Otmar Scherzer

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.

Inverse Problems Light: Numerical Differentiation

involving a nonlinear diffusion coefficient a: IR iR+. Problem (1.1) also serves as a model for the saturation of porous media by liquid flow, in which case a (u) is related to the capillary pressure of the pores. In certain industrial applications a numerical simulation may require solving (1.1) for u. We call this the direct problem. In these simulations it is crucial that a coefficient a(u) be used that is not only qualitatively correct but also reasonably accurate. Unfortunately, tabulated values for a (u) from the literature often provide only a rough guess of the true coefficient; in this case simulations are not likely to be reliable. Consequently, identification of the diffusion coefficient a (u) from experimental data (typically, u(x, t) for some abscissa x E (0, 1) and 0 < t < T) is often the first hurdle to clear. This is the associated inverse problem. A standard method to solve the inverse problem is the output least squares method, which tries to match the given data with simulated quantities using a gradient or Newton type method for updating the diffusion coefficient. Alternatively, one can consider (1.1) as a linear equation for a (u). To set up this equation requires numerical differentiation of the data [6]. This approach is called the equation error method. It must be emphasized that inverse problems are often very ill-conditioned: for example, small changes in a (.) have little effect on the solution u in (1.1), and consequently one cannot expect high resolution reconstructions of a in the presence of measurement errors in u. Indeed, small errors in u may cause large errors in the computed a if they are not taken into account appropriately. Numerical differentiation of the data encompasses many subtleties and pitfalls that a complex (linear) inverse problem can exhibit; yet it is very easy to understand and analyze. For this reason one could say that numerical differentiation itself is an ideal model for inverse problems in a basic numerical analysis course. To support this statement we revisit a well-known algorithm for numerical differentiation of noisy data and present a new error bound for it. The method and the error bound can be interpreted as an instance of one of the most important results in regularization theory for ill-posed problems. Still, our presentation is on a very basic level and requires no prior knowledge besides standard n-dimensional calculus and the notion of cubic splines. Groetschs book [4] presents other realistic inverse problems on an elementary technical level. Further examples and a rigorous introduction to regularization theory for the computation of stable solutions to these examples can be found in [1].

Relations Between Regularization and Diffusion Filtering

Regularization may be regarded as diffusion filtering with an implicit time discretization where one single step is used. Thus, iterated regularization with small regularization parameters approximates a diffusion process. The goal of this paper is to analyse relations between noniterated and iterated regularization and diffusion filtering in image processing. In the linear regularization framework, we show that with iterated Tikhonov regularization noise can be better handled than with noniterated. In the nonlinear framework, two filtering strategies are considered: the total variation regularization technique and the diffusion filter technique of Perona and Malik. It is shown that the Perona-Malik equation decreases the total variation during its evolution. While noniterated and iterated total variation regularization is well-posed, one cannot expect to find a minimizing sequence which converges to a minimizer of the corresponding energy functional for the Perona–Malik filter. To overcome this shortcoming, a novel regularization technique of the Perona–Malik process is presented that allows to construct a weakly lower semi-continuous energy functional. In analogy to recently derived results for a well-posed class of regularized Perona–Malik filters, we introduce Lyapunov functionals and convergence results for regularization methods. Experiments on real-world images illustrate that iterated linear regularization performs better than noniterated, while no significant differences between noniterated and iterated total variation regularization have been observed.

Optimal a posteriori parameter choice for Tikhonov regularization for solving nonlinear ill-posed problems

The authors propose an a-posteriors strategy for choosing the regularization parameter in Tikhonov regularization for solving nonlinear ill-posed problems and show that under certain conditions, the convergence rate obtained with this strategy is optimal. As a by-product, a new stability estimate for the regularized solutions is given which applies to a class of parameter identification problems. The authors compare the parameter choice strategy with Morozov’s Discrepancy Principle. Finally, numerical results are presented.

Thermoacoustic tomography with integrating area and line detectors

Thermoacoustic (optoacoustic, photoacoustic) tomography is based on the generation of acoustic waves by illumination of a sample with a short electromagnetic pulse. The absorption density inside the sample is reconstructed from the acoustic pressure measured outside the illuminated sample. So far measurement data have been collected with small detectors as approximations of point detectors. Here, a novel measurement setup applying integrating detectors (e.g., lines or planes made of piezoelectric films) is presented. That way, the pressure is integrated along one or two dimensions, enabling the use of numerically efficient algorithms, such as algorithms for the inverse radon transformation, for thermoacoustic tomography. To reconstruct a three-dimensional sample, either an area detector has to be moved tangential around a sphere that encloses the sample or an array of line detectors is rotated around a single axis. The line detectors can be focused on cross sections perpendicular to the rotation axis using a synthetic aperture (SAFT) or by scanning with a cylindrical lens detector. Measurements were made with piezoelectric polyvinylidene fluoride film detectors and evaluated by comparison with numerical simulations. The resolution achieved in the resulting tomography images is demonstrated on the example of the reconstructed cross section of a grape.

A convergence analysis of iterative methods for the solution of nonlinear ill-posed problems under affinely invariant conditions

For iterative methods for well-posed problems, invariance properties have been used to provide a unified framework for convergence analysis. We carry over this approach to iterative methods for nonlinear ill-posed problems and prove convergence with rates for the Landweber and the iteratively regularized Gauss-Newton methods. The conditions needed are weaker as far as the nonlinearity is concerned than those needed in earlier papers and apply also to severely ill-posed problems. With no additional effort, we can also treat multilevel versions of our methods.

Thermoacoustic computed tomography with large planar receivers

Thermoacoustic imaging is a promising new modality for nondestructive evaluation. So far point measurement data for thermoacoustic imaging are used. In this paper we propose a novel measurement set-up with relatively large piezo foils (planar receivers) and an according real time imaging algorithm based on the Radon transform. We present numerical simulations for simulated and real world data.

Sparse regularization with lq penalty term

We consider the stable approximation of sparse solutions to nonlinear operator equations by means of Tikhonov regularization with a subquadratic penalty term. Imposing certain assumptions, which for a linear operator are equivalent to the standard range condition, we derive the usual convergence rate of the regularized solutions in dependence of the noise level δ. Particular emphasis lies on the case, where the true solution is known to have a sparse representation in a given basis. In this case, if the differential of the operator satisfies a certain injectivity condition, we can show that the actual convergence rate improves up to O(δ).

Error estimates for non-quadratic regularization and the relation to enhancement

In this paper error estimates for non-quadratic regularization of nonlinear ill-posed problems in Banach spaces are derived. Our analysis is based on a few novel features: in comparison with the classical analysis of regularization methods for inverse and ill-posed problems where a Lipschitz continuity for the Frechet derivative is required, we use a differentiability condition with respect to the Bregman distance. Also, a stability result for the regularized solutions in terms of Bregman distances is proven. Moreover, a source-wise representation of the solution as used in standard theory is interpreted in terms of data enhancement. It is also shown that total variation Bregman distance regularization for image analysis, as developed recently, can be considered as a two-step regularization method consisting of a combination of total variation regularization and additional enhancement. This technique can also be applied for filtering.