Kolmogorov widths and low-rank approximations of parametric elliptic PDEs

Kolmogorov $n$-widths and low-rank approximations are studied for families of elliptic diffusion PDEs parametrized by the diffusion coefficients. The decay of the $n$-widths can be controlled by that of the error achieved by best $n$-term approximations using polynomials in the parametric variable. However, we prove that in certain relevant instances where the diffusion coefficients are piecewise constant over a partition of the physical domain, the $n$-widths exhibit significantly faster decay. This, in turn, yields a theoretical justification of the fast convergence of reduced basis or POD methods when treating such parametric PDEs. Our results are confirmed by numerical experiments, which also reveal the influence of the partition geometry on the decay of the $n$-widths.Comment: 27 pages, 6 figure

Sparse polynomial approximation of parametric elliptic PDEs. Part I: affine coefficients

We consider elliptic partial differential equations with diffusion coefficients that depend affinely on countably many parameters. We study the summability properties of polynomial expansions of the function mapping parameter values to solutions of the PDE, considering both Taylor and Legendre series. Our results considerably improve on previously known estimates of this type, in particular taking into account structural features of the affine parametrization of the coefficient. Moreover, the results carry over to more general Jacobi polynomial expansions. We demonstrate that the new bounds are sharp in certain model cases and we illustrate them by numerical experiments

Sparse polynomial approximation of parametric elliptic PDEs. Part II: lognormal coefficients

Elliptic partial differential equations with diffusion coefficients of lognormal form, that is $a=exp(b)$, where $b$ is a Gaussian random field, are considered. We study the $\ell^p$ summability properties of the Hermite polynomial expansion of the solution in terms of the countably many scalar parameters appearing in a given representation of $b$. These summability results have direct consequences on the approximation rates of best $n$-term truncated Hermite expansions. Our results significantly improve on the state of the art estimates available for this problem. In particular, they take into account the support properties of the basis functions involved in the representation of $b$, in addition to the size of these functions. One interesting conclusion from our analysis is that in certain relevant cases, the Karhunen-Lo\`eve representation of $b$ may not be the best choice concerning the resulting sparsity and approximability of the Hermite expansion

Notebookeinsatz beim selbstgesteuerten Lernen. Mehrwert für Motivation, Lernklima und Qualität des Lernens?

In einer empirischen Studie mit 152 TeilnehmerInnen eines einsemestrigen Softwarepraktikums wurde untersucht, welche Zusammenhänge zwischen dem Notebookeinsatz in einem universitätsweiten Funknetz und der Qualität selbstgesteuerten Lernens bestehen. Eingenommen wurde eine intrapsychische Perspektive auf die subjektive Interpretation der ubiquitären Lernsituation und die individuellen Parameter des Lernens. Die Ergebnisse zeigen Vorteile des Notebookeinsatzes für das selbstgesteuerte Lernen. Die Effekte fielen beim Erleben von Autonomie beim Lernen verhältnismäßig groß, bei Lernklima, Motivation und Lern-Leistung nur klein bis moderat aus. Darüber hinaus zeigte sich, dass die beobachtbar veränderten Nutzungsmuster ohne Einfluss auf die Qualität des Lernens waren. Zusammenhänge mit der Lern-Leistung zeigten sich dagegen mit Motivation und Lernklima. Auf der Basis dieser Befunde werden in diesem Beitrag Implikationen für die Gestaltung von Lernkontexten an einer Notebook-Universität diskutiert. (DIPF/Orig.

Reproducibility, accuracy and performance of the Feltor code and library on parallel computer architectures

Feltor is a modular and free scientific software package. It allows developing platform independent code that runs on a variety of parallel computer architectures ranging from laptop CPUs to multi-GPU distributed memory systems. Feltor consists of both a numerical library and a collection of application codes built on top of the library. Its main target are two- and three-dimensional drift- and gyro-fluid simulations with discontinuous Galerkin methods as the main numerical discretization technique. We observe that numerical simulations of a recently developed gyro-fluid model produce non-deterministic results in parallel computations. First, we show how we restore accuracy and bitwise reproducibility algorithmically and programmatically. In particular, we adopt an implementation of the exactly rounded dot product based on long accumulators, which avoids accuracy losses especially in parallel applications. However, reproducibility and accuracy alone fail to indicate correct simulation behaviour. In fact, in the physical model slightly different initial conditions lead to vastly different end states. This behaviour translates to its numerical representation. Pointwise convergence, even in principle, becomes impossible for long simulation times. In a second part, we explore important performance tuning considerations. We identify latency and memory bandwidth as the main performance indicators of our routines. Based on these, we propose a parallel performance model that predicts the execution time of algorithms implemented in Feltor and test our model on a selection of parallel hardware architectures. We are able to predict the execution time with a relative error of less than 25% for problem sizes between 0.1 and 1000 MB. Finally, we find that the product of latency and bandwidth gives a minimum array size per compute node to achieve a scaling efficiency above 50% (both strong and weak)

Force production capacity and functional reflex activity in young and elderly men

Background and aims: There is growing evidence that biological aging results in impaired force production of the lower extremities and deficits in reflex activity. This study therefore investigated maximal and explosive force production capacity, functional reflex activity (FRA) during gait perturbations, and the relationship between variables of force production capacity and FRA in young and elderly men. Methods: Twenty-eight young (age 27±3 yrs, n=14) and old (age 67±4 yrs, n=14) healthy active men were tested for decelerating impulses on a treadmill and for their maximal isometric leg extension force (MIF) and rate of force development (RFD) on a leg-press. Results: MIF and RFD were significantly lower in the elderly than in younger participants (MIF ≈45%, p<0.01; RFD ≈50%, p<0.01). Elderly subjects showed significant decreases in FRA in the prime mover which compensated for the decelerating impulse (FRA ≈29%, p<0.05). No significant correlations were found between FRA, MIF or RFD. Conclusions: Lower MIF, RFD and impaired FRA were found in old compared with young men. The absence of significant correlations between measures of strength performance and FRA may indicate that different mechanisms within the neuromuscular system are responsible for these capacities. This result may imply that force production and functional reflex activity are independent of each other and may have to be trained complementaril