Dynamic Load Balancing Using Hilbert Space-Filling Curves for Parallel Reservoir Simulations

Abstract New reservoir simulators designed for parallel computers enable us to overcome performance limitations of workstations and personal computers and to simulate large-scale reservoir models with billions of grid cells. With development of parallel reservoir simulators, more complex physics and detailed models can be studied. The key to design efficient parallel reservoir simulators is not to improve the performance of individual CPUs drastically but to utilize the aggregation of computing power of all requested nodes through high speed networks. An ideal scenario is that when the number of MPI processors is doubled, the running time of parallel reservoir simulators is reduced by half. In real simulation, numerical difficulties and performance problems appear when the number of MPI processors grows due to the deteriorating linear solver efficiency and increasing communication overhead, which are determined by a grid distribution. The goal of load balancing (grid partitioning) is to minimize overall computations and to make sure that all MPI processors have a similar workload. Geometric methods divide a grid by using a location of cells while topological methods work with connectivity of cells, which is generally described as a graph. The geometric methods are much faster than the topological methods. This paper introduces a Hilbert space-filling curve method. A space-filling curve is a continuous curve and defines a map between a onedimensional space and a multi-dimensional space. A Hilbert space-filling curve is one special space-filling curve discovered by Hilbert and has many useful characteristics, such as good locality, which means that two objects that are close to each other in a multi-dimensional space are also close to each other in a one dimensional space. This property can model communications in grid-based parallel applications. The idea of the Hilbert space-filling curve method is to map a computational domain into a one-dimensional space, partition the one-dimensional space to certain intervals, and assign all cells in a same interval to a MPI processor. To implement a dynamic load balancing method, we need a mapping kernel that converts high-dimensional coordinates to a scalar value, and an efficient one-dimensional partitioning module that divides a one-dimensional space and makes sure that all intervals have a similar workload. The Hilbert space-filling curve method is compared with other load balancing methods, such as the K-way method from ParMETIS and other geometric methods from Zoltan. The results show that our Hilbert space-filling curve is much faster than graph methods. It also has good partition quality. This method has been applied to reservoir models with billions of grid cells and linear scalability has been obtained on many parallel computing systems.

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