Indirect regularization in non-Hilbert spaces

In the present project we are going to deal with inverse problems where the features of the interest should be inferred from derived observable quantities. It is customary to model the features of the interest by elements of some Hilbert space, and the derived quantities by elements of another Hilbert space. Such inverse problems often arise in scientific context, raging from stereological microscopy to satellite geodesy. Usually they are ill-posed in the sense that the element of the interest depends in a discontinuous way on the element representing observations. Therefore, a numerical treatment of inverse problems requires the application of special regularization methods. Classical regularization theory proposes the methods which were originally designed for performing in Hilbert spaces. At the same time, in several practically important cases one is interested in recovering an element of interest in some other space which is not equipped with the Hilbert space structure. For example, the use of non- Hilbert spaces is often advocated when one expects the element of interest to have a sparse expansion with respect to a preassigned basis. In such cases general theory does not provide us with recipes for the direct use of standard regularization methods. On the other hand, these methods are attractive, because they allow a rather simple numerical realization, while methods recently developed for a regularization in non-Hilbert spaces are nonlinear even for linear problems and demand sophisticated numerical procedures. Moreover, our preliminary numerical study shows that in several cases standard regularization methods are competitive with methods developed recently. But until now the methods of classical regularization theory have never been systematically used for regularization in non-Hilbert spaces. The aim of the project is to lower the gap between the well understood regularization in Hilbert spaces and non- Hilbert space regularization, where many questions are still open. Toward this end we are going to develop a methodology for estimating the stability of regularization methods in non-Hilbert spaces that is a key to their successful application.

The project was aimed at the study of inverse problems where the features of the interest should be inferred from derived observable quantities. It is customary to model the features of the interest by elements of some Hilbert space, and the derived quantities by elements of another Hilbert space. Such inverse problems often arise in scientific context, ranging from stereological microscopy to satellite geodesy. Usually they are ill-posed in the sense that the element of the interest depends in a discontinuous way on the element representing observations. Therefore, a numerical treatment of inverse problems requires the application of special regularization methods. Classical regularization theory proposes the methods which were originally designed for performing in Hilbert spaces. At the same time, in several practically important cases one is interested in recovering an element of interest in some other space which is not equipped with the Hilbert space structure. For example, the use of non- Hilbert spaces is often advocated when one expects the element of interest to have a sparse expansion with respect to a preassigned basis. In such cases general theory does not provide us with recipes for the direct use of standard regularization methods. On the other hand, these methods are attractive, because they allow a rather simple numerical realization, while methods recently developed for regularization in non-Hilbert spaces are nonlinear even for linear problems and demand sophisticated numerical procedures. Note that the methods of classical regularization theory were never used systematically for regularization in non-Hilbert spaces before the project began. The project results in lowering the gap between the well understood regularization in Hilbert spaces and non- Hilbert space regularization. For example, it has been shown that a recently developed parameter choice rule called the balancing principle allows the effective use of the standard Tikhonov regularization in the reconstruction of a sparse structure. Moreover, it has been found out that multi-penalty version of the classical Tikhonov regularization scheme allows the effective control of the regularization performance in several spaces simultaneously.