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)

Paying attention to working memory: similarities in the spatial distribution of attention in mental and physical space

Selective attention is not limited to information that is physically present in the external world, but can also operate on mental representations in the internal world. However, it is not known whether mechanisms of attentional selection in mental space operate in a similar fashion as in physical space. We studied the spatial distribution of attention for items in physical and in mental space by comparing how successfully distracters were rejected at varying distances from the attended location. The results indicate very similar distribution characteristics of spatial attention in physical and mental space. Specifically, we found that performance monotonically improved with increasing distracter distance relative to the attended location suggesting that distracter confusability is particularly pronounced for nearby distracters relative to further away distracters. The present findings suggest that mental representations preserve their spatial configuration in working memory, and that similar mechanistic principles underlie selective attention in physical and mental space

The role of phonological and executive working memory resources in simple arithmetic strategies

The current study investigated the role of the central executive and the phonological loop in arithmetic strategies to solve simple addition problems (Experiment 1) and simple subtraction problems (Experiment 2). The choice/no-choice method was used to investigate strategy execution and strategy selection independently. The central executive was involved in both retrieval and procedural strategies, but played a larger role in the latter than in the former. Active phonological processes played a role in procedural strategies only. Passive phonological resources, finally, were only needed when counting was used to solve subtraction problems. No effects of working memory load on strategy selection were observed

Cultural differences in complex addition: efficient Chinese versus adaptive Belgians and Canadians

In the present study, the authors tested the effects of working-memory load on math problem solving in 3 different cultures: Flemish-speaking Belgians, English-speaking Canadians, and Chinese-speaking Chinese currently living in Canada. Participants solved complex addition problems (e.g., 58 + 76) in no-load and working-memory load conditions, in which either the central executive or the phonological loop was loaded. The authors used the choice/no-choice method to obtain unbiased measures of strategy selection and strategy efficiency. The Chinese participants were faster than the Belgians, who were faster and more accurate than the Canadians. The Chinese also required fewer working-memory resources than did the Belgians and Canadians. However, the Chinese chose less adaptively from the available strategies than did the Belgians and Canadians. These cultural differences in math problem solving are likely the result of different instructional approaches during elementary school (practice and training in Asian countries vs. exploration and flexibility in non-Asian countries), differences in the number language, and informal cultural norms and standards. The relevance of being adaptive is discussed as well as the implications of the results in regards to the strategy choice and discovery simulation model of strategy selection (J. Shrager & R. S. Siegler, 1998)

Do multiplication and division strategies rely on executive and phonological working memory resources?

The role of executive and phonological working-memory resources in simple arithmetic was investigated in two experiments. Participants had to solve simple multiplication problems (e.g., 4 x 8; Experiment 1) or simple division problems (e.g., 42 : 7; Experiment 2) under no-load, phonological-load, and executive-load conditions. The choice/no-choice method was used to investigate strategy execution and strategy selection independently. Results on strategy execution showed that executive working memory resources were involved in direct memory retrieval of both multiplication and division facts. Executive working-memory resources were also needed to execute nonretrieval strategies. Phonological working-memory resources, on the other hand, tended to be involved in non-retrieval strategies only. Results on strategy selection showed no effects of working-memory load. Finally, correlation analyses showed that both strategy execution and strategy selection correlated with individual-difference variables such as gender, math anxiety, associative strength, calculator use, arithmetic skill, and math experience

The collapse of cooperation in evolving games

Game theory provides a quantitative framework for analyzing the behavior of rational agents. The Iterated Prisoner's Dilemma in particular has become a standard model for studying cooperation and cheating, with cooperation often emerging as a robust outcome in evolving populations. Here we extend evolutionary game theory by allowing players' strategies as well as their payoffs to evolve in response to selection on heritable mutations. In nature, many organisms engage in mutually beneficial interactions, and individuals may seek to change the ratio of risk to reward for cooperation by altering the resources they commit to cooperative interactions. To study this, we construct a general framework for the co-evolution of strategies and payoffs in arbitrary iterated games. We show that, as payoffs evolve, a trade-off between the benefits and costs of cooperation precipitates a dramatic loss of cooperation under the Iterated Prisoner's Dilemma; and eventually to evolution away from the Prisoner's Dilemma altogether. The collapse of cooperation is so extreme that the average payoff in a population may decline, even as the potential payoff for mutual cooperation increases. Our work offers a new perspective on the Prisoner's Dilemma and its predictions for cooperation in natural populations; and it provides a general framework to understand the co-evolution of strategies and payoffs in iterated interactions.Comment: 33 pages, 13 figure

Evolutionary Fitness in Variable Environments

One essential ingredient of evolutionary theory is the concept of fitness as a measure for a species' success in its living conditions. Here, we quantify the effect of environmental fluctuations onto fitness by analytical calculations on a general evolutionary model and by studying corresponding individual-based microscopic models. We demonstrate that not only larger growth rates and viabilities, but also reduced sensitivity to environmental variability substantially increases the fitness. Even for neutral evolution, variability in the growth rates plays the crucial role of strongly reducing the expected fixation times. Thereby, environmental fluctuations constitute a mechanism to account for the effective population sizes inferred from genetic data that often are much smaller than the census population size.Comment: main: 5 pages, 4 figures; supplement: 7 pages, 7 figue

Insight and search in Katona’s five-square problem

Insights are often productive outcomes of human thinking. We provide a cognitive model that explains insight problem solving by the interplay of problem space search and representational change, whereby the problem space is constrained or relaxed based on the problem representation. By introducing different experimental conditions that either constrained the initial search space or helped solvers to initiate a representational change, we investigated the interplay of problem space search and representational change in Katona’s five-square problem. Testing 168 participants, we demonstrated that independent hints relating to the initial search space and to representational change had little effect on solution rates. However, providing both hints caused a significant increase in solution rates. Our results show the interplay between problem space search and representational change in insight problem solving: The initial problem space can be so large that people fail to encounter impasse, but even when representational change is achieved the resulting problem space can still provide a major obstacle to finding the solution