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Graphics Processing Units (GPUs) exhibit significantly higher peak performance than conventional CPUs. However, in general only highly parallel algorithms can exploit their potential. In this scenario, the iterative solution to sparse linear systems of equations could be carried out quite efficiently on a GPU as it requires only matrix-by-vector products, dot products, and vector updates. However, to be really effective, any iterative solver needs to be properly preconditioned and this represents a major bottleneck for a successful GPU implementation. Due to its inherent parallelism, the factored sparse approximate inverse (FSAI) preconditioner represents an optimal candidate for the conjugate gradient-like solution of sparse linear systems. However, its GPU implementation requires a nontrivial recasting of multiple computational steps. We present our GPU version of the FSAI preconditioner along with a set of results that show how a noticeable speedup with respect to a highly tuned CPU counterpart is obtained. [ABSTRACT FROM AUTHOR]

In recent years the growing popularity of supercomputers has fostered the development of algorithms able to take advantage of the massive parallelism offered by multiple processors. Direct methods, though robust and computationally efficient, hardly exploit high degrees of parallelism. By contrast, Krylov methods preconditioned by Factored Sparse Approximate Inverses (FSAI) provide, at least in principle, a perfectly parallel approach but are often thwarted by an excessive set-up cost. In this paper we extend the concept of supernode from sparse LU factorizations to approximate inverses, and use it to accelerate the computation of an FSAI-type preconditioner. The numerical experiments on real-world problems show that the overall FSAI efficiency can be significantly increased while preserving its intrinsic parallelism. [ABSTRACT FROM AUTHOR]

Abstract: Contact mechanics can be addressed numerically by finite elements using either a penalty formulation or the Lagrange multipliers. The penalty approach leads to a linearized symmetric positive definite system which can prove severely ill-conditioned, with the iterative solution to large 3D problems requiring expensive preconditioners to accelerate, or even to allow for, convergence. In the present paper a mixed constraint preconditioner (MCP) is developed and implemented for the iterative solution of contact problems based on the penalty approach. A theoretical bound for the eigenspectrum of the preconditioned matrix is provided depending on the user-specified parameters that control the preconditioner fill-in. The performance of MCP combined with the preconditioned conjugate gradient is studied in two real 3D test cases describing the geomechanics of faulted porous media and compared to a well-established preconditioner such as the incomplete Cholesky decomposition ILLT with optimal fill-in. Numerical results show that typically MCP outperforms ILLT especially in the most challenging problems where very large penalty parameters must be used. Finally, a modification aimed at simplifying the MCP implementation and increasing its robustness is developed and experimented with, showing that the modified MCP can be viewed as an efficient and viable alternative to both MCP and ILLT, especially in problems where the size of the contact surfaces is large with respect to the global domain. [Copyright &y& Elsevier]