An Evaluation of Partitioners for Parallel SAMR Applications Sumir Chandra & Manish Parashar ECE Dept., Rutgers University Submitted to: Euro-Par 2001 : European Conference on Parallel Computing

Introduction AMR – Adaptive Mesh Refinement AMR used for solving PDEs for dynamic applications Challenges involved: Dynamic resource allocation Dynamic data distribution and load balancing Communication and co-ordination Partitioning of adaptive grid hierarchy Evaluation of dynamic domain-based partitioning strategies with an application-centric approach

Motivation & Goal Even for a single application, the most suitable partitioning technique depends on input parameters and its run-time state Application-centric characterization of partitioners as a function of number of processors, problem size, and granularity Enable the run-time selection of partitioners based on input parameters and application state

Adaptive Mesh Refinement Start with a base coarse grid with minimum acceptable resolution Tag regions in the domain requiring additional resolution, cluster the tagged cells, and fit finer grids over these clusters Proceed recursively so that regions on the finer grid requiring more resolution are similarly tagged and even finer grids are overlaid on these regions Resulting grid structure is a dynamic adaptive grid hierarchy The Berger-Oliger Algorithm Recursive Procedure Integrate(level) If (RegridTime) Regrid Step  t on all grids at level “level” If (level + 1 exists) Integrate (level + 1) Update(level, level + 1) End if End Recursion level = 0 Integrate(level) Partitioning Adaptive Grid Hierarchies

SAMR 2-D Grid Hierarchy Time Step 40Time Step 80 Time Step 0 Time Step 160Time Step 120Time Step 182 Level 1: Level 0: Level 3:Level 2:Level 4: Legend

Partitioning Techniques Static or Dynamic techniques Geometric or Non-geometric Dynamic partitioning – global or local approaches Partitioners for SAMR grid applications Patch-based Domain-based Hybrid

Partitioners Evaluated SFC: Space Filling Curve based partitioning G-MISP: Geometric Multi-level Inverse Space filling curve Partitioning G-MISP+SP: Geometric Multi-level Inverse Space filling curve Partitioning with Sequence Partitioning pBD-ISP: p-way Binary Dissection Inverse Space filling curve Partitioning SP-ISP: “Pure” Sequence Partitioning with Inverse Space filling curve Partitioning WD: Wavefront Diffusion based on global work load

SFC Recursive linear representation of multi-dimensional grid hierarchy using space-filling mappings (N-to-1D mapping) Computational load determined by segment length and recursion level

G-MISP & G-MISP+SP G-MISP Multi-level algorithm views matrix of workloads from SAMR grid hierarchy as a one-vertex graph, refined recursively Speed at expense of load balance G-MISP+SP “Smarter” variant of G-MISP – uses sequence partitioning to assign consecutive portions of one- dimensional list to processors Load balance improves but scheme is computationally more expensive

pBD-ISP Generalization of binary dissection – domain partitioned into p partitions Each split divides load as evenly as possible, considering processors

SP-ISP Domain sub-divided into p*b equally sized blocks Dual-level algorithm - parameter settings for each level Fine granularity scheme: good load balance but increased overhead, communication and computational cost

WD Part of ParMetis suite based on global workload Used for repartitioning graphs with scattered refinements Results in fine grain partitionings with jagged boundaries and increased communication costs and overheads Metis integration extremely expensive, dedicated SAMR partitioners performed much better Two extra steps needed for Metis in our interface Metis graph generated from grid before partitioning, clustering used to regenerate grid blocks from graph partitions after partitioning

Experimental Setup Application – RM3D 3-D “real world” compressible turbulence application solving Richtmyer-Meshkov instability Fingering instability which occurs at a material interface accelerated by a shock wave Machine – NPACI IBM SP2 Blue Horizon at SDSC Teraflop-scale Power3 based SMP cluster 1152 processors and 512GB of main memory AIX operating system Peak bi-directional data transfer rate of approx. 115 MBps

Experimental Setup (contd.) Base coarse grid – 128 * 32 * 32 3 levels of factor 2 space-time refinements Application ran for 150 coarse level time-steps Experiments consisted of varying – Partitioner (from the set of evaluated partitioners) Number of processors (16 – 128) Granularity, i.e. the atomic unit (2*2*2 – 8*8*8) Metrics used – total run-time, maximum load imbalance, AMR efficiency

Experimental Results RM3D application on 16 processors with granularity 2 PartitionerRun-time (s)Max. Load Imbalance (%) AMR Efficiency (%) SFC G-MISP G-MISP+SP pBD-ISP SP-ISP

Run-times

Max. Load Imbalance

AMR Efficiency

Experimental Evaluation RM3D needs rapid refinement and efficient redistribution pBD-ISP, G-MISP+SP, SFC best suited for RM3D – fast partitioners with low imbalance and maintaining good communication patterns pBD-ISP fastest, but average load imbalance G-MISP+SP and SFC generate lowest imbalance but are relatively slower Evaluated partitioning techniques scale reasonably well

Evaluation (contd.) Coarse granularity produces high load imbalance Fine granularity leads to greater synchronization and coordination overheads and higher execution times Optimal partitioning granularity requires a trade-off between execution speed and load imbalance For RM3D application, granularity of 4 gives lowest execution time with acceptable load imbalance

Conclusions Experimental evaluation of dynamic domain-based partitioning and load-balancing techniques RM3D compressible turbulence application Effect of choice of partitioner and granularity on execution time Formulation of application-centric characterization of the partitioners as a function of number of processors, problem size, and partitioning granularity