5,985 research outputs found
Kolmogorov widths and low-rank approximations of parametric elliptic PDEs
Kolmogorov -widths and low-rank approximations are studied for families of
elliptic diffusion PDEs parametrized by the diffusion coefficients. The decay
of the -widths can be controlled by that of the error achieved by best
-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 -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 -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 , where is a Gaussian random field, are
considered. We study the summability properties of the Hermite
polynomial expansion of the solution in terms of the countably many scalar
parameters appearing in a given representation of . These summability
results have direct consequences on the approximation rates of best -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 , 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 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
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