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

Smirnova0207050 研究者和她的同事旨在从理论上开发一种新方法，即 DSM 动力系统方法，用于解决各种线性和非线性病态问题，实现基于该方法的算法，并展示该方法在效率和准确性方面的优势。 DSM方法被用作构建解决病态问题的正则化算法的通用方法，即制定停止规则：在给定数据存在一定误差的情况下，选择基本演化方程的解的值稳定地逼近原始病态方程的解的时刻的规则。 不同版本的 DSM 的应用考虑了计算数学的经典病态问题，例如噪声数据的稳定微分、病态矩阵的稳定求逆，以及地球物理学、量子物理、医学、技术遥感和其他应用领域中出现的非线性逆问题。 同时更新逆导数算子而不实际求逆该算子的 DSM 是为了解决非线性不适定问题而开发的。 DSM 被用作构造收敛迭代过程以求解不适定算子方程的通用方法。 即求解DSM基本演化方程的收敛离散化方案为求解原方程提供了收敛迭代方法。 DSM 是为无界算子（没有连续逆算子）以及导数为具有非平凡零空间的 Fredholm 算子的非线性算子开发的。 不适定（不稳定）问题的领域极其困难，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，不适定问题无法通过经典方法解决：它们相应的数值程序结果是不同的。 然而，自然科学和工程学的许多分支中经常遇到不适定问题：天体物理学、地球物理学、光谱学、等离子体诊断、计算机断层扫描、天线设计、技术系统和工程建设的优化设计、优化规划、各种过程的优化控制等许多领域。 这些问题的数学表述以第一类算子方程、函数最小化问题、无界算子值确定问题、变分不等式等形式给出。 该项目开发的计算方法可为各种不适定问题提供更准确的解决方案。 研究人员还在佐治亚州立大学教授的不适定问题和逆问题研究生课程中使用了这些结果。最后，由于不适定问题在应用中具有基本重要性，因此结果在工程和应用科学中得到广泛使用。

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.