Volume 17, Issue 1

This minisymposium was held at the International Conference – Inverse Problems: Modeling and Simulation, May 26–30, 2008, in ldeniz, Fethiye, Turkey.

We give a survey of results concerning the convergence and convergence rates and oracle inequalities for the quasi-optimality parameter choice for the most common regularization methods.

This paper addresses the problem of computing the minimizers for Tikhonov functionals associated with inverse problems with sparsity constraints in general Banach spaces. We present, based on splitting the Tikhonov functional into a smooth and a non-smooth part, a general iterative procedure for the Banach-space setting. In case of sparsity constraints, this algorithm yields a successive application of thresholding-like functions which generalizes the well-known iterative soft-thresholding procedure. The convergence properties of the proposed method are studied. Depending on the smoothness and convexity of the underlying spaces, convergence of asymptotic rate is obtained with the help of Bregman and Bregman–Taylor distance estimates. In particular, strong convergence can be achieved for a large class of linear inverse problems with sparsity constraints in Banach space.

In this paper we deal with convergence rates for a Tikhonov regularization approach for linear and nonlinear ill-posed problems in Banach spaces. Here, we deal with so-called distance functions which quantify the violation of a given reference source condition. With the aid of these functions we present error bounds and convergence rates for regularized solutions of linear and nonlinear problems when the reference source condition is violated. Introducing this topic for linear problems we extend the theory also to nonlinear problems. Finally an a-posteriori choice of the regularization parameter is suggested yielding the optimal convergence rate.

We present numerical results concerning the identification of the Lamé constants λ and μ in linear elasticity from measured deformation of a test body. For the numerical solution of the identification problem deterministic Gauss – Newton and multi-parameter regularization approaches are compared with evolutionary algorithms. Furthermore we deal with practical questions such as the experimental design for obtaining sufficient data for recovering the unknown parameters.

In piezoelectric applications, especially when the devices are used as actuators, the piezoelectric materials are driven under large signals which cause a nonlinear behavior. One way to model the nonlinearities is by functional dependencies of the material parameters on the electric field strength or the mechanical strain, respectively. The focus lies in the inverse problem, namely the identification of the parameter curves by appropriate measurements of charge signals over time. The problem is assumed to be ill-posed, since in general measured data are contaminated with noise. The solution process requires regularizing methods where modified Landweber iterations are in the focus. Implementations of modified Landweber iterations, namely the steepest descent and minimal error method can be shown to perform much faster than classical Landweber iterations due to the flexible handling of the relaxation parameter. In our application, parameter curve identification in nonlinear piezoelectricity, the sought-for quantities require to be discretized. Therefore, an iterative multilevel algorithm as proposed by Scherzer [Numer.Math.: 579–600, 1998] is investigated where the iterations begin with coarse discretizations of the parameter curves profitting from the inherent regularization property of coarse discretization. At an advanced state of the iterations the algorithm switches according to an inner discrepancy principle to finer levels of discretization. By this, a sufficient smooth resolution of the sought-for quantities can be achieved. Convergence results and the regularizing property of such an iterative multilevel algorithm are proven. Numerical identification results are presented at the end of this article.

This paper addresses the regularization of linear inverse problems with so-called sparsity constraints in terms of ℓ P -penalty terms. Error estimates and convergence rates are derived. The application of the generalized gradient projection method and the semismooth Newton method to the according minimization problem is shown. Numerical experiments show the efficiency of the semismooth Newton method.

In this paper we deal with linear inverse problems and convergence rates for Tikhonov regularization. We consider regularization in a scale of Banach spaces, namely the scale of Besov spaces. We show that regularization in Banach scales differs from regularization in Hilbert scales in the sense that it is possible that stronger source conditions may lead to weaker convergence rates and vice versa. Moreover, we present optimal source conditions for regularization in Besov scales.

There exists a vast literature on convergence rates results for Tikhonov regularized minimizers. The first convergence rates results for non-linear problems have been developed by Engl, Kunisch and Neubauer in 1989 [Inverse Problems 5: 523–540, 1989]. While these results apply for operator equations formulated in Hilbert spaces, the results of Burger and Osher from 2004 [Inverse Problems 20: 1411–1421, 2004], more generally, apply to operators formulated in Banach spaces. Recently, Resmerita and Scherzer [Inverse Problems 22: 801–814, 2006] presented a modification of the convergence rates result of Burger and Osher which turns out a complete generalization of the result of Engl et. al. In all these papers relatively strong regularity assumptions are made. However, it has been observed numerically, that violations of the smoothness assumptions on the operator do not necessarily affect the convergence rate negatively. We have taken this observation and weakened the smoothness assumptions on the operator and have proved a novel convergence rate result published in [Hofmann, Kaltenbacher, Pöschl, Scherzer, Inverse Problems 23: 987–1010, 2007]. The most significant difference of this result to the previous ones is that the source condition is formulated as a variational inequality and not as an equation as before, which is necessary due to the lack of smoothness assumptions on F .

This paper surveys the concept of conditional stability for linear ill-posed problems with error estimates based on the modulus of continuity. Combining the ideas of [Hofmann, Mathé, Schieck, J. Inv. Ill-Posed Problems 16: 569–587, 2008] and [Kabanikhin, Schieck, J. Inv. Ill-Posed Problems 16: 267–282, 2008] fundamental properties are formulated and convergence rates results for regularized solutions concerning linear regularization schemes are carried out.

The Landweber method is a well-known iterative procedure to compute regularized solutions of linear operator equations in Hilbert spaces. Unfortunately it is also known to be very slow. Likewise its generalization to Banach spaces has good regularizing properties but slow convergence. This article intends to give a short survey about the use of sequential subspace optimization to accelerate this method while preserving its regularizing properties.