Thomas Schuster

Regularization methods in Banach spaces

Regularization methods aimed at finding stable approximate solutions are a necessary tool to tackle inverse and ill-posed problems. Usually the mathematical model of an inverse problem consists of an operator equation of the first kind and often the associated forward operator acts between Hilbert spaces. However, for numerous problems the reasons for using a Hilbert space setting seem to be based rather on conventions than on an approprimate and realistic model choice, so often a Banach space setting would be closer to reality. Furthermore, sparsity constraints using general Lp-norms or the BV-norm have recently become very popular. Meanwhile the most well-known methods have been investigated for linear and nonlinear operator equations in Banach spaces. Motivated by these facts the authors aim at collecting and publishing these results in a monograph.

Nonlinear iterative methods for linear ill-posed problems in Banach spaces

We introduce and discuss nonlinear iterative methods to recover the minimum-norm solution of the operator equation Ax = y in Banach spaces X, Y, where A is a continuous linear operator from X to Y. The methods are nonlinear due to the use of duality mappings which reflect the geometrical aspects of the underlying spaces. The space X is required to be smooth and uniformly convex, whereas Y can be an arbitrary Banach space. The case of exact as well as approximate and disturbed data and operator are taken into consideration and we prove the strong convergence of the sequence of the iterates.

Minimization of Tikhonov Functionals in Banach Spaces

Tikhonov functionals are known to be well suited for obtaining regularized solutions of linear operator equations. We analyze two iterative methods for finding the minimizer of norm-based Tikhonov functionals in Banach spaces. One is the steepest descent method, whereby the iterations are directly carried out in the underlying space, and the other one performs iterations in the dual space. We prove strong convergence of both methods.

The 3D Doppler transform: elementary properties and computation of reconstruction kernels

The 3D Doppler transform maps a vector field to its line integrals over that component of the field which is parallel to the line. In this paper we consider only lines aligned to the coordinate axes. Since the Doppler transform describes the mathematical model for the vector tomography, efficient inversion formulae are necessary in order to solve the reconstruction problem. The approximate inverse represents a numerical inversion scheme based on scalar products of the data with so-called reconstruction kernels. We characterize these reconstruction kernels as solutions of a normal equation connected with the Doppler transform and a mollifier. To solve this equation elementary properties of the underlying operator are investigated and a smoothing property is proved. We succeed in computing a reconstruction kernel for one special mollifier and give a representation with the help of the singular value decomposition of the 2D Radon transform.

An efficient mollifier method for three-dimensional vector tomography: convergence analysis and implementation

We consider the problem of three-dimensional vector tomography, that means the reconstruction of vector fields and their curl from line integrals over certain components of the field. It is well known that only the solenoidal part of the field can be recovered from these data. In this paper the method of approximate inverse is modified for vector fields and applied to this problem, leading to an efficient solver of filtered backprojection type. We prove convergence of the reconstructed solution, if the number of data tends to infinity, which means the method is exact. Finally, numerical results are presented for a straight flow through a cylinder.

The Approximate Inverse in Action with an Application to Computerized Tomography

The approximate inverse is a scheme to obtain stable numerical inversion formulae for linear operator equations of the first kind. Yet, in some applications the computation of a crucial ingredient, the reconstruction kernel, is time-consuming and instable. It may even happen that the kernel does not exist for a particular semidiscrete system. To cure this dilemma we propose and analyze a technique that is based on a singular value decomposition of the underlying operator. The results are applied to the reconstruction problem in 2D-computerized tomography where they enable the design of reconstruction filters and lead to a novel error analysis of the filtered backprojection algorithm.

Local Inversion Of The Sonar Transform Regularized By The Approximate Inverse

A new reconstruction method is given for the spherical mean transform with centers on a plane in R 3 which is also called the sonar transform. Standard inversion formulas require data over all spheres, but typically, the data are limited in the sense that the centers and radii are in a compact set. Our reconstruction operator is local because, to reconstruct at x, one needs only spheres that pass near x, and the operator reconstructs singularities, such as object boundaries. The microlocal properties of the reconstruction operator, including its symbol as a pseudodifferential operator, are given. The method is implemented using the approximate inverse, and reconstructions are given. They are evaluated in light of the microlocal properties of the reconstruction operator. (Some figures in this article are in colour only in the electronic version)

Metric and Bregman Projections onto Affine Subspaces and their Computation via Sequential Subspace Optimization Methods

Abstract In this article we investigate and prove relationships between metric and Bregman projections induced by powers of the norm of a Banach space. We consider Bregman projections onto affine subspaces of Banach spaces and deduce some interesting analogies to results which are well known for Hilbert spaces. Using these concepts as well as ideas from sequential subspace optimization techniques we construct efficient iterative methods to compute Bregman projections onto affine subspaces that are connected to linear, bounded operators between Banach spaces. Especially these methods can be used to compute minimum-norm solutions of linear operator equations or best approximations in the range of a linear operator. Numerical experiments illuminate the performance of our iterative algorithms and demonstrate a significant acceleration compared to the Landweber method.

The approximate inverse in action II: convergence and stability

The approximate inverse is a scheme for constructing stable inversion formulas for operator equations. Originally, it is defined on L2-spaces. In the present article we extend the concept of approximate inverse to more general settings which allow us to investigate the discrete version of the approximate inverse which actually underlies numerical computations. Indeed, we show convergence if the discretization parameter tends to zero. Further, we prove stability, that is, we show the regularization property. Finally we apply the results to the filtered backprojection algorithm in 2D-tomography to obtain convergence rates.

Solving linear operator equations in Banach spaces non-iteratively by the method of approximate inverse

The method of approximate inverse is a mollification method for stably solving inverse problems. In its original form it has been developed to solve operator equations in L 2 -spaces and general Hilbert spaces. We show that the method of approximate inverse can be extended to solve linear, ill-posed problems in Banach spaces. This paper is restricted to function spaces. The method itself consists of evaluations of dual pairings of the given data with reconstruction kernels that are associated with mollifiers and the dual of the operator. We first define what we mean by a mollifier in general Banach spaces and then investigate two settings more exactly: the case of L p -spaces and the case of the Banach space of continuous functions on a compact set. For both settings we present the criteria turning the method of approximate inverse into a regularization method and prove convergence with rates. As an application we refer to x-ray diffractometry which is a technique of non-destructive testing that is concerned with computing the stress tensor of a specimen. Since one knows that the stress tensor is smooth, x-ray diffractometry can appropriately be modelled by a Banach space setting using continuous functions.