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Event Details

  • Monday, May 22, 2017
  • 09:00 - 09:55

Design of Numerical Algorithms for Partial Differential Equations on Next-Generation Computer Architectures

The end of Dennard scaling, resulting in the use of low-power processor technologies in HPC, is leading to the deployment of large-scale parallelism on a single compute node to obtain high performance. By 2024, the cost of flops on HPC-capable systems, in terms of dollars and power, is expected to decrease by a factor of 100, relative to what it was in 2014. However, the performance of memory systems for these processors will not keep pace, due to power constraints. The ratio of the peak aggregate flop rate to the bandwidth between DRAM and the floating point units, as well as ratio of flop rate to overall size of DRAM, will be significantly less than what we have been used to for the last 20 years. Traditional numerical methods for partial differential equations have low arithmetic intensity (AI). Such methods can only obtain a small fraction of peak performance on such systems: performance is limited by rate at which data can get to the floating point units. In this talk, we will give several examples of how these new architectures lead to new tradeoffs in the design of numerical algorithms for PDE, trading flops for bytes in a way that reduces the overall data traffic and memory footprint required to obtain a given level of accuracy in a calculation. We will focus our attention on structured-grid and particle methods; in those cases, the most obvious example of this approach is the use of methods that are higher-order accurate than the ones customarily used (mostly second order) in multi-physics calculations. High-order methods can, in principle, obtain a given level of accuracy for fewer degrees of freedom, and the arithmetic intensity increases with the order of accuracy of the method. The design of new discretization methods is central to the success of this endeavor. However, there are other issues that need to be addressed in order for this approach to be successful. The design of high-order accurate methods that will retain their apparent advantages for realistic applications is nontrivial, and significant effort is required to realize the theoretically predicted performance by careful implementation.