Simultaneous Estimation of Noise Level and Solution Smoothness for Ill-Posed Problems

在逆问题中，任务是根据观察到的结果确定原因。这类问题的解决特点是对数据的敏感性。这类正则化方法通过解决接近原始问题的辅助问题来解决这一困难，在该问题中，“贴近度”由所谓的正则化参数控制。该参数的选择决定了所获得的重建的质量。噪声水平，即对数据不可避免地包含的测量噪声大小的估计，对其选择起着至关重要的作用。正则化理论的目标是发展根据噪声水平选择正则化参数的方法，使得在消除噪声的限制下，获得对应于无噪声数据的解。为了保证精确解的最佳逼近，需要额外的假设。一种可能性是源条件，在经典公式中，源条件要求无噪声解包含在前向算子的某个(最好是高)幂的范围内。如果不知道源条件中的噪声电平或指数，就不能保证恢复解的高质量。特别是，最佳正则化参数的选择取决于这两个值。由于理论上的考虑，假设这两个参数都是已知的，但在实践中往往无法获得。在这个项目中，将开发一种方法，该方法自动提取给定逆问题的源条件下的测量噪声和光滑性参数的估计，从而可以应用现有的正则化理论来构造最优重建方法。源条件的估计基于Kurdyka-Lojasiewicz不等式(Kli)，例如从凸分析或偏微分方程的渐近研究中已知。与反问题的联系在于，大多数正则化方法可以被表示为最小化问题，而KLI描述了泛函在其临界点附近的行为。这可以使用迭代方法来恢复，例如Landweber迭代，允许近似震源条件。噪声水平的估计也可以使用诸如Krylov子空间投影的迭代方法来实现。该技术基于迭代过程中对正向算子的连续应用，并利用其光滑性，可以构造不需要上述参数的先验知识的反问题的求解方法，从而自动地最优地适应给定的正向算子对和噪声数据。它的目的是将这些技术推广到Tikhonov泛函、Banach空间背景和非线性向前算子，并对各自的方法进行数值实现。

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.