Continuous Regularization for Nonlinear Ill-Posed Problems

在不规则（病态、不稳定）问题的现代理论中，已知许多正则化计算方法。这些方法正在不断改进，并补充新的算法。应用逆问题是这一发展的主要来源。构造和研究求解病态算子方程的稳定方法的主要方法之一是迭代正则化。许多收敛定理描述了迭代正则化算法对不同类别不稳定问题的有效性，并给出了存在性结果。然而，在离散格式和相应的收敛定理之间导航是相当困难的。这些定理的证明通常基于收缩映射原理，有时相当复杂。在这个项目中，PI基于Banach和Hilbert空间中非线性动力系统的渐近行为分析，进行了连续正则化的研究。当一个连续方法的收敛定理被证明后，人们可以研究由这个连续过程产生的各种离散格式。因此，离散数值格式的构建分为两个部分：连续过程的开发和相应的非线性算子微分方程的数值积分。因此，当涉及到离散格式的收敛定理时，人们可以区分连续过程收敛的充分条件，这些条件源于不适定问题的性质，以及源于特定数值积分方法的条件。这项研究将对许多科学学科（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）产生广泛的影响，因为相应的应用逆问题可以在本提案的框架内进行理论和数值研究。这些问题是“病态的”，因为它们的解相对于观测到的图像数据中的噪声是不稳定的。因此，经典的计算数学方法不能应用。为了克服这种不稳定性并同时结合先验信息，人们使用了称为正则化方法的特殊技术。PI的研究兴趣在于这些正则化方法的发展和分析。

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.