The stable solution of ill-posed non-linear operator equations in Banach space requires regularization. One important approach is based on Tikhonov regularization, in which case a one-parameter family of regularized solutions is obtained. It is crucial to choose the parameter appropriately. Here, a variant of the discrepancy principle is analyzed. In many cases such parameter choice exhibits the feature, called regularization property below, that the chosen parameter tends to zero as the noise tends to zero, but slower than the noise level. Here we shall show such regularization property under two natural assumptions. First, exact penalization must be excluded, and secondly, the discrepancy principle must stop after a finite number of iterations. We conclude this study with a discussion of some consequences for convergence rates obtained by the discrepancy principle under the validity of some kind of variational inequality, a recent tool for the analysis of inverse problems.

In der Arbeit werden effektive Verfahren zur Auswertung der Hilbert-Transformation entwickelt und zur Lösung nichtlinearer Randwertaufgaben der Funktionentheorie eingesetzt. Die Verwendung polynomialer Spline-Wavelets und geeignet modifizierter Wavelet-Algorithmen ermöglichen die schnelle Berechnung auf gleichmäßigen und ungleichmäßigen Gittern sowie deren automatische Anpassung an lokale Besonderheiten der Lösung. Die detaillierte Untersuchung des Zusammenhangs zwischen der Glattheit, der Größe des Trägers des Splines, der Anzahl verschwindender Momente und des asymptotischen Verhaltens der Hilbert-Transformierten erlaubt die Anpassung der Parameter des Verfahrens in Bezug auf Genauigkeit und Effektivität. Im zweiten Teil der Arbeit werden verschiedene Algorithmen zur Lösung von Riemann-Hilbert Probleme vorgeschlagen und deren Konvergenzverhalten untersucht. Die theoretischen Ergebnisse werden durch numerische Experimente bestätigt.

Working with problems and making investigations is an activity one has to learn already very early. Therefore in primary school children should not only learn concepts and solve given tasks. They also should find out knowledge and reasons by themselves. Here you will find some problem fields in elementary arithmetic within children of primary school can make different investigations and find as well as give reasons for special statements. The topics concerned are partitions of numbers, sums of consecutive numbers, figured numbers, sequences and chains, table of hundred and numberwalls.

Objective: This study aims to ascertain once and for all whether children and adolescents affected by ADHD show a higher risk for accidents, as well as investigating a possible association between the administration of ADHD-specific medication and the occurrence of accidents. Method: Two exceptionally large sets of data were implemented in this analysis. Participants included children and adolescents representative of the entire German population. Data for Survey 1 was collected through extensive administration of questionnaires. Data for Survey 2 stemmed from the records of a leading German health insurance company. In terms of statistical analysis, chi-square tests as well as logistic regression analyses were applied and odds ratios (ORs) were determined. Results: Innovative results are presented showing a significantly higher likelihood for ADHD-affected youngsters to be involved in accidents compared with their nonaffected counterparts (Survey 1: OR = 1.60; Survey 2: OR = 1.89) but lacking an overall significant influence of medication regarding the occurrence of accidents (Survey 1: OR = 1.28; Survey 2: OR = 0.97). Frequency of accidents could be predicted by ADHD, gender, and age in both samples. Medication intake served as a weak predictor only in Survey 2. Conclusion: It has been determined in two representative and independent German samples that youngsters with ADHD are at a significantly higher risk of being involved in accidents. In the future, this should always be considered when setting up a treatment plan to ensure a safer and healthier coming of age without relying solely on specific effects of medication.

Inhomogeneous and changing temperature distributions in machine tools lead to sometimes considerable quality problems in the manufacturing process. In addition, the switching on and off of aggregates, for example, leads to further fluctuations in the temperature field of machine tools. More than 100 specialists discussed these and other topics from the field of thermal research at the 1st Conference on Termal Issues in Machine Tools in Dresden from 22 to 23 March.:Efficient modelling and computation of structure-variable thermal behavior of machine tools S. Schroeder, A. Galant, B. Kauschinger, M. Beitelschmidt Parameter identification software for various thermal model types B. Hensel, S. Schroeder, K. Kabitzsch Minimising thermal error issues on turning centre M. Mareš, O. Horejš, J. Hornych The methods for controlled thermal deformations in machine tools A. P. Kuznetsov, H.-J. Koriath, A.O. Dorozhko Efficient FE-modelling of the thermo-elastic behaviour of a machine tool slide in lightweight design C. Peukert, J. Müller, M. Merx, A. Galant, A. Fickert, B. Zhou, S. Städtler, S. Ihlenfeldt, M. Beitelschmidt Development of a dynamic model for simulation of a thermoelectric self-cooling system for linear direct drives in machine tools E. Uhlmann, L. Prasol, S.Thom, S. Salein, R. Wiese System modelling and control concepts of different cooling system structures for machine tools J. Popken, L. Shabi, J. Weber, J. Weber The electric drive as a thermo-energetic black box S. Winkler, R. Werner Thermal error compensation on linear direct drive based on latent heat storage I. Voigt, S. Winkler, R. Werner, A. Bucht, W.-G. Drossel Industrial relevance and causes of thermal issues in machine tools M. Putz, C. Richter, J. Regel, M. Bräunig Clustering by optimal subsets to describe environment interdependencies J. Glänzel, R. Unger, S. Ihlenfeldt Using meta models for enclosures in machine tools F. Pavliček, D. P. Pamies, J. Mayr, S. Züst, P. Blaser, P. Hernández-Becerro, K. Wegener Model order reduction of thermal models of machine tools with varying boundary conditions P. Hernández-Becerro, J. Mayr, P. Blaser, F. Pavliček, K. Wegener Effectiveness of modelling the thermal behaviour of the ball screw unit with moving heat sources taken into account J. Jedrzejewski, Z. Kowal, W. Kwasny, Z. Winiarski Analyzing and optimizing the fluidic tempering of machine tool frames A. Hellmich, J. Glänzel, A. Pierer Thermo-mechanical interactions in hot stamping L. Penter, N. Pierschel Experimental analysis of the heat flux into the grinding tool in creep feed grinding with CBN abrasives C. Wrobel, D. Trauth, P. Mattfeld, F. Klocke Development of multidimensional characteristic diagrams for the real-time correction of thermally caused TCP-displacements in precise machining M. Putz, C. Oppermann, M. Bräunig Measurement of near cutting edge temperatures in the single point diamond turning process E. Uhlmann, D. Oberschmidt, S. Frenzel, J. Polte Investigation of heat flows during the milling processes through infrared thermography and inverse modelling T. Helmig, T. Augspurger, Y. Frekers, B. Döbbeler, F. Klocke, R. Kneer Thermally induced displacements of machine tool structure, tool and workpiece due to cutting processes O. Horejš, M. Mareš, J. Hornych A new calibration approach for a grey-box model for thermal error compensation of a C-Axis C. Brecher, R. Spierling, M. Fey Investigation of passive torque of oil-air lubricated angular contact ball bearing and its modelling J. Kekula, M. Sulitka, P. Kolář, P. Kohút, J. Shim, C. H. Park, J. Hwang Cooling strategy for motorized spindle based on energy and power criterion to reduce thermal errors S. Grama, A. N. Badhe, A. Mathur Cooling potential of heat pipes and heat exchangers within a machine tool spindleo B. Denkena, B. Bergman, H. Klemme, D. Dahlmann Structure model based correction of machine tools X. Thiem, B. Kauschinger, S. Ihlenfeldt Optimal temperature probe location for the compensation of transient thermal errors G. Aguirre, J. Cilla, J. Otaegi, H. Urreta Adaptive learning control for thermal error compensation on 5-axis machine tools with sudden boundary condition changes P. Blaser, J. Mayr, F. Pavliček, P. Hernández-Becerro, K. Wegener Hybrid correction of thermal errors using temperature and deformation sensors C. Naumann, C. Brecher, C. Baum, F. Tzanetos, S. Ihlenfeldt, M. Putz Optimal sensor placement based on model order reduction P. Benner, R. Herzog, N. Lang, I. Riedel, J. Saak Workpiece temperature measurement and stabilization prior to dimensional measurement N. S. Mian, S. Fletcher, A. P. Longstaff Measurement of test pieces for thermal induced displacements on milling machines H. Höfer, H. Wiemer Model reduction for thermally induced deformation compensation of metrology frames J. v. d. Boom Local heat transfer measurement A. Kuntze, S. Odenbach, W. Uffrecht Thermal error compensation of 5-axis machine tools using a staggered modelling approach J. Mayr, T. Tiberini. P. Blaser, K. Wegener Design of a Photogrammetric Measurement System for Displacement and Deformation on Machine Tools M. Riedel, J. Deutsch, J. Müller. S. Ihlenfeldt Thermography on Machine Tools M. Riedel, J. Deutsch, J. Müller, S. Ihlenfeldt Test piece for thermal investigations of 5-axis machine tolls by on-machine measurement M. Wiesener. P. Blaser, S. Böhl, J. Mayr, K. Wegener

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Quantification of stochastic or quantum systems by a joint probability density or wave function is a notoriously difficult computational problem, since the solution depends on all possible states (or realizations) of the system. Due to this combinatorial flavor, even a system containing as few as ten particles may yield as many as $10^{10}$ discretized states. None of even modern supercomputers are capable to cope with this curse of dimensionality straightforwardly, when the amount of quantum particles, for example, grows up to more or less interesting order of hundreds. A traditional approach for a long time was to avoid models formulated in terms of probabilistic functions, and simulate particular system realizations in a randomized process. Since different times in different communities, data-sparse methods came into play. Generally, they aim to define all data points indirectly, by a map from a low amount of representers, and recast all operations (e.g. linear system solution) from the initial data to the effective parameters. The most advanced techniques can be applied (at least, tried) to any given array, and do not rely explicitly on its origin. The current work contributes further progress to this area in the particular direction: tensor product methods for separation of variables. The separation of variables has a long history, and is based on the following elementary concept: a function of many variables may be expanded as a product of univariate functions. On the discrete level, a function is encoded by an array of its values, or a tensor. Therefore, instead of a huge initial array, the separation of variables allows to work with univariate factors with much less efforts. The dissertation contains a short overview of existing tensor representations: canonical PARAFAC, Hierarchical Tucker, Tensor Train (TT) formats, as well as the artificial tensorisation, resulting in the Quantized Tensor Train (QTT) approximation method. The contribution of the dissertation consists in both theoretical constructions and practical numerical algorithms for high-dimensional models, illustrated on the examples of the Fokker-Planck and the chemical master equations. Both arise from stochastic dynamical processes in multiconfigurational systems, and govern the evolution of the probability function in time. A special focus is put on time propagation schemes and their properties related to tensor product methods. We show that these applications yield large-scale systems of linear equations, and prove analytical separable representations of the involved functions and operators. We propose a new combined tensor format (QTT-Tucker), which descends from the TT format (hence TT algorithms may be generalized smoothly), but provides complexity reduction by an order of magnitude. We develop a robust iterative solution algorithm, constituting most advantageous properties of the classical iterative methods from numerical analysis and alternating density matrix renormalization group (DMRG) techniques from quantum physics. Numerical experiments confirm that the new method is preferable to DMRG algorithms. It is as fast as the simplest alternating schemes, but as reliable and accurate as the Krylov methods in linear algebra.

This paper proposes and investigates a metaheuristic tabu search algorithm (TSA) that generates optimal or near optimal solutions sequences for the feedback length minimization problem (FLMP) associated to a design structure matrix (DSM). The FLMP is a non-linear combinatorial optimization problem, belonging to the NP-hard class, and therefore finding an exact optimal solution is very hard and time consuming, especially on medium and large problem instances. First, we introduce the subject and provide a review of the related literature and problem definitions. Using the tabu search method (TSM) paradigm, this paper presents a new tabu search algorithm that generates optimal or sub-optimal solutions for the feedback length minimization problem, using two different neighborhoods based on swaps of two activities and shifting an activity to a different position. Furthermore, this paper includes numerical results for analyzing the performance of the proposed TSA and for fixing the proper values of its parameters. Then we compare our results on benchmarked problems with those already published in the literature. We conclude that the proposed tabu search algorithm is very promising because it outperforms the existing methods, and because no other tabu search method for the FLMP is reported in the literature. The proposed tabu search algorithm applied to the process layer of the multidimensional design structure matrices proves to be a key optimization method for an optimal product development. / Ce mémoire présente un nouvel algorithme métaheuristique de recherche taboue pour trouver des solutions optimales ou sous-optimales au problème de minimisation de la longueur des dépendances d’une matrice de conception (FLMP). Ce problème comporte une fonction économique non-linéaire et il appartient à la classe NP-ardu. Il s’ensuit qu’il est très difficile à trouver une solution optimale exacte en temps réel pour les problèmes de taille moyenne ou grande. D’abord, on présente le problème et une revue de la littérature associée. Ensuite, on analyse le problème, et on présente les détails du nouvel algorithme de recherche taboue produisant des solutions au problème de réduction de l’effet de retour en utilisant deux voisinages différents, le premier basé sur l’échange des positions de deux activités ("swap"), et le second sur le déplacement d’une activité à une position différente ("shift"). Des résultats numériques permettent d’analyser le comportement de l’algorithme et de comparer les deux voisinages. La première étape consiste à déterminer de bonnes valeurs pour les paramètres en utilisant des problèmes générés aléatoirement. Ensuite nos résultats sont comparés avec ceux obtenus dans la littérature. On conclut que l’algorithme de recherche taboue proposé est très prometteur, car nos résultats sont meilleurs que ceux publiés dans la litérature. D’autant plus que la recherche taboue semble avoir été utilisée pour la première fois sur ce problème.

Sei (X(t), t>= -r) ein stationärer stochastischer Prozess, der die affine stochastische Differentialgleichung mit Gedächtnis dX(t)=L(X(t+s))dt+sigma dW(t), t>= 0, löst, wobei sigma>0, (W(t), t>=0) eine Standard-Brownsche Bewegung und L ein stetiges lineares Funktional auf dem Raum der stetigen Funktionen auf [-r,0], dargestellt durch ein endliches signiertes Maß a, bezeichnet. Wir nehmen an, dass eine Trajektorie (X(t), -r 0, konvergiert. Diese Rate ist schlechter als in vielen klassischen Fällen. Wir beweisen jedoch eine untere Schranke, die zeigt, dass keine Schätzung eine bessere Rate im Minimax-Sinn aufweisen kann. Für zeit-diskrete Beobachtungen von maximalem Abstand Delta konvergiert die Galerkin-Schätzung immer noch mit obiger Rate, sofern Delta is in etwa von der Ordnung T^(-1/2). Hingegen wird bewiesen, dass für festes Delta unabhängig von T die Rate sich signifikant verschlechtern muss, indem eine untere Schranke von T^(-s/(2s+6)) gezeigt wird. Außerdem wird eine adaptive Schätzung basierend auf Wavelet-Thresholding-Techniken für das assoziierte schlechtgestellte Problem konstruiert. Diese nichtlineare Schätzung erreicht die obige Minimax-Rate sogar für die allgemeinere Klasse der Besovräume B^s_(p,infinity) mit p>max(6/(2s+3),1). Die Restriktion p>=max(6/(2s+3),1) muss für jede Schätzung gelten und ist damit inhärent mit dem Schätzproblem verknüpft. Schließlich wird ein Hypothesentest mit nichtparametrischer Alternative vorgestellt, der zum Beispiel für das Testen auf Gedächtnis verwendet werden kann. Dieser Test ist anwendbar für eine L^2-Trennungsrate zwischen Hypothese und Alternative der Ordnung T^(-s/(2s+2.5)). Diese Rate ist wiederum beweisbar optimal für jede mögliche Teststatistik. Für die Beweise müssen die Parameterabhängigkeit der stationären Lösungen sowie die Abbildungseigenschaften der assoziierten Kovarianzoperatoren detailliert bestimmt werden. Weitere Resultate von allgemeinem Interessen beziehen sich auf die Mischungseigenschaft der stationären Lösung, eine Fallstudie zu exponentiellen Gewichtsfunktionen sowie der Approximation des stationären Prozesses durch autoregressive Prozesse in diskreter Zeit. / Let (X(t), t>= -r) be a stationary stochastic process solving the affine stochastic delay differential equation dX(t)=L(X(t+s))dt+sigma dW(t), t>= 0, with sigma>0, (W(t), t>=0) a standard one-dimensional Brownian motion and with a continuous linear functional L on the space of continuous functions on [-r,0], represented by a finite signed measure a. Assume that a trajectory (X(t), -r 0. This rate is worse than those obtained in many classical cases. However, we prove a lower bound, stating that no estimator can attain a better rate of convergence in a minimax sense. For discrete time observations of maximal distance Delta, the Galerkin estimator still attains the above asymptotic rate if Delta is roughly of order T^(-1/2). In contrast, we prove that for observation intervals Delta, with Delta independent of T, the rate must deteriorate significantly by providing the rate estimate T^(-s/(2s+6)) from below. Furthermore, we construct an adaptive estimator by applying wavelet thresholding techniques to the corresponding ill-posed inverse problem. This nonlinear estimator attains the above minimax rate even for more general classes of Besov spaces B^s_(p,infinity) with p>max(6/(2s+3),1). The restriction p >= 6/(2s+3) is shown to hold for any estimator, hence to be inherently associated with the estimation problem. Finally, a hypothesis test with a nonparametric alternative is constructed that could for instance serve to decide whether a trajectory has been generated by a stationary process with or without time delay. The test works for an L^2-separation rate between hypothesis and alternative of order T^(-s/(2s+2.5)). This rate is again shown to be optimal among all conceivable tests. For the proofs, the parameter dependence of the stationary solutions has to be studied in detail and the mapping properties of the associated covariance operators have to be determined exactly. Other results of general interest concern the mixing properties of the stationary solution, a case study for exponential weight functions and the approximation of the stationary process by discrete time autoregressive processes.

In dieser Arbeit präsentieren wir eine neue Art von Newton-Verfahren mit Liniensuche, basierend auf Interpolation im Bildbereich nach Wedin et al. [LW84]. Von dem resultierenden stabilisierten Newton-Algorithmus wird theoretisch und praktisch gezeigt, dass er effizient ist im Falle von nichtsingulären Lösungen. Darüber hinaus wird beobachtet, dass er eine superlineare Rate von Konvergenz bei einfachen Singularitäten erhält. Hingegen ist vom Newton-Verfahren ohne Liniensuche bekannt, dass es nur linear von fast allen Punkten in der Nähe einer singulären Lösung konvergiert. In Hinsicht auf Anwendungen auf Komplementaritätsprobleme betrachten wir auch Systeme, deren Jacobimatrix nicht differenzierbar sondern nur semismooth ist. Auch hier erreicht unser stabilisiertes und beschleunigtes Newton- Verfahren Superlinearität bei einfachen Singularitäten. / In this thesis we present a new type of line-search for Newton’s method, based on range space interpolation as suggested by Wedin et al. [LW84]. The resulting stabilized Newton algorithm is theoretically and practically shown to be efficient in the case of nonsingular roots. Moreover it is observed that it maintains a superlinear rate of convergence at simple singularities. Whereas Newton’s method without line-search is known to converge only linearly from almost all points near the singular root. In view of applications to complementarity problems we also consider systems, whose Jacobian is not differentiable but only semismooth. Again, our stabilized and accelerated Newton’s method achieves superlinearity at simple singularities.