TY - GEN

T1 - Efficiency of High Order Spectral Element Methods on Petascale Architectures

AU - Hutchinson, Maxwell

AU - Heinecke, Alexander

AU - Pabst, Hans

AU - Henry, Greg

AU - Parsani, Matteo

AU - Keyes, David E.

N1 - KAUST Repository Item: Exported on 2020-10-01

PY - 2016/6/15

Y1 - 2016/6/15

N2 - High order methods for the solution of PDEs expose a tradeoff between computational cost and accuracy on a per degree of freedom basis. In many cases, the cost increases due to higher arithmetic intensity while affecting data movement minimally. As architectures tend towards wider vector instructions and expect higher arithmetic intensities, the best order for a particular simulation may change. This study highlights preferred orders by identifying the high order efficiency frontier of the spectral element method implemented in Nek5000 and NekBox: the set of orders and meshes that minimize computational cost at fixed accuracy. First, we extract Nek’s order-dependent computational kernels and demonstrate exceptional hardware utilization by hardware-aware implementations. Then, we perform productionscale calculations of the nonlinear single mode Rayleigh-Taylor instability on BlueGene/Q and Cray XC40-based supercomputers to highlight the influence of the architecture. Accuracy is defined with respect to physical observables, and computational costs are measured by the corehour charge of the entire application. The total number of grid points needed to achieve a given accuracy is reduced by increasing the polynomial order. On the XC40 and BlueGene/Q, polynomial orders as high as 31 and 15 come at no marginal cost per timestep, respectively. Taken together, these observations lead to a strong preference for high order discretizations that use fewer degrees of freedom. From a performance point of view, we demonstrate up to 60% full application bandwidth utilization at scale and achieve ≈1PFlop/s of compute performance in Nek’s most flop-intense methods.

AB - High order methods for the solution of PDEs expose a tradeoff between computational cost and accuracy on a per degree of freedom basis. In many cases, the cost increases due to higher arithmetic intensity while affecting data movement minimally. As architectures tend towards wider vector instructions and expect higher arithmetic intensities, the best order for a particular simulation may change. This study highlights preferred orders by identifying the high order efficiency frontier of the spectral element method implemented in Nek5000 and NekBox: the set of orders and meshes that minimize computational cost at fixed accuracy. First, we extract Nek’s order-dependent computational kernels and demonstrate exceptional hardware utilization by hardware-aware implementations. Then, we perform productionscale calculations of the nonlinear single mode Rayleigh-Taylor instability on BlueGene/Q and Cray XC40-based supercomputers to highlight the influence of the architecture. Accuracy is defined with respect to physical observables, and computational costs are measured by the corehour charge of the entire application. The total number of grid points needed to achieve a given accuracy is reduced by increasing the polynomial order. On the XC40 and BlueGene/Q, polynomial orders as high as 31 and 15 come at no marginal cost per timestep, respectively. Taken together, these observations lead to a strong preference for high order discretizations that use fewer degrees of freedom. From a performance point of view, we demonstrate up to 60% full application bandwidth utilization at scale and achieve ≈1PFlop/s of compute performance in Nek’s most flop-intense methods.

UR - http://hdl.handle.net/10754/622140

UR - http://link.springer.com/10.1007/978-3-319-41321-1_23

UR - http://www.scopus.com/inward/record.url?scp=84977586700&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-41321-1_23

DO - 10.1007/978-3-319-41321-1_23

M3 - Conference contribution

SN - 9783319413204

SP - 449

EP - 466

BT - Lecture Notes in Computer Science

PB - Springer Nature

ER -