Theoretical and Numerical Investigation of Dynamical Systems Method for Solving Linear and Nonlinear Ill-Posed Problems

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Smirnova0207050 The investigator and her colleague aim to developtheoretically a novel method, the DSM-dynamical system method,for solving a wide variety of linear and nonlinear ill-posedproblems, to implement algorithms based on this method, and todemonstrate the advantages of this method in efficiency andaccuracy. The DSM method is used as a general approach to theconstruction of regularizing algorithms for solving ill-posedproblems, i.e. a stopping rule is developed: a rule for choosingthat moment of time at which the value of the solution to thebasic evolution equation stably approximates the solution to theoriginal ill-posed equation in the case when the data are givenwith some error. Applications of different versions of the DSMare considered to classical ill-posed problems of computationalmathematics, such as stable differentiation of noisy data, stableinversion of ill-conditioned matrices, and to nonlinear inverseproblems arising in geophysics, quantum physics, medicine, remotesensing in technology, and other applied areas. The DSM withsimultaneous updates of the inverse derivative operator withoutactual inverting of this operator is developed for solvingnonlinear ill-posed problems. The DSM is used as a general methodfor constructing convergent iterative processes for solvingill-posed operator equations. Namely, convergent discretizationschemes for solving the basic evolution equation of the DSMprovide convergent iterative methods for solving the originalequation. The DSM is developed for unbounded operators, which donot have continuous inverse operators, and also for a nonlinearoperators whose derivative is a Fredholm operator with nontrivialnull-space. The area of ill-posed (unstable) problems is extremelydifficult, because solutions to ill-posed problems are verysensitive to small variation in input data. For that reasonill-posed problems cannot be solved by classical methods: thecorresponding numerical procedures for them turn out to bedivergent. However, ill-posed problems are frequently encounteredin many branches of natural sciences and engineering:astrophysics, geophysics, spectroscopy, plasma diagnostics,computerized tomography, antenna design, optimal design oftechnical systems and engineering constructions, optimalplanning, optimal control over various processes, and many otherfields. Mathematical statements of these problems are given inthe form of operator equations of the first kind, problems offunctional minimization, problems of determining values ofunbounded operators, variational inequalities, and so on. Theproject develops computational methods that provide more accuratesolutions to a wide range of ill-posed problems. The investigatoralso uses the results in the graduate courses on ill-posed andinverse problems she teaches at Georgia State University.Finally, because ill-posed problems are of basic importance inapplications, the results are of wide use in engineering andapplied sciences.