Continuous Regularization for Nonlinear Ill-Posed Problems

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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.