The investigator and her colleague aim to develop theoretically a novel method, the DSM-dynamical system method, for solving a wide variety of linear and nonlinear ill-posed problems, to implement algorithms based on this method, and to demonstrate the advantages of this method in efficiency and accuracy. The DSM method is used as a general approach to the construction of regularizing algorithms for solving ill-posed problems, i.e. a stopping rule is developed: a rule for choosing that moment of time at which the value of the solution to the basic evolution equation stably approximates the solution to the original ill-posed equation in the case when the data are given with some error. Applications of different versions of the DSM are considered to classical ill-posed problems of computational mathematics, such as stable differentiation of noisy data, stable inversion of ill-conditioned matrices, and to nonlinear inverse problems arising in geophysics, quantum physics, medicine, remote sensing in technology, and other applied areas. The DSM with simultaneous updates of the inverse derivative operator without actual inverting of this operator is developed for solving nonlinear ill-posed problems. The DSM is used as a general method for constructing convergent iterative processes for solving ill-posed operator equations. Namely, convergent discretization schemes for solving the basic evolution equation of the DSM provide convergent iterative methods for solving the original equation. The DSM is developed for unbounded operators, which do not have continuous inverse operators, and also for a nonlinear operators whose derivative is a Fredholm operator with nontrivial null-space. The area of ill-posed (unstable) problems is extremely difficult, because solutions to ill-posed problems are very sensitive to small variation in input data. For that reason ill-posed problems cannot be solved by classical methods: the corresponding numerical procedures for them turn out to be divergent. However, ill-posed problems are frequently encountered in many branches of natural sciences and engineering: astrophysics, geophysics, spectroscopy, plasma diagnostics, computerized tomography, antenna design, optimal design of technical systems and engineering constructions, optimal planning, optimal control over various processes, and many other fields. Mathematical statements of these problems are given in the form of operator equations of the first kind, problems of functional minimization, problems of determining values of unbounded operators, variational inequalities, and so on. The project develops computational methods that provide more accurate solutions to a wide range of ill-posed problems. The investigator also uses the results in the graduate courses on ill-posed and inverse problems she teaches at Georgia State University. Finally, because ill-posed problems are of basic importance in applications, the results are of wide use in engineering and applied sciences.
