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The paper presents the results of a study done to find the optimal shapes of matrix element partitioning on three abstract heterogeneous processors when performing multiplication operations. An abstract processor model allows applying the research results in systems with different heterogeneous architectures. To determine the optimal partitioning shape, the work uses non-rectangular candidate shapes identified by Ashley DeFlumere in her work as a result of applying the technology of redistribution of matrix elements between the processors «push»: Square Corner, Rectangle Corner, Square Rectangle, Block Rectangle, L-Rectangle, Traditional 1D Rectangular. The optimality of shapes is determined for four classes of matrix multiplication algorithms: Serial Communication with Barrier (SCB), Parallel Communication with Barrier (PCB), Serial Communication with Bulk Overlap (SCO) and Parallel Communication with Overlap (PCO). The Hockney model was used to evaluate the communication complexity of algorithms. Mathematical models of the algorithm execution time were introduced in the paper for each considered candidate shape in all algorithms. Based on the developed mathematical models, software was developed that allows to select the form of elements partitioning between processors, depending on the ratio of their speeds and latency of the transmission medium.

Purpose – The purpose of this paper is to develop an efficient numerical scheme for non-linear two-dimensional (2D) parabolic partial differential equations using modified bi-cubic B-spline functions. As a test case, method has been applied successfully to 2D Burgers equations. Design/methodology/approach – The scheme is based on collocation of modified bi-cubic B-Spline functions. The authors used these functions for space variable and for its derivatives. Collocation form of the partial differential equation results into system of first-order ordinary differential equations (ODEs). The obtained system of ODEs has been solved by strong stability preserving Runge-Kutta method. The computational complexity of the method is O(p log(p)), where p denotes total number of mesh points. Findings – Obtained numerical solutions are better than those available in literature. Ease of implementation and very small size of computational work are two major advantages of the present method. Moreover, this method provides approximate solutions not only at the grid points but also at any point in the solution domain. Originality/value – First time, modified bi-cubic B-spline functions have been applied to non-linear 2D parabolic partial differential equations. Efficiency of the proposed method has been confirmed with numerical experiments. The authors conclude that the method provides convergent approximations and handles the equations very well in different cases.

Previous studies of application usage show that the performance of collective communications are critical for high-performance computing. Despite active research in the field, both general and feasible solution to the optimization of collective communication problem is still missing. In this paper, we analyze and attempt to improve intra-cluster collective communication in the context of the widely deployed MPI programming paradigm by extending accepted models of point-to-point communication, such as Hockney, LogP/LogGP, and PLogP, to collective operations. We compare the predictions from models against the experimentally gathered data and using these results, construct optimal decision function for broadcast collective. We quantitatively compare the quality of the model-based decision functions to the experimentally-optimal one. Additionally, in this work, we also introduce a new form of an optimized tree-based broadcast algorithm, splitted-binary. Our results show that all of the models can provide useful insights into various aspects of the different algorithms as well as their relative performance. Still, based on our findings, we believe that the complete reliance on models would not yield optimal results. In addition, our experimental results have identified the gap parameter as being the most critical for accurate modeling of both the classical point-to-point-based pipeline and our extensions to fan-out topologies.

Describes a very successful writing assignment the author uses with his high school English students, in which students write about a letter of the alphabet. Notes that this assignment grants freedom for exploration and voice, the raw material for shaping and can be used to teach everything from the writing of poetry to essays for college admission. (SR)

On Dec. 6, 1987, following six months of concentrated effort, the curtain rose, revealing Hockney's spectacular designs-- bold, colorful, sculptural interpretations of Tristan's ship under full sail, King Marke's turreted castle, the shadowy forest and sun-parched cliffs of Cornwall. [...] Hockney (looking dapper in dark suit, checked tie, scarf and cap) says he's been enjoying rediscovering the opera and watching the production come together, though, he admits, it's a rigorous (even tedious) process requiring hours and hours of technical rehearsals.