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

Smirnova 0207050和她的同事致力于从理论上发展一种新的方法-DSM-动力系统方法，用于解决各种线性和非线性不适定问题，实现基于该方法的算法，并展示该方法在效率和精度方面的优势。用DSM方法作为构造求解不适定问题的正则化算法的一般方法，即发展了一个停止规则：在给定的数据有一定误差的情况下，选择基本发展方程的解的值稳定地逼近原始不适定方程的解的时刻的规则。不同版本的DSM被考虑应用于计算数学的经典不适定问题，如含噪数据的稳定微分、病态矩阵的稳定求逆，以及在地球物理、量子物理、医学、遥感技术等应用领域中出现的非线性逆问题。针对非线性不适定问题，提出了一种同时更新逆导数算子而不求逆的DSM方法。DSM被用作构造解不适定算子方程的收敛迭代过程的一般方法。也就是说，求解DSM基本发展方程的收敛离散格式为求解原始方程提供了收敛的迭代方法。DSM是针对不具有连续逆算子的无界算子，以及其导数为具有非平凡零空间的Fredholm型算子的非线性算子而发展的。不适定(不稳定)问题的领域是非常困难的，因为不适定问题的解决方案对输入数据的微小变化非常敏感。因此，不适定问题不能用经典方法来解决：相应的数值方法是发散的。然而，在自然科学和工程的许多分支中经常遇到不适定问题：天体物理、地球物理、光谱学、等离子体诊断、计算机断层扫描、天线设计、技术系统和工程结构的优化设计、最优规划、各种过程的最优控制等许多其他领域。这些问题的数学表述以第一类算子方程、泛函极小化问题、无界算子的取值问题、变分不等式等形式给出。该项目开发了计算方法，为广泛的不适定问题提供更准确的解决方案。这位研究人员还使用了她在佐治亚州立大学教授的关于不适定问题和逆问题的研究生课程的结果。最后，由于不适定问题在应用中具有基本的重要性，这些结果在工程和应用科学中具有广泛的用途。

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.