BackgroundAlthough the popularity of workplace health promotion (WHP) has considerably increased over the years, there are still concerns about the way this concept is being implemented by the companies. There is, however, a seeming lack of empirical knowledge about variations in WHP practice. Therefore, the aim of this study was to determine the prevalence of different patterns (and related quality levels) of WHP activity and the effect of organisational predictors on the chances of these WHP activity levels being implemented.MethodsData from an establishment survey (N = 6,500) were used to calculate the prevalences of four configurations of WHP among German companies. Furthermore, multinominal logistic regressions were performed to determine odds ratios for these WHP activity levels according to several organisational characteristics.Results9% of companies exhibited the most comprehensive type of WHP including analysis, individual-directed prevention measures and participatory groups concerned with working conditions improvement (level A), 18% featured a combination of analysis and individual-directed prevention (level B), 29% had reported measures from only one of these categories (level C), and 44% showed no WHP activity at all (level D). In the multivariate analysis company size turned out to be the strongest predictor of WHP at all levels. WHP was also predicted by a good economic situation of the company, the availability of safety specialist assistance, the availability of specialist assistance in occupational health and the presence of an employee representative body. These effects usually became stronger when moving up in the hierarchy of WHP levels. For the two sector-level variables (private vs. public, production vs. services) no statistically significant associations with WHP were found.ConclusionsWHP still shows great potential for improvement both in quantitative and qualitative terms. Especially required are additional efforts in developing and implementing WHP practice models and dissemination strategies which are tailored to the particular conditions and needs of small companies. However, findings suggest that the chances for achieving progress in WHP also depend on developments in adjacent policy areas such as labour relations or occupational safety and health.

We introduce the tensor numerical method for solving optimal control problems that are constrained by fractional two-(2D) and three-dimensional (3D) elliptic operators with variable coefficients. We solve the governing equation for the control function which includes a sum of the fractional operator and its inverse, both discretized over large 3D n × n × n spacial grids. Using the diagonalization of the arising matrix-valued functions in the eigenbasis of the one-dimensional Sturm-Liouville operators, we construct the rank-structured tensor approximation with controllable precision for the discretized fractional elliptic operators and the respective preconditioner. The right-hand side in the constraining equation (the optimal design function) is supposed to be represented in a form of a low-rank canonical tensor. Then the equation for the control function is solved in a tensor structured format by using preconditioned CG iteration with the adaptive rank truncation procedure that also ensures the accuracy of calculations, given an 𝜀-threshold. This method reduces the numerical cost for solving the control problem to O(n log n) (plus the quadratic term O(n 2 ) with a small weight), which outperforms traditional approaches with O(n 3 log n) complexity in the 3D case. The storage for the representation of all 3D nonlocal operators and functions involved is also estimated by O(n log n).This essentially outperforms the traditional methods operating with fully populated n 3 × n 3 matrices and vectors in R n 3 . Numerical tests for 2D/3D control problems indicate the almost linear complexity scaling of the rank truncated preconditioned conjugate gradient iteration in the univariate grid size n.

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