Continuous Regularization for Nonlinear Ill-Posed Problems

非线性不适定问题的连续正则化

• 批准号: 1112897
• 负责人: Alexandra Smirnova
• 金额: $ 25万
• 依托单位: Georgia State University Research Foundation, Inc.
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-10-01 至 2015-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1112897&HistoricalAwards=false
• 关键词: Continuous; Regularization; Nonlinear; Ill; Posed

In the modern theory of irregular (ill-posed, unstable) problems, numerous regularized computational methods are known. These methods are being constantly improved and supplemented with new algorithms. Applied inverse problems are the main sources of this development. One of the primary approaches to the construction and investigation of stable methods for solving ill-posed operator equations is iterative regularization. Numerous convergence theorems describe the efficiency of iteratively regularized algorithms for different classes of unstable problems, and give existence results. However it is quite hard to navigate among discrete schemes and the corresponding convergence theorems. Proofs of these theorems are usually based on the contraction mapping principle and are sometimes rather complicated. In this project, PI conducts research on continuous regularization, which is based on the analysis of asymptotic behavior of nonlinear dynamical systems in Banach and Hilbert spaces. When a convergence theorem is proven for a continuous method, one can investigate various discrete schemes generated by this continuous process. Thus, construction of a discrete numerical scheme is split into two parts: development of a continuous process and numerical integration of the corresponding nonlinear operator-differential equation. Consequently, when it comes to a convergence theorem for a discrete scheme, one can differentiate between the sufficient conditions for the convergence of a continuous process, which stem from the nature of the ill-posed problem, and the conditions that originate from a specific method of numerical integration.The research will have a broad impact on a large number of scientific disciplines (biomedical imaging, gravitational sounding, chaos theory, spectroscopy, computerized tomography, and other areas of science and engineering) since the corresponding applied inverse problems can be investigated in the framework of this proposal both, theoretically and numerically. These problems are "ill-posed" in the sense that their solutions are unstable with respect to noise in the observed image data. For this reason, classical methods of computational mathematics cannot be applied. To overcome this instability and to simultaneously incorporate a priori information, one uses special techniques known as regularization methods. PI's research interests lie in the development and analysis of these regularization methods.

在现代理论的不规则（不适定，不稳定）的问题，许多正规化的计算方法是已知的。这些方法正在不断改进，并补充了新的算法。应用反问题是这一发展的主要来源。构造和研究求解不适定算子方程的稳定方法的主要途径之一是迭代正则化。许多收敛定理描述了迭代正则化算法对不同类型的不稳定问题的效率，并给出存在性结果。然而，它是相当困难的离散格式和相应的收敛性定理之间的导航。这些定理的证明通常基于压缩映射原理，有时相当复杂。在这个项目中，PI进行了连续正则化的研究，这是基于Banach和Hilbert空间中的非线性动力系统的渐近行为的分析。当连续方法的收敛定理被证明时，人们可以研究由这个连续过程产生的各种离散格式。因此，一个离散的数值方案的建设分为两个部分：一个连续的过程和相应的非线性算子微分方程的数值积分的发展。因此，当涉及到离散格式的收敛定理时，可以区分源于不适定问题的性质的连续过程收敛的充分条件和源于特定数值积分方法的条件，该研究将对许多科学学科产生广泛的影响（生物医学成像、重力探测、混沌理论、光谱学、计算机断层扫描以及其他科学和工程领域），因为相应的应用逆问题可以在该提议的框架中在理论上和数值上进行研究。这些问题是“不适定”的意义上说，他们的解决方案是不稳定的，在所观察到的图像数据中的噪声。由于这个原因，计算数学的经典方法不能应用。为了克服这种不稳定性，并同时纳入先验信息，人们使用特殊的技术称为正则化方法。PI的研究兴趣在于这些正则化方法的开发和分析。