Parallel preconditioned conjugate gradient method for large sparse and highly ill-conditioned systems arising in computational geomechanics

Abstract

The efficiency of parallel preconditioned conjugate gradient PCG algorithm for solving large sparse linear systems arising from application of interior point methods to conic optimisation problems in the context of nonlinear finite element limit analysis FELA for computational geomechanics is studied. For large 3D problems, the use of direct solvers in general becomes prohibitively expensive owing to exponentially growing memory requirements and computational time. And the so-called saddle-point systems resulting from use of optimisation framework is not an exemption. On the other hand, although preconditioned iterative methods have moderate storage requirements and therefore can be applied to much larger problems than direct methods, they usually exhibit high number of iterations to reach convergence. In the present paper, we show that this problem can be effectively tackled using efficient variants of sparse approximate inverse preconditioners along with an elaborate parallel implementation on multicore CPUs and significant improvements can be achieved by parallel implementation on graphic processing unit GPU. Furthermore, the efficiency of our proposed implementation is verified by the presented numerical results.