Project Details
Projekt
Simultaneous Estimation of Noise Level and Solution Smoothness for Ill-Posed Problems
Applicant
Dr. Daniel Gerth
Subject Area
Mathematics
Term
from 2019 to 2023
Project identifier
Deutsche Forschungsgemeinschaft (DFG) - Project number 416552794
In inverse problems, the task is to determine a cause from observed effects. The solution of such problems is characterized by sensitivity with respect to the data. The class of regularization methods addresses this difficulty by solving an auxilliary problem close to the original one, where "closeness" is controlled by a so-called regularization parameter. The choice of this parameter determines the quality of the obtained reconstructions. The noise level, i.e. an estimate of the size of the measurement noise the data inevitably contains, plays a crucial role for its choice. The goal of regularization theory is to develop methods that choose the regularization parameter in dependance of the noise level in such a way that, in the limit of vanishing noise, the solution corresponding to noise-free data is obtained. In order to guarantee such an optimal approximation of the exact solution, additional assumptions are necessary. One possibility for this are source conditions which, in the classical formulation, require the noise-free solution to be contained in the range of a certain (preferably high) power of the forward operator. Without knowledge of either the noise level or the exponent in the source condition, a high quality of the recovered solution can not be guaranteed. In particular, the choice of the optimal regularization parameter depends on both values. For theoretical considerations, knowledge of both parameters is assumed, while in practice these are often unavailable.In this project, a method is to be developed which automatically extracts an estimate for the measurement noise and the smoothness parameter in the source condition for a given inverse problem, such that existing regularization theory can be applied to construct optimal reconstruction methods. The estimation of the source condition is based on the Kurdyka-Lojasiewicz inequality (KLI), known for example from convex analysis or the asymptotic study of partial differential equations. The link to inverse problems lies in the fact that most regularization methods can be formulated as a minimization problem and the KLI describes the behaviour of functionals around their critical points. This can be recovered using iterative methods, e.g. Landweber iteration, allowing approximation of the source condition. Estimation of the noise level can also be achieved using iterative methods such as Krylov subspace projection. The technique is based on a successive application of the forward operator during the iteration and exploiting its smoothing property.With these approaches, one can construct methods for the solution of inverse problems that do not require a priori knowledge of the parameters mentioned above to automatically and optimally adapt to a given pair of forward operator and noisy data. It is intended to generalize the techniques to Tikhonov functionals, to a Banach space setting, and to nonlinear forward operators and to implement the respective methods numerically.
DFG Programme
Research Grants
International Connection
Austria, Czech Republic
