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Many decision-making queries are based on aggregating massive amounts of data, where sampling is an important approximation technique for reducing execution times. It is important to estimate error bounds when sampling to help users balance between accuracy and performance. However, error bound estimation is challenging because data processing pipelines often transform the input dataset in complex ways before computing the final aggregated values. In this paper, we introduce a sampling framework to support approximate computing with estimated error bounds in Spark. Our framework allows sampling to be performed at multiple arbitrary points within a sequence of transformations preceding an aggregation operation. The framework constructs a data provenance tree to maintain information about how transformations are clustering output data items to be aggregated. It then uses the tree and multistage sampling theories to compute the approximate aggregate values and corresponding error bounds. When information about output keys are available early, the framework can also use adaptive stratified reservoir sampling to avoid (or reduce) key losses in the final output and to achieve more consistent error bounds across popular and rare keys. Finally, the framework includes an algorithm to dynamically choose sampling rates to meet user-specified constraints on the CDF of error bounds in the outputs. We have implemented a prototype of our framework called ApproxSpark and used it to implement five approximate applications from different domains. Evaluation results show that ApproxSpark can (a) significantly reduce execution time if users can tolerate small amounts of uncertainties and, in many cases, loss of rare keys, and (b) automatically find sampling rates to meet user-specified constraints on error bounds. We also explore and discuss extensively tradeoffs between sampling rates, execution time, accuracy and key loss.

This article focuses on the elucidation of a three-dimensional model of the structure of anhydrite crystal (CaSO4). The structure parameters of anhydrite crystal were obtained by means of first principles after structure optimization at 0~120 MPa. In comparison with previous experimental and theoretical calculation values, the results we obtained are strikingly similar to the previous data. The elastic constants and physical parameters of anhydrite crystal were also studied by the first-principles method. Based on this, we further studied the Young’s modulus and Poisson’s ratio of anhydrite crystal, the anisotropy factor, the speed of sound, the minimum thermal conductivity and the hardness of the material. It was shown that the bulk modulus and Poisson’s ratio of anhydrite crystal rose slowly with increasing pressure. The anisotropy characteristics of the Young’s modulus and shear modulus of anhydrite crystal were consistent under various pressure levels, while the difference in the anisotropy characteristics of the bulk modulus appeared. The acoustic velocities of anhydrite crystal tended to be stable with increasing pressure. The minimum thermal conductivity remained relatively unchanged with increasing pressure. However, the material hardness declined gradually with increasing pressure.

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