Parallel Solution of Sparse Linear Systems

Abstract

Many simulations in science and engineering give rise to sparse linear systems of equations. It is a well known fact that the cost of the simulation process is almost always governed by the solution of the linear systems especially for largescale problems. The emergence of extreme-scale parallel platforms, along with the increasing number of processing cores available on a single chip pose significant challenges for algorithm development. Machines with tens of thousands of multicore processors place tremendous constraints on the communication as well as memory access requirements of algorithms. The increase in number of cores in a processing unit without an increase in memory bandwidth aggravates an already significant memory bottleneck. Sparse linear algebra kernels are well-known for their poor processor utilization. This is a result of limited memory reuse, which renders data caching less effective. In view of emerging hardware trends, it is necessary to develop algorithms that strike a more meaningful balance between memory accesses, communication, and computation. Specifically, an algorithm that performs more floating point operations at the expense of reduced memory accesses and communication is likely to yield better performance. We present two alternative variations of DS factorization based methods for solution of sparse linear systems on parallel computing platforms. Performance comparisons to traditional LU factorization based parallel solvers are also discussed. We show that combining iterative methods with direct solvers and using DS factorization, one can achieve better scalability and shorter time to solution. IntroductionMany simulations in science and engineering give rise to sparse linear systems of equations. It is a well known fact that the cost of the solution process is almost always governed by the solution of the linear systems especially for large-scale problems. The emergence of extreme-scale parallel platforms, along with the increasing number of processing cores available on a single chip pose significant challenges M. Manguoglu ( )

Parallel computational methods for solving large linear systems

“…Hence, the two-level domain decomposition methods are effective and suitable methods for solving large linear systems on supercomputing infrastructure. The domain decomposition methods that partition the domain Ω based on the algebraic properties of the graph corresponding to the coefficient matrix A, are called algebraic domain decomposition methods, [62,124,190]. The graph partitioning is commonly performed by graph partitioning algorithms, such as the METIS algorithm, [100,101], or the SCOTCH algorithm, [35,140].…”