Methods for Solving Incorrectly Posed Problems Some problems of mathematical physics and analysis can be formulated as the problem of solving the equation f € F (1) Au = f where A: DA C U + F is an operator with a non empty domain of definition D in a metric space U with range in a metric space F. The metrics A on U and F will be denoted by P and P  respectively. Relative u F to the twin spaces U and F J. Hadamard P 06] gave the following defini­ tion of correctness: the problem (1) is said to be well posed (correct properly posed) if the following conditions are satisfied: (1) The range of the value Q of the operator A coincides with A F ( sol vabi li ty condition); (2) The equality AU = AU for any u u € DA implies the I 2 l 2 equality u = u ( uniqueness condition); l 2 (3) The inverse operator A I is continuous on F ( stability condition). Any reasonable mathematical formulation of a physical problem requires that conditions (1) (3) be satisfied. That is why Hadamard postulated that any ill posed (improperly posed) problem that is to say one which does not satisfy conditions (1) (3) is non physical. Hadamard also gave the now classical example of an ill posed problem namely the Cauchy problem for the Laplace equation. 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 From Approximate 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.