. The Theory of Accuracy of Regular Methods. - Section 13. The Computation of the Estimation Function. - Section 14. Examples of Regular Methods. - Section 15. The Principle of Residual Optimality for Approximate Solutions of Equations with Nonlinear Operators. - Section 16. The Regularization Method for Nonlinear Equations. - 4. The Problem of Computation and the General Theory of Splines. - Section 17. The Problem of Computation and the Parameter Identification Problem. - Section 18. Properties of Smoothing Families of Operators. - Section 19. The Optimality of Smoothing Algorithms. - Section 20. The Differentiation Problem and Algorithms of Approximation of the Experimental Data. - Section 21. The Theory of Splines and the Problem of Stable Computation of Values of an Unbounded Operator. - Section 22. Approximate Solution of Operator Equations Using Splines. - Section 23. Recovering the Solution of the Basic Problem FromApproximate Values of the Functiona1s. - 5. Regular Methods for Special Cases of the Basic Problem. Algorithms for Choosing the Regularization Parameter. - Section 24. Pseudosolutions. - Section 25. Optimal Regularization. - Section 26. Numerical Algorithms for Regularization Parameters. - Section 27. Heuristic Methods for Choosing a Parameter. - Section 28. The Investigation of Adequacy of Mathematical Models. Language: English