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IPscatt is a free, open-source MATLAB toolbox facilitating the solution for time-independent scattering (also known as time-harmonic scattering) in two- and three-dimensional settings. The toolbox has three main application cases: simulation of the scattered field for a given transmitter-receiver geometry; the generation of simulated data as well as the handling of the real-world data from Institute Fresnel; and the reconstruction of the contrast from several measured, scattered fields. In each case, a variety of options tailored to the needs of practitioners is provided. For example, the toolbox allows the simulation of the scattered near field as well as of the far field. Also, it provides methods for the modeling of the incident field as point sources as well as plane waves. Finally, many common geometries of transmitters and receivers are included out of the box. Regarding the reconstruction, the provided functions implement the regularization scheme that relies on a primal-dual algorithm and was introduced by F. Bürgel, K. S. Kazimierski, and A. Lechleiter [ Journal of Computational Physics 339 (2017), 1–30]. This article provides a survey of the mathematical concepts in scattering, connects them with the provided implementation, gives an overview of the software framework as well as its application areas, and compares it with existing software packages solving the same problem.

This paper presents the results of an investigation of an unusual unpainted area in the Corinthian vase painting silhouettes painting technique during the Archaic Period (late 7th to 6th century BC). A painting of two panthers in the heraldic scheme done by the Erlenmeyer Painter on an alabastron contains a striking unpainted circular area inside the shoulder of one of the panthers. Since this feature cannot be explained by iconographical constraints, it has previously been referred to as an “opening”. However, by combining X-ray methods, particularly computed tomography (CT) and related 3D surface renderings, with mineralogical and geochemical methods (SEM-EDS, XRD) and stylistic arguments, we have arrived at a new interpretation. The CT data revealed a feature directly beneath the unpainted circular area that was shaped like an angular plug and had similar structural properties and attenuation coefficients as the surrounding clay matrix. A visible inclusion on the surface of this alabastron was identified as a rock fragment, most likely sandstone or sandy shale, which suggests that a problematic large inclusion had been removed from the vessel and was subsequently filled with clay during the firm leather-hard stage. We argue that the painter intentionally deviated from the usual compositional form by leaving the area of the repair unpainted and adapting the drawing of the panther accordingly. One advantage of this method, particularly for vase collections, is that CT is a non-invasive technique that collects data from the entire ceramic body, which in combination with geoscience methods such as SEM-EDX and XRD allows for minimally invasive analysis that can strengthen the X-ray-based evaluation. Therefore, we contend that integrating CT data combined with mineralogical and geochemical data into pottery studies will open new avenues for the study of ancient vase painting, particularly for interpreting unpainted areas and other irregularities within black-glazed silhouette figures.

The DjVu file format and image compression techniques are widely used in the archival of digital documents. Its key ingredients are the separation of the document into fore- and background layers and a binary switching mask, followed by a lossy, transform-based compression of the former and a dictionary-based compression of the latter. The lossy compression of the layers is based on a wavelet decomposition and bit truncation, which leads, in particular at higher compression rates, to severe compression artifacts in the standard decompression of the layers. The aim of this paper is to break ground for the variational decompression of DjVu files. To this aim, we provide an in-depth analysis and discussion of the compression standard with a particular focus on modeling data constraints for decompression. This allows to carry out DjVu decompression as regularized inversion of the compression procedure. As particular example, we evaluate the performance of such a framework using total variation and total generalized variation regularization. Furthermore, we provide routines for obtaining the necessary data constraints from a compressed DjVu file and for the forward and adjoint transformation operator involved in DjVu compression.

In arterial spin labeling (ASL) a perfusion weighted image is achieved by subtracting a label image from a control image. This perfusion weighted image has an intrinsically low signal to noise ratio and numerous measurements are required to achieve reliable image quality, especially at higher spatial resolutions. To overcome this limitation various denoising approaches have been published using the perfusion weighted image as input for denoising. In this study we propose a new spatio-temporal filtering approach based on total generalized variation (TGV) regularization which exploits the inherent information of control and label pairs simultaneously. In this way, the temporal and spatial similarities of all images are used to jointly denoise the control and label images. To assess the effect of denoising, virtual ground truth data were produced at different SNR levels. Furthermore, high-resolution in-vivo pulsed ASL data sets were acquired and processed. The results show improved image quality, quantitative accuracy and robustness against outliers compared to seven state of the art denoising approaches.

We present a fast computational framework for the inverse medium problem in scattering, i.e. we look at discretization, reconstruction and numerical performance. The Helmholtz equation in two and three dimensions is used as a physical model of scattering including point sources and plane waves as incident fields as well as near and far field measurements. For the reconstruction of the medium, we set up a rapid variational regularization scheme and indicate favorable choices of the various parameters. The underlying paradigm is, roughly speaking, to minimize the discrepancy between the reconstruction and measured data while, at the same time, taking into account various structural a-priori information via suitable penalty terms. In particular, the involved penalty terms are designed to promote information expected in real-world environments. To this end, a combination of sparsity promoting terms, total variation, and physical bounds of the inhomogeneous medium, e.g. positivity constraints, is employed in the regularization penalty. A primal-dual algorithm is used to solve the minimization problem related to the variational regularization. The computational feasibility, performance and efficiency of the proposed approach is demonstrated for synthetic as well as experimentally measured data.

Precise measurement of mechanical forces is crucial to efficient micro-manufacturing. The quality of such measurements depends heavily on the properties of the noise inevitably accompanying every measurement process. In the micro-range, the signal-to-noise ratio tends to be very low, and the noise dynamic varies for different frequencies. In result, common denoising methods that assume white noise perform poorly in this setting. In this paper, a novel, easily implementable denoising method based on a local statistic of the measured data’s spectrum is proposed. By testing it on a representative dataset, it is shown that the proposed method is robust and stable. Particularly, it allows for an efficient retrieval of the force signal encountered in micro-milling processes.

In this paper we deal with Besov spaces . We show that generalized forms of the polarization identity hold for the wavelet-characterization of the Besov space norm. In particular we show that these spaces are smooth of power type and convex of power type for .

We introduce and discuss an iterative method of modified Landweber type for regularization of nonlinear operator equations in Banach spaces. Under smoothness and convexity assumptions on the solution space we present convergence and stability results. Furthermore, we will show that under the so-called approximate source conditions convergence rates may be achieved by a proper a-priori choice of the parameter of the presented algorithm. We will illustrate these theoretical results with a numerical example.

For Tikhonov functionals of the form ?(x)=?Ax?y? Y r +??x? X q we investigate a steepest descent method in the dual of the Banach space X. We show convergence rates for the proposed method and present numerical tests.

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.

In this paper, we present results that have been obtained in the DFG-SPP project “Adaptive Wavelet Frame Methods for Operator Equations: Sparse Grids, Vector-Valued Spaces and Applications to Nonlinear Inverse Problems”. This project has been concerned with (nonlinear) elliptic and parabolic operator equations on nontrivial domains as well as with related inverse parameter identification problems. In this paper we study analytic properties of the underlying parameter-to-state map, which is motivated by a parabolic model for the embryonal development of drosophila melanogaster.

We construct adaptive Mann iterations for finding fixed points of strongly pseudocontractive mappings and solving nonlinear strongly accretive (not necessary continuous) operator equation Sx = f in p-smooth Banach spaces.

One feasible way to minimize a non-smooth functional is to replace it by some smoothed version, which leads to a surrogate minimization problem that is easily treated by standard means. A prominent example of such a problem is given by a Tikhonov-type functional incorporating a sparsity-enforcing penalty term. It has received enormous attention in recent years, yet its efficient minimization remains challenging. In this paper we consider general Tikhonov-type functionals and show, under mild conditions, the stability of their minimizer with respect to the replacement of the penalty term with an appropriate approximation. In particular, we consider the case of separable penalty terms. Finally, we apply the proposed strategy to the inverse medium problem and demonstrate numerical results that indicate the efficiency of the approach.

This paper presents Tikhonov- and iterated soft-shrinkage regularization methods for nonlinear inverse medium scattering problems. Motivated by recent sparsity-promoting reconstruction schemes for inverse problems, we assume that the contrast of the medium is supported within a small subdomain of a known search domain and minimize Tikhonov functionals with sparsity-promoting penalty terms based on Lp-norms. Analytically, this is based on scattering theory for the Helmholtz equation with the refractive index in Lp, 1 < p < ∞, and on crucial continuity and compactness properties of the contrast-to-measurement operator. Algorithmically, we use an iterated soft-shrinkage scheme combined with the differentiability of the forward operator in Lp to approximate the minimizer of the Tikhonov functional. The feasibility of this approach together with the quality of the obtained reconstructions is demonstrated via numerical examples.

We consider the Tikhonov functional incorporating a lp-norm as a penalty term. The minimizer of this functional is investigated in regard to its sensitivity with respect to the parameter p. To quantify the sensitivity of the minimizer, we determine the derivative with respect to p. The underlying idea is based on the implicit function theorem. We show that the minimizer of the Tikhonov functional is differentiable with respect to p for any 1 < p < 2. For p = 1, the derivative exists only under additional assumptions.

We investigate a method of accelerated Landweber type for the iterative regularization of nonlinear ill-posed operator equations in Banach spaces. Based on an auxiliary algorithm with a simplified choice of the step-size parameter we present a convergence and stability analysis of the algorithm under consideration. We will close our discussion with the presentation of a numerical example.