Barbara Kaltenbacher

Iterative regularization methods for nonlinear ill-posed problems

Nonlinear inverse problems appear in many applications, and typically they lead to mathematical models that are ill-posed, i.e., they are unstable under data perturbations. Those problems require a regularization, i.e., a special numerical treatment. This book presents regularization schemes which are based on iteration methods, e.g., nonlinear Landweber iteration, level set methods, multilevel methods and Newton type methods.

A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators

There exists a vast literature on convergence rates results for Tikhonov regularized minimizers. We are concerned with the solution of nonlinear ill-posed operator equations. The first convergence rates results for such problems were developed by Engl, Kunisch and Neubauer in 1989. While these results apply for operator equations formulated in Hilbert spaces, the results of Burger and Osher from 2004, more generally, apply to operators formulated in Banach spaces. Recently, Resmerita and co-workers presented a modification of the convergence rates result of Burger and Osher which turns out to be a complete generalization of the rates result of Engl and co-workers. In all these papers relatively strong regularity assumptions are made. However, it has been observed numerically that violations of the smoothness assumptions of the operator do not necessarily affect the convergence rate negatively. We take this observation and weaken the smoothness assumptions on the operator and prove a novel convergence rate result. The most significant difference in this result from the previous ones is that the source condition is formulated as a variational inequality and not as an equation as previously. As examples, we present a phase retrieval problem and a specific inverse option pricing problem, both previously studied in the literature. For the inverse finance problem, the new approach allows us to bridge the gap to a singular case, where the operator smoothness degenerates just when the degree of ill-posedness is minimal.

Some Newton-type methods for the regularization of nonlinear ill-posed problems

In this paper we consider a combination of Newtons method with linear Tikhonov regularization, linear Landweber iteration and truncated SVD, for regularizing an abstract, nonlinear, ill-posed operator equation. We show that under certain smoothness conditions on the nonlinear operator, these methods converge locally. For perturbed data we propose an a priori stopping rule, that guarantees convergence of the iterates to a solution, as the noise level goes to zero. Under appropriate closeness and smoothness assumptions on the starting value and the solution, we obtain convergence rates.

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.

Iterative methods for nonlinear ill-posed problems in Banach spaces: convergence and applications to parameter identification problems

In this paper, we study convergence of two different iterative regularization methods for nonlinear ill-posed problems in Banach spaces. One of them is a Landweber type iteration, the other one the iteratively regularized Gauss–Newton method with an a posteriori chosen regularization parameter in each step. We show that a discrepancy principle as a stopping rule renders these iteration schemes regularization methods, i.e., we prove their convergence as the noise level tends to zero. The theoretical findings are illustrated by two parameter identification problems for elliptic PDEs.

Regularization by projection with a posteriori discretization level choice for linear and nonlinear ill-posed problems

In this paper we study the regularization of linear and nonlinear ill-posed operator equations by projection onto finite-dimensional spaces with a posteriori chosen space dimension. We show that this results in a regularization method, i.e. a stable solution method for the ill-posed problem, that converges to an exact solution as the data noise level goes to zero, with optimal rates under additional regularity conditions.

Convergence rates for the iteratively regularized Gauss–Newton method in Banach spaces*

In this paper we consider the iteratively regularized Gauss–Newton method (IRGNM) in a Banach space setting and prove optimal convergence rates under approximate source conditions. These are related to the classical concept of source conditions that is available only in Hilbert space. We provide results in the framework of general index functions, which include, e.g. Holder and logarithmic rates. Concerning the regularization parameters in each Newton step as well as the stopping index, we provide both a priori and a posteriori strategies, the latter being based on the discrepancy principle.

FEM-Based determination of real and complex elastic, dielectric, and piezoelectric moduli in piezoceramic materials

We propose an enhanced iterative scheme for the precise reconstruction of piezoelectric material parameters from electric impedance and mechanical displacement measurements. It is based on finite-element simulations of the full three-dimensional piezoelectric equations, combined with an inexact Newton or nonlinear Landweber iterative inversion scheme. We apply our method to two piezoelectric materials and test its performance. For the first material, the manufacturer provides a full data set; for the second one, no material data set is available. For both cases, our inverse scheme, using electric impedance measurements as input data, performs well.

Efficient Modeling of Ferroelectric Behavior for the Analysis of Piezoceramic Actuators

This work proposes a method of efficiently modeling the hysteresis of ferroelectric materials. Our approach includes the additive combination of a reversible and an irreversible portion of the polarization and strain, respectively. Whereas the reversible parts correspond to the common piezoelectric linear equations, the irreversible parts are modeled by hysteresis operators. These operators are based on Preisach and Jiles-Atherton hysteresis models which are well-established tools in ferromagnetic modeling. In contrast to micromechanical approaches, a Preisach or a Jiles-Atherton hysteresis operator can be efficiently numerically evaluated. A comparison of the resulting simulations to measured data concludes the article.

Efficient computation of the Tikhonov regularization parameter by goal-oriented adaptive discretization

Parameter identification problems for partial differential equations (PDEs) often lead to large-scale inverse problems. For their numerical solution it is necessary to repeatedly solve the forward and even the inverse problem, as it is required for determining the regularization parameter, e.g., according to the discrepancy principle in Tikhonov regularization. To reduce the computational effort, we use adaptive finite-element discretizations based on goal-oriented error estimators. This concept provides an estimate of the error in a so-called quantity of interest, which is a functional of the searched for parameter q and the PDE solution u. Based on this error estimate, the discretizations of q and u are locally refined. The crucial question for parameter identification problems is the choice of an appropriate quantity of interest. A convergence analysis of the Tikhonov regularization with the discrepancy principle on discretized spaces for q and u provides a possible answer: it shows, that in order to determine the correct regularization parameter, one has to guarantee sufficiently high accuracy in the squared residual norm—which is therefore our quantity of interest—whereas q and u themselves need not be computed precisely everywhere. This fact allows for relatively low dimensional adaptive meshes and hence for a considerable reduction of the computational effort. In this paper, we study an efficient inexact Newton algorithm for determining an optimal regularization parameter in Tikhonov regularization according to the discrepancy principle. With the help of error estimators we guide this algorithm and control the accuracy requirements for its convergence. This leads to a highly efficient method for determining the regularization parameter.