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ZusammenfassungDie COVID-19-Pandemie hat vielfältige Herausforderungen für Public Health und den Öffentlichen Gesundheitsdienst (ÖGD) in Deutschland offenbart bzw. verstärkt. Sie bietet jedoch auch ein Gelegenheitsfenster für eine langfristig wirksame Transformation im Bereich der öffentlichen Gesundheit. Vor diesem Hintergrund erfolgte im Oktober und November 2020 eine Online-Befragung der Mitglieder des Nachwuchsnetzwerkes Öffentliche Gesundheit (NÖG), in welcher die Erfahrungen mit und der Blick auf Public Health während der COVID-19-Pandemie eruiert wurden und sich erste Erkenntnisse aus der Pandemie für den deutschen Public-Health-Kontext abzeichneten. In diesem Beitrag werden diese vorgestellt und ausgehend von den Ergebnissen der Befragung Desiderate formuliert, welche zielgerichtete und konkrete Anhaltspunkte für die Weiterentwicklung und Förderung der öffentlichen Gesundheit geben sollen. Zentrale Themen, welche die befragten Nachwuchsfachkräfte beschäftigten, waren die erhöhte öffentliche und politische Aufmerksamkeit für öffentliche Gesundheit mit einer Fokussierung auf den Infektionsschutz, der Stellenwert von Public Health in Deutschland sowie Stärken und Schwächen von Public-Health-Strukturen und -Fachkräften. Die Desiderate zielen auf eine langfristige und holistische Stärkung von Public Health ab, in der die Ausbildung interdisziplinärer Nachwuchsfachkräfte einen hohen Stellenwert einnimmt.

While the term ‘social field’ has surfaced sporadically in various disciplines throughout the twentieth Century, it has largely lain dormant as a conceptual framework. In this article, we re-introduce the social field as a foundational concept both for understanding collective lived experience and for developing methodologies to effect systems change. We explore and expand on three inter-related properties that we consider to be phenomena common to all social fields: intercorporeality, autonomy, and affordance. Drawing on recent and emerging intervention methodologies focusing on these properties, we illustrate the potential of taking a social field perspective for both diagnosis and intervention in the change process. We make the case that the social field is a distinct entity and a powerful leverage point for effecting systems change, and that the re-invigoration of social field theory and practice can make a significant contribution to the field of organizational and systems change.

In recent work it has been established that deep neural networks (DNNs) are capable of approximating solutions to a large class of parabolic partial differential equations without incurring the curse of dimension. However, all this work has been restricted to problems formulated on the whole Euclidean domain. On the other hand, most problems in engineering and in the sciences are formulated on finite domains and subjected to boundary conditions. The present paper considers an important such model problem, namely the Poisson equation on a domain $D\subset \mathbb {R}^d$ subject to Dirichlet boundary conditions. It is shown that DNNs are capable of representing solutions of that problem without incurring the curse of dimension. The proofs are based on a probabilistic representation of the solution to the Poisson equation as well as a suitable sampling method.

In this piece we share our discovery process as action researchers in an online, global change initiative that emerged during the first lockdown of the Covid-19 pandemic, 2020. In the spirit of sharing our work "in the making" we aim to make visible our own reflection process and the questions that surface from it. In particular, we share and explore our realization that in order to fully serve the transformational intention of the initiative and the research itself, we needed to expand our research framework mid-process. The framework that best serves the emergent and transformative nature of the initiative is one that both supports awareness-based action and generates widely applicable knowledge; that integrates a variety of perspectives on social phenomena (first-, second-, and third-person); and that aims to bring systematic inquiry both to the observable phenomena and the deeper underlying dimensions. The approach requires us to make visible our assumptions and to integrate and validate different epistemologies, including relational, intuitive and aesthetic knowing. As such, the approach to research we suggest here can be thought of as an epistemological framework itself.

Structural features and the heterogeneity of disease transmissions play an essential role in the dynamics of epidemic spread. But these aspects can not completely be assessed from aggregate data or macroscopic indicators such as the effective reproduction number. We propose an index of effective aggregate dispersion (EffDI) that indicates the significance of infection clusters and superspreading events in the progression of outbreaks by carefully measuring the level of stochasticity in time series of reported case numbers. This allows to detect the transition from predominantly clustered spreading to a diffusive regime with diminishing significance of singular clusters, which can be decisive for the future progression of outbreaks and relevant in the planning of containment measures. We evaluate EffDI for SARS-CoV-2 case data in different countries and compare the results with a quantifier for the socio-demographic heterogeneity in disease transmissions in a case study to substantiate that EffDI qualifies as a measure for the heterogeneity in transmission dynamics.

We propose and analyze several multilevel algorithms for the fast simulation of possibly nonstationary Gaussian random fields (GRFs) indexed, for example, by the closure of a bounded domain [Formula: see text] or, more generally, by a compact metric space [Formula: see text] such as a compact [Formula: see text]-manifold [Formula: see text]. A colored GRF [Formula: see text], admissible for our algorithms, solves the stochastic fractional-order equation [Formula: see text] for some [Formula: see text], where [Formula: see text] is a linear, local, second-order elliptic self-adjoint differential operator in divergence form and [Formula: see text] is white noise on [Formula: see text]. We thus consider GRFs on [Formula: see text] with covariance operators of the form [Formula: see text]. The proposed algorithms numerically approximate samples of [Formula: see text] on nested sequences [Formula: see text] of regular, simplicial partitions [Formula: see text] of [Formula: see text] and [Formula: see text], respectively. Work and memory to compute one approximate realization of the GRF [Formula: see text] on the triangulation [Formula: see text] of [Formula: see text] with consistency [Formula: see text], for some consistency order [Formula: see text], scale essentially linearly in [Formula: see text], independent of the possibly low regularity of the GRF. The algorithms are based on a sinc quadrature for an integral representation of (the application of) the negative fractional-order elliptic “coloring” operator [Formula: see text] to white noise [Formula: see text]. For the proposed numerical approximation, we prove bounds of the computational cost and the consistency error in various norms.

The COVID-19 pandemic has both exposed and intensified various challenges for Public Health and the Public Health service (ÖGD) in Germany. However, it also offers a window of opportunity for effective long-term transformation of the country's Public Health system. Against this backdrop, an online survey was carried out among the members of the German Network of Young Professionals in Public Health (Nachwuchsnetzwerk Öffentliche Gesundheit (NÖG)) in October and November 2020. It sought to elicit members' experiences and views related to Public Health during the COVID-19 pandemic. The resulting preliminary "lessons learned" for the German Public Health context are presented in this article. Based on the results of the survey, recommendations were formulated which are intended to provide targeted and concrete advice for the strengthening and transformation of Public Health in Germany. The main issues that preoccupied the young professionals were the increased public and political attention to Public Health and the narrow focus on infectious disease control, the standing of Public Health in Germany and the strengths and weaknesses of Public Health structures and workforce. The recommendations are aimed at promoting long-term and holistic strengthening of Public Health, with the training of an interdisciplinary workforce of young professionals presenting a key focus.Die COVID-19-Pandemie hat vielfältige Herausforderungen für Public Health und den Öffentlichen Gesundheitsdienst (ÖGD) in Deutschland offenbart bzw. verstärkt. Sie bietet jedoch auch ein Gelegenheitsfenster für eine langfristig wirksame Transformation im Bereich der öffentlichen Gesundheit. Vor diesem Hintergrund erfolgte im Oktober und November 2020 eine Online-Befragung der Mitglieder des Nachwuchsnetzwerkes Öffentliche Gesundheit (NÖG), in welcher die Erfahrungen mit und der Blick auf Public Health während der COVID-19-Pandemie eruiert wurden und sich erste Erkenntnisse aus der Pandemie für den deutschen Public-Health-Kontext abzeichneten. In diesem Beitrag werden diese vorgestellt und ausgehend von den Ergebnissen der Befragung Desiderate formuliert, welche zielgerichtete und konkrete Anhaltspunkte für die Weiterentwicklung und Förderung der öffentlichen Gesundheit geben sollen. Zentrale Themen, welche die befragten Nachwuchsfachkräfte beschäftigten, waren die erhöhte öffentliche und politische Aufmerksamkeit für öffentliche Gesundheit mit einer Fokussierung auf den Infektionsschutz, der Stellenwert von Public Health in Deutschland sowie Stärken und Schwächen von Public-Health-Strukturen und -Fachkräften. Die Desiderate zielen auf eine langfristige und holistische Stärkung von Public Health ab, in der die Ausbildung interdisziplinärer Nachwuchsfachkräfte einen hohen Stellenwert einnimmt.

Social-emotional education and the relational competence of school staff and leaders are emphasized in research since they strongly impact childrens’ social, emotional, and cognitive development. In a longitudinal project—Empathie macht Schule (EmS)—we aim at evaluating the outcome and process of an empathy training for the whole school staff, including leaders. We compare three treatments to three control elementary schoolsviaa mixed-methods approach employing qualitative and quantitative research methods targeting both, the school staff and the schoolchildren. Since the start of the project in 2019, the COVID-19 pandemic has disrupted the global education process, that is, the range of training activities for school staff in an unprecedented manner. First the lockdown and then the hygienic measures impact the habits and certainties in schools on multiple levels, including artifacts (e.g., physical distancing measures and virtual platforms), processes (e.g., virtual learning and home-schooling), social structures (e.g., separation of a high-risk group), and values (e.g., difficulties in building relations and showing empathy due to physical distance). Leaders and staff are facing an uncertain situation, while their actions and decisions may—also unintentionally—shape the social reality that will be inhabited to a significant extent. In this context, a number of questions become salient. How does the disruption of the pandemic affect interpersonal relationships, interactions, and the social field—the sum of relationships within the system of a school—as a whole? And specifically, how do the actors reflect on changes in the social field, their relationships, and the schools’ and classrooms’ overall relationship quality due to the crisis? The assessment combines qualitative interviews with leaders and teachers (N= 10) along with a self-report survey (N= 80) addressing the effects of the pandemic on interpersonal aspects in schools. Surprisingly, a number of positive effects were mentioned regarding the learning environment in the smaller-sized classes, which were caused by hygienic measures, as well as increased cohesion among faculty. The potential influence of these effects by consciously shaping relationships and cultivating empathy is discussed in the article.

Objective Microsurgical diskectomy/sequestrectomy is the standard procedure for the surgical treatment of lumbar disk herniations. The transforaminal endoscopic sequestrectomy technique is a minimally invasive alternative with potential advantages such as minimal blood loss and tissue damage, as well as early mobilization of the patient. We report the implementation of this technique in a German university hospital setting. Methods One single surgeon performed transforaminal endoscopic sequestrectomy from February 2013 to July 2016 for lumbar disk herniation in 44 patients. Demographic as well as perioperative, clinical, and radiologic data were analyzed from electronic records. Furthermore, we investigated complications, intraoperative change of the procedure to microsurgery, and reoperations. The postoperative course was analyzed using the Macnab criteria, supplemented by a questionnaire for follow-up. Pre- and postoperative magnetic resonance imaging volumetric analyses were performed to assess the radiologic efficacy of the technique. Results Our study population had a median age of 52 years. The median follow-up was 15 months, and the median length of hospital stay was 4 days. Median duration of surgery was 100 minutes with a median blood loss of 50 mL. Surgery was most commonly performed at the L4–L5 level (63%) and in caudally migrated disk herniations (44%). In six patients, surgery was performed for recurrent disk herniations. The procedure had to be changed to conventional microsurgery in four patients. We observed no major complications. Minor complications occurred in six patients, and in four patients a reoperation was performed. Furthermore, a significantly lower Oswestry Disability Index score (p = 0.03), a lower Short Form 8 Health Survey (SF-8) score (p = 0.001), a lower visual analog scale (VAS) lower back pain score (p = 0.03) and VAS leg pain score (p = 0.0008) at the 12-month follow-up were observed in comparison with the preoperative examination. In MRI volumetry, we detected a median postoperative volume reduction of the disk herniation of 57.1% (p = 0.02). Conclusions The transforaminal endoscopic sequestrectomy can be safely implemented in a university hospital setting in selected patients with primary and recurrent lumbar disk herniations, and it leads to good clinical and radiologic results. However, learning curve, caseload, and residents' microsurgical training requirements clearly affect the implementation process.

We analyze the convergence rate of a multilevel quasi-Monte Carlo (MLQMC) Finite Element Method (FEM) for a scalar diffusion equation with log-Gaussian, isotropic coefficients in a bounded, polytopal domain D ⊂ ℝd. The multilevel algorithm QL* which we analyze here was first proposed, in the case of parametric PDEs with sequences of independent, uniformly distributed parameters in Kuo et al. (Found. Comput. Math. 15 (2015) 411–449). The random coefficient is assumed to admit a representation with locally supported coefficient functions, as arise for example in spline- or multiresolution representations of the input random field. The present analysis builds on and generalizes our single-level analysis in Herrmann and Schwab (Numer. Math. 141 (2019) 63–102). It also extends the MLQMC error analysis in Kuo et al. (Math. Comput. 86 (2017) 2827–2860), to locally supported basis functions in the representation of the Gaussian random field (GRF) in D, and to product weights in QMC integration. In particular, in polytopal domains D ⊂ ℝd, d=2,3, our analysis is based on weighted function spaces to describe solution regularity with respect to the spatial coordinates. These spaces allow GRFs and PDE solutions whose realizations become singular at edges and vertices of D. This allows for non-stationary GRFs whose covariance operators and associated precision operator are fractional powers of elliptic differential operators in D with boundary conditions on ∂D. In the weighted function spaces in D, first order, Lagrangian Finite Elements on regular, locally refined, simplicial triangulations of D yield optimal asymptotic convergence rates. Comparison of the ε-complexity for a class of Matérn-like GRF inputs indicates, for input GRFs with low sample regularity, superior performance of the present MLQMC-FEM with locally supported representation functions over alternative representations, e.g. of Karhunen–Loève type. Our analysis yields general bounds for the ε-complexity of the MLQMC algorithm, uniformly with respect to the dimension of the parameter space.

We survey the numerical analysis of a class of deterministic, higher-order QMC integration methods in forward and inverse uncertainty quantification algorithms for advection-diffusion-reaction (ADR) equations in polygonal domains D⊂R2 with distributed uncertain inputs. We admit spatially heterogeneous material properties. For the parametrization of the uncertainty, we assume at hand systems of functions which are locally supported in D. Distributed uncertain inputs are written in countably parametric, deterministic form with locally supported representation systems. Parametric regularity and sparsity of solution families and of response functions in scales of weighted Kontrat’ev spaces in D are quantified using analytic continuation. ISSN:2194-1009 ISSN:2194-1017

We analyze convergence rates of quasi-Monte Carlo (QMC) quadratures for countably-parametric solutions of linear, elliptic partial differential equations (PDE) in divergence form with log-Gaussian diffusion coefficient, based on the error bounds in Nichols and Kuo (J Complex 30(4):444–468, 2014. https://doi.org/10.1016/j.jco.2014.02.004 ). We prove, for representations of the Gaussian random field PDE input with locally supported basis functions, and for continuous, piecewise polynomial finite element discretizations in the physical domain novel QMC error bounds in weighted spaces with product weights that exploit localization of supports of the basis elements representing the input Gaussian random field. In this case, the cost of the fast component-by-component algorithm for constructing the QMC points scales linearly in terms of the integration dimension. The QMC convergence rate $$\mathcal {O}(N^{-1+\delta })$$ (independent of the parameter space dimension s) is achieved under weak summability conditions on the expansion coefficients.

For the numerical solution of linear systems that arise from discretized linear partial differential equations, multigrid and domain decomposition methods are well established. Multigrid methods are known to have optimal complexity and domain decomposition methods are in particular useful for parallelization of the implemented algorithm. For linear random operator equations, the classical theory is not directly applicable, since condition numbers of system matrices may be close to degenerate due to non-uniform random input. It is shown that iterative methods converge in the strong, i.e. $$L^p$$, sense if the random input satisfies certain integrability conditions. As a main result, standard multigrid and domain decomposition methods are applicable in the case of linear elliptic partial differential equations with lognormal diffusion coefficients and converge strongly with deterministic bounds on the computational work which are essentially optimal. This enables the application of multilevel Monte Carlo methods with rigorous, deterministic bounds on the computational work.

This paper deals with the derivation and the simulation of flatness based control and tracking control for a ball in tube system. Flatness based control is a control loop with a two-degree-of-freedom structure in terms of a feedforward (precontrol) and a feedback control path. Tracking control is characterized as a control loop in terms of a feedback control path only where the state vector needs to be measured or estimated. The control laws for both methods are computed for the nonlinear and the linearized system. The reference trajectory for the flat system output, the ball position, is defined as a polynomial transition between two stationary setpoints. The control quality of the closed-loop system is evaluated with the given reference trajectory in a simulation study. The disturbance rejection of the ball in tube system is examined by an external force impulse. In the literature nonlinearity measures are defined in order to answer the question how nonlinear a nonlinear system is. The nonlinearity measure gives therefore a decision if it is a mildly or a highly nonlinear system. The introduced nonlinearity measure is valid for nonlinear input-affine systems represented as a state space model and is calculated for the ball in tube system.

Understanding others’ feelings, intentions, and beliefs is a crucial social skill both for our personal lives and for meeting the challenges of a globalized world. Recent evidence suggests that the ability to represent and infer others’ mental states (Theory of Mind, ToM) can be enhanced by mental training in healthy adults. The present study investigated the role of training-induced understanding of oneself for the enhanced understanding of others. In a large-scale longitudinal study, two independent participant samples (N = 80 and N = 81) received a 3-month contemplative training. This training focused on perspective taking and was inspired by the Internal Family Systems model that conceives the self as being composed of a complex system of inner personality aspects. Specifically, participants practiced perspective taking on their own inner states by learning to identify and classify different inner personality parts. Results revealed that the degree to which participants improved their understanding of themselves—reflected in the number of different inner parts they could identify—predicted their improvements in high-level ToM performance over training. Especially the number of identified parts that were negatively valenced showed a strong relation with enhanced ToM capacities. This finding suggests a close link between getting better in understanding oneself and improvement in social intelligence. Electronic supplementary material The online version of this article (doi:10.1007/s41465-017-0023-6) contains supplementary material, which is available to authorized users.

The usage of unmanned civil and military drones is steadily rising. Essential components of every multicopter are its electric motors and the electronic speed controllers (ESC). These also determine, besides the flight dynamics and the stability behavior, the flight duration in a crucial way. Therefore in this contribution, an ESC for permanent magnet synchronous motors with a sinusoidal back electromotive force (PMSM) is presented, which was designed, under consideration of all these points. The basis for the precise and dynamic speed control is a high-resolution position sensor in combination with a 72MHz microcontroller. Under these circumstances, the speed control can operate at a high frequency of 20 kHz together with a high resolution. Using a field oriented control in combination with a flatness based control concept theoretically guarantees a maximum torque per ampere and thus a low energy consumption. The required speed signal is gained from the rotor position using a proportional integral observer. An exact asymptotic regulation is achieved implementing a state feedback integral controller.

We survey recent convergence rate bounds for single-level and multilevel QMC Finite Element (FE for short) algorithms for the numerical approximation of linear, second order elliptic PDEs in divergence form in a bounded, polygonal domain D. The diffusion coefficient a is assumed to be an isotropic, log-Gaussian random field (GRF for short) in D. The representation of the GRF \(Z = \log a\) is assumed affine-parametric with i.i.d. standard normal random variables, and with locally supported functions \(\psi _j\) characterizing the spatial variation of the GRF Z. The goal of computation is the evaluation of expectations (i.e., of so-called “ensemble averages”) of (linear functionals of) the random solution. The QMC rules employed are randomly shifted lattice rules proposed in Nichols, Kuo (J Complex 30:444–468, 2014, [19]) as used and analyzed previously in a similar setting (albeit for globally in D supported spatial representation functions \(\psi _j\) as arise in Karhunen-Loeve expansions) in Graham et al. (Numer Math 131:329–368, 2015, [9]), Kuo et al. (Math Comput 86:2827–2860, 2017, [14]). The multilevel QMC-FE approximation \(Q^*_L\) analyzed here for locally supported \(\psi _j\) was proposed first in Kuo, Schwab, Sloan (Found Comput Math 15:411–449, 2015, [17]) for affine-parametric operator equations. As shown in Gantner, Herrmann, Schwab (SIAM J Numer Anal 56:111–135, 2018, [7]), Gantner, Herrmann, Schwab (Contemporary computational mathematics - a celebration of the 80th birthday of Ian Sloan. Springer, Cham, 2018, [6]), Herrmann, Schwab (QMC integration for lognormal-parametric, elliptic PDEs: local supports and product weights. Technical Report 2016-39, Seminar for Applied Mathematics, ETH Zurich, Switzerland, 2016, [10]), Herrmann, Schwab (Multilevel quasi-Monte Carlo integration with product weights for elliptic PDEs with lognormal coefficients. Technical Report 2017-19, Seminar for Applied Mathematics, ETH Zurich, Switzerland, 2017, [11]) localized supports of the \(\psi _j\) (which appear in multiresolution representations of GRFs Z of Levy–Ciesielski type in D) allow for the use of product weights, originally proposed in construction of QMC rules in Sloan, Woźniakowski (J Complex 14:1–33, 1998, [23]) (cf. the survey (Dick, Kuo, Sloan in Acta Numer 22:133–288, 2013, [4]) and references there). The present results from Herrmann, Schwab (Multilevel quasi-Monte Carlo integration with product weights for elliptic PDEs with lognormal coefficients. Technical Report 2017-19, Seminar for Applied Mathematics, ETH Zurich, Switzerland, 2017, [11]) on convergence rates for the multilevel QMC FE algorithm allow for general polygonal domains D and for GRFs Z whose realizations take values in weighted spaces containing \(W^{1,\infty }(D)\). Localized support assumptions on \(\psi _j\) are shown to allow QMC rule generation by the fast, FFT based CBC constructions in Nuyens, Cools (J Complex 22:4–28, 2006, [21]), Nuyens, Cools (Math Comput 75:903–920, 2006, [20]) which scale linearly in the integration dimension which, for multiresolution representations of GRFs, is proportional to the number of degrees of freedom used in the FE discretization in the physical domain D. We show numerical experiments based on public domain QMC rule generating software in Gantner (A generic c++ library for multilevel quasi-Monte Carlo. In: Proceedings of the Platform for Advanced Scientific Computing Conference, PASC ’16, ACM, New York, USA, pp 11:1–11:12 2016, [5]), Kuo, Nuyens (Found Comput Math 16:1631–1696, 2016, [13]).

Numerical solutions of stationary diffusion equations on the unit sphere with isotropic lognormal diffusion coefficients are considered. H\"older regularity in $L^p$ sense for isotropic Gaussian random fields is obtained and related to the regularity of the driving lognormal coefficients. This yields regularity in $L^p$ sense of the solution to the diffusion problem in Sobolev spaces. Convergence rate estimates of multilevel Monte Carlo Finite and Spectral Element discretizations of these problems are then deduced. Specifically, a convergence analysis is provided with convergence rate estimates in terms of the number of Monte Carlo samples of the solution to the considered diffusion equation and in terms of the total number of degrees of freedom of the spatial discretization, and with bounds for the total work required by the algorithm in the case of Finite Element discretizations. The obtained convergence rates are solely in terms of the decay of the angular power spectrum of the (logarithm) of the diffusion coefficient. Numerical examples confirm the presented theory., Comment: 35 pages, 1 figure; rewritten Sections 2 and 3, added numerical experiments

We analyze convergence rates of first-order quasi--Monte Carlo (QMC) integration with randomly shifted lattice rules and for higher-order, interlaced polynomial lattice rules for a class of countably parametric integrands that result from linear functionals of solutions of linear, elliptic diffusion equations with affine-parametric, uncertain coefficient function $a(x,{y}) = \bar{a}(x) + \sum_{j\geq 1} y_j \psi_j(x)$ in a bounded domain $D\subset \mathbb{R}^d$. Extending the result in [F. Y. Kuo, C. Schwab, and I. H. Sloan, SIAM J. Numer. Anal., 50 (2012), pp. 3351--3374], where $\psi_j$ was assumed to have global support in the domain $D$, we assume in the present paper that ${supp}(\psi_j)$ is localized in $D$ and that we have control on the overlaps of these supports. Under these conditions we prove dimension-independent convergence rates in [1/2,1) of randomly shifted lattice rules with product weights and corresponding higher-order convergence rates by higher-order, interlaced polynomial lattice rule...

We present an error analysis of higher order Quasi-Monte Carlo (QMC) integration and of randomly shifted QMC lattice rules for parametric operator equations with uncertain input data taking values in Banach spaces. Parametric expansions of these input data in locally supported bases such as splines or wavelets was shown in Gantner et al. (SIAM J Numer Anal 56(1):111–135, 2018) to allow for dimension independent convergence rates of combined QMC-Galerkin approximations. In the present work, we review and refine the results in that reference to the multilevel setting, along the lines of Kuo et al. (Found Comput Math 15(2):441–449, 2015) where randomly shifted lattice rules and globally supported representations were considered, and also the results of Dick et al. (SIAM J Numer Anal 54(4):2541–2568, 2016) in the particular situation of locally supported bases in the parametrization of uncertain input data. In particular, we show that locally supported basis functions allow for multilevel QMC quadrature with product weights, and prove new error vs. work estimates superior to those in these references (albeit at stronger, mixed regularity assumptions on the parametric integrand functions than what was required in the single-level QMC error analysis in the first reference above). Numerical experiments on a model affine-parametric elliptic problem confirm the analysis.

This paper deals with the derivation and the implementation of a flatness based control for a nonlinear ball in tube system as a testbed. For this purpose the system equations are quoted, considering that the system properties are changing in dependence on a positive or a negative fan acceleration. The proof for the ball position as chosen output shows that the system has the attribute flat. On the basis of this proof a precontrol law is calculated, in order to track the output to a predefined reference trajectory exactly, in case of an ideal model and non-occurring disturbances. Within the whole feedforward control the nonlinear static gain of the DC motor is taken into account by the use of an inverse measured characteristic. The feedforward control is combined with a state feedback and an integral controller. The control quality of the closed-loop system is evaluated with a setpoint transition in form of three different trajectories. In this evaluation the performance is compared using the control concept with and without the integral controller. Concluding the disturbance rejection of the ball in tube system is examined.